Massive S-matrix from minimal coupling to black holes (and ......Massive S-matrix from minimal...

Massive S-matrix from minimal coupling

to black holes (and stars)

Sangmin LeeSeoul National University

Amplitudes 2019, Trinity College Dublin05 July 2019

[M. Chung, Y. Huang, J. Kim, S. Lee, 1812.08752] + 1907.nnnnn + poster by Jung-Wook

Talks by Plefka, Kosower, Ochirov, Bern, (Rothstein)

PN & PM

1PM

2PM

3PM

0PN 1PN 2PN 3PN

...<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

✓Gm

r

◆( 1 + v2 + v4 + v6 + v8 + · · · )

<latexit sha1_base64="3jRiP2CucKtJE9eStlqJ6lYBfLM=">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</latexit>

✓Gm

r

◆2( 1 + v2 + v4 + v6 + · · · )

<latexit sha1_base64="kAq74CkLquODFqpiMPugwYG3CGk=">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</latexit>

✓Gm

r

◆3( 1 + v2 + v4 + · · · )

<latexit sha1_base64="FSxFOdHvkZp+9QjgVRa+0IQjXis=">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</latexit>

✓Gm

r

◆4( 1 + v2 + · · · )

<latexit sha1_base64="PF0VZJeckYVEoPhwU/VtLlO2AZk=">AAACP3icdVBNS8NAEN34bf2qevSyWISKUJJavxCk4EGPFWwVTC2b7aZdusmG3Umhhv4d/4d3rwr+Aj2JV29u2gpV9MEwjzczDO95keAabPvFmpicmp6ZnZvPLCwuLa9kV9dqWsaKsiqVQqprj2gmeMiqwEGw60gxEniCXXmd03R+1WVKcxleQi9i9YC0Qu5zSsBIjWzZFcyHvOsrQpOzoJ+ovqt4qw3btyWcd4+xg93jHVO4e1tM24AbuLQpQePtRjZnF+wB8BjZs52jfQc7IyWHRqg0sm9uU9I4YCFQQbS+cewI6glRwKlg/YwbaxYR2iEtdmNoSAKm68nAaR9vGaWJfalMhYAH6vhFQgKte4FnNgMCbf17lop/zlIlInfy13/wD+sJD6MYWEiH7/1YYJA4DRM3uWIURM8QQhU3DjBtExMlmMgzJppv//h/UisWnN2CfVHKlU9GIc2hDbSJ8shBB6iMzlEFVRFF9+gRPaFn68F6td6tj+HqhDW6WUc/YH1+AUzAq/o=</latexit>

4PM

Our contribution

Spin-dependent terms up to 1-loop (2PM)

Massive spinor helicity variable

Vertices : 3-point, 4-point (Compton)

Leading singularity and Holomorphic classical limit

Overlap with Ochirov's talk Complementary to talks by Plefka, Kosower, Bern

On-shell approach :

Massive spinor helicity SO(1, 3) = SL(2, C)/Z2<latexit sha1_base64="q923UKAH2Yu9jyOgn5SSWRhTC30=">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</latexit>

laI ! UI

JlaJ , U 2 SU(2)

<latexit sha1_base64="++7CrANpJevJvfB5iPLTrN5KPxw=">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</latexit>

[Arkani-Hamed,Huang,Huang, 17]

pµsµaa =

✓p0 + p3 p1 � ip2

p1 + ip2 p0 � p3

◆⌘ paa

<latexit sha1_base64="0zb+XLeoontkeVAM1d80X4/knQU=">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</latexit>

[Kleiss, Stirling, Schwinn, Weinzierl, Badger, Glover, Khoze, Svrcek, Ozeren, Stirling, ... ]

det(paa) = m2 =) paa = laI lIa , det(la

I) = m<latexit sha1_base64="cUUI4BGB5Z5bUYlm0BZsPoA0ISE=">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</latexit>

pµ pµ = m2 = det(paa)<latexit sha1_base64="18alX3Zb+pTy0b2GNo4OoU7tMuA=">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</latexit>

[Arkani-Hamed,Huang,Huang, 17]Massive particles

Spin-1, polarization vector :

Spin-S :

#I Jaa ⌘ 1

ml(Ia la

J)<latexit sha1_base64="/5bhLnezjb/V757XwlOIWKvbJqw=">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</latexit>

no need for auxiliary spinor

BOLD : h12i ⌘ (l1)aI1(t1)I1(l2)a

I2(t2)I2<latexit sha1_base64="EAjon330Zi+EXGkgRne5hhxOG20=">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</latexit>

Amplitudes are homogeneous polynomials in t’s

Ex) #(P2)⇤ · #(P1) =

h21iS[21]S

m2s<latexit sha1_base64="aKYQo3iL5+WwH4FcsLAB/D8DMnA=">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</latexit>

(#⇤)aaI J #KL

aa = d(K(I d

L)J) , (#⇤)aa

I J #I Jbb

=12

✓da

bdab� 1

m2 pab pa

b

◆.

<latexit sha1_base64="XM9mQUmjD01A/odeiU1aqkiYJT0=">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</latexit>

3

Âi=1

(#⇤i )µ(#i)n = hµn �pµ pn

m2

!

<latexit sha1_base64="UoUh8GIGjabRzX9sFTuXtZ79E7c=">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</latexit>

(total symmetrization of SU(2)-doublet)

#I1...I2Sa1...aS ,a1...aS

!✓

ll

m

◆�S

<latexit sha1_base64="jqZNyLmpG9qBKJLumhfaVKn4zz8=">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</latexit>

[Arkani-Hamed,Huang,Huang, 17]Compton amplitudes

+<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

1<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>








M(1S, 2+1, 3�1, 4S) =h3|p1|2]2�2S(h34i[12] + h31i[42])2S

(s � m2)(u � m2)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Amplitudes are Lorentz invariant and little-group covariant !!!

A naïve extrapolation to S > 1 fails, reflecting the fact that such particles cannot be elementary.

[Arkani-Hamed,Huang,Huang, 17]3-point on-shell


3+<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

dim-less, helicity +1The x-factor :

2<latexit sha1_base64="ZpFtvL32MVI7AcjOcIFukl4vyxA=">AAACB3icbVDLSgMxFL3js9ZX1aWbYBFclZkq6EoKblxWsA9sh5JJM21oJhmSjFCHfoB7t/oL7sStn+Ef+Blm2lnY1gOBw7mvkxPEnGnjut/Oyura+sZmYau4vbO7t186OGxqmShCG0RyqdoB1pQzQRuGGU7bsaI4CjhtBaObrN56pEozKe7NOKZ+hAeChYxgY6WHboTNMAjT6qRXKrsVdwq0TLyclCFHvVf66fYlSSIqDOFY647nxsZPsTKMcDopdhNNY0xGeEA7lgocUe2nU8cTdGqVPgqlsk8YNFX/TqQ40nocBbYzc6gXa5n4X62TmPDKT5mIE0MFmR0KE46MRNn3UZ8pSgwfW4KJYtYrIkOsMDE2pOLcmWx5jJ9kFo23GMQyaVYr3nnFvbso167zkApwDCdwBh5cQg1uoQ4NICDgBV7hzXl23p0P53PWuuLkM0cwB+frF3EZmjM=</latexit>

(analytic continuation if necessary)

Ex) "parity symmetric" amplitude

P1 + P2 + k = 0<latexit sha1_base64="G1VoEJ7yVx+ZfDyzvsodzcciCxY=">AAACD3icdVDdSsMwGE3n35x/nV56ExyCIIx2/l8oA2+8rOA2YSslzdItLG1Kkiqz7CF8AG/1EbwTb30En8DXMN0qbKIHEg7nfF++L8ePGZXKsj6Nwtz8wuJScbm0srq2vmGWN5uSJwKTBuaMi1sfScJoRBqKKkZuY0FQ6DPS8geXmd+6I0JSHt2oYUzcEPUiGlCMlJY8s+x4NtyHjlfT9wCeQ8szK1bVGgNOkSPLPju2oZ0rFZDD8cyvTpfjJCSRwgxJ2batWLkpEopiRkalTiJJjPAA9Uhb0wiFRLrpePUR3NVKFwZc6BMpOFanO1IUSjkMfV0ZItWXv71M/MtrJyo4dVMaxYkiEZ4MChIGFYdZDrBLBcGKDTVBWFC9K8R9JBBWOq2ZKdnbMXrgI53Mz/fh/6RZq9oHVev6sFK/yDMqgm2wA/aADU5AHVwBBzQABvfgCTyDF+PReDXejPdJacHIe7bADIyPbyF2mmc=</latexit>

mx ⌘ #+(k) · (P1 � P2)��on�shell =) x =

[k|P1|zimhkzi .

<latexit sha1_base64="n9lPUw8MdD2at26BtRY1yL5L0Qs=">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</latexit>

A3(1S, 2S, 3+) = k⇥#+(k) · (p1 � p2)

⇤[#1 · #⇤2] = k(mx)

✓h12iS[12]S

m2S

◆.

<latexit sha1_base64="M+SKJPluzXBcdKd6Sc6vmA9+2m4=">AAAC83icdVLNjtMwEHbC31J+tsCRi0WFlLJLlaT8HkC74sJxUenuijqJHNdprTqJZTsrShSJK7wCN7RXHog34AF4AJw0XbUIRrI08833zYzHjgVnSrvuT8u+dPnK1Ws71zs3bt66vdu9c/dY5YUkdExynsvTGCvKWUbHmmlOT4WkOI05PYkXb+r8yRmViuXZe70UNEjxLGMJI1gbKOr+RinWc4J5eVhFQ6eJ4qT0qnC0vw78OhiGe334CqIFFgJDxGmiJ+gMSyoU43kW7jmLPiLTXDsi8uBjKCK/D5Fks7kOWjrc4BtOw97C/PDRWnLRyUk/9ld6B6JEYlIijrMZp/BiVr8yqgYLR3CyAQfhqCrTsPRHVdUWNrX2B1G35w7cxuCG89T1Xj7zoNciPdDaUdT9haY5KVKaacKxUhPPFToosdSMcFp1UKGowGSBZ3Ri3AynVAVl8zoVfGiQKUxyaU6mYYNuKkqcKrVMY8OsR1d/52rwX7lJoZMXQckyUWiakVWjpOBQ57B+ajhlkhLNl8bBRDIzKyRzbFaozYfobLWpiwv8Ka/Matb3h/93jv2BNxy47570Dl63S9oB98ED4AAPPAcH4C04AmNArA/WZ+uL9dUu7G/2d/t8RbWtVnMPbJn94w9aaOz4</latexit>

Minimal coupling for small spin

A(11, 21, 3+) = (mx)✓h12i

m

◆2.

<latexit sha1_base64="6irzmKDQjCIXyUUh9SQwSmIOlak=">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</latexit>

(W-boson) :

(Proca) :


3+<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

2<latexit sha1_base64="ZpFtvL32MVI7AcjOcIFukl4vyxA=">AAACB3icbVDLSgMxFL3js9ZX1aWbYBFclZkq6EoKblxWsA9sh5JJM21oJhmSjFCHfoB7t/oL7sStn+Ef+Blm2lnY1gOBw7mvkxPEnGnjut/Oyura+sZmYau4vbO7t186OGxqmShCG0RyqdoB1pQzQRuGGU7bsaI4CjhtBaObrN56pEozKe7NOKZ+hAeChYxgY6WHboTNMAjT6qRXKrsVdwq0TLyclCFHvVf66fYlSSIqDOFY647nxsZPsTKMcDopdhNNY0xGeEA7lgocUe2nU8cTdGqVPgqlsk8YNFX/TqQ40nocBbYzc6gXa5n4X62TmPDKT5mIE0MFmR0KE46MRNn3UZ8pSgwfW4KJYtYrIkOsMDE2pOLcmWx5jJ9kFo23GMQyaVYr3nnFvbso167zkApwDCdwBh5cQg1uoQ4NICDgBV7hzXl23p0P53PWuuLkM0cwB+frF3EZmjM=</latexit>

P1 + P2 + k = 0<latexit sha1_base64="G1VoEJ7yVx+ZfDyzvsodzcciCxY=">AAACD3icdVDdSsMwGE3n35x/nV56ExyCIIx2/l8oA2+8rOA2YSslzdItLG1Kkiqz7CF8AG/1EbwTb30En8DXMN0qbKIHEg7nfF++L8ePGZXKsj6Nwtz8wuJScbm0srq2vmGWN5uSJwKTBuaMi1sfScJoRBqKKkZuY0FQ6DPS8geXmd+6I0JSHt2oYUzcEPUiGlCMlJY8s+x4NtyHjlfT9wCeQ8szK1bVGgNOkSPLPju2oZ0rFZDD8cyvTpfjJCSRwgxJ2batWLkpEopiRkalTiJJjPAA9Uhb0wiFRLrpePUR3NVKFwZc6BMpOFanO1IUSjkMfV0ZItWXv71M/MtrJyo4dVMaxYkiEZ4MChIGFYdZDrBLBcGKDTVBWFC9K8R9JBBWOq2ZKdnbMXrgI53Mz/fh/6RZq9oHVev6sFK/yDMqgm2wA/aADU5AHVwBBzQABvfgCTyDF+PReDXejPdJacHIe7bADIyPbyF2mmc=</latexit>

A(11, 21, 3+) = (mx)h12i[12]

m2 .

= (mx)

"✓h12i

m

◆2�x

h12ih13ih32im3

#.

<latexit sha1_base64="UoUnjPKUnj+5pc8naLx82WcCXMM=">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</latexit>

worse UV behavior!

On-shell minimal coupling

c1 = 0 () g = 2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

[Arkani-Hamed,Huang,Huang, 17]

Most general 3-point amplitude

Zeeman coupling :

"on-shell minimal" : most well-behaved in the UV

VZ = �~µ · ~B , ~µ = g⇣ e

2m

⌘~S

<latexit sha1_base64="+ZH5LggpfiajjY8yUKFCpADMSiU=">AAACV3icdVDLbhMxFHUGKCE8msKyG4sIqUglmmlLHwuqim5YFkHSiDiKPJ47iVV7PLXvVEpH82v9j+67YAO/QD1JKrUIrmT5+NxzfO0T50o6DMPrRvDo8ZOVp81nrecvXr5aba+97jtTWAE9YZSxg5g7UDKDHkpUMMgtcB0rOI3Pjuv+6QVYJ032HWc5jDSfZDKVgqOnxu1Bf/yDfqIf2AWIkumiYiIxSOfHzxXb3KQtdl7wxG93Ci+fUKYgxQ2WWi5KqMotXTErJ1N8v7B+q8btTtgN50XvgY9hdLAb0WjJdMiyTsbtnywxotCQoVDcuWEU5jgquUUpFFQtVjjIuTjjExh6mHENblTOE6joO88kNDXWrwzpnL3vKLl2bqZjr9Qcp+7vXk3+qzcsMN0flTLLC4RMLAalhaJoaB0nTaQFgWrmARdW+rdSMeU+FfShtx6MqS/P+aWpo7n7P/0/6G91o+1u+HWnc3S4DKlJ1slbskEiskeOyBdyQnpEkCtyQ36R343rxp9gJWgupEFj6XlDHlSwdgvVe7W6</latexit>

"g = 2 as the natural value of the tree-level gyromagnetic ratio" [Ferrara,Porrati,Telegdi, 92][Holstein 06]HE Compton scattering

Gravity amplitude factorization GDH sum rule

A+h3 = khmxh

✓h12i2S

m2S + c1xh12i2S�1h13ih32i

m2S+1 + · · ·◆

.<latexit sha1_base64="X9SCNOPZrnTuPgmgDfSmLDBqhfU=">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</latexit>

Spin-dependence

(Jµn)ab =

i2(s[µsn])a

b ,

(Jµn)a

b =i2(s[µsn])

ab .

<latexit sha1_base64="jmi6j2CEjvUjsCfCqdrLd7wxXyc=">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</latexit>

(q · S)ab =

x2

lalb =x2|qihq| ,

(q · S)ab = � 1

2xlalb = � 1

2x|q][q| .

<latexit sha1_base64="Rz5OWs6073GyPcVSKD9f0pZGR28=">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</latexit>

S =12

:<latexit sha1_base64="s2lvLkL353FY7AOMsxGBM0Yc7dc=">AAACFnicbZDLSgMxFIYz9VbrbdSFCzfBIrgqM1VQFKXgxmVFe4FOKZk004ZmkiHJCHWY93DvVl/Bnbh16xv4GGbaWWjrgYSf/z+Hw/n8iFGlHefLKiwsLi2vFFdLa+sbm1v29k5TiVhi0sCCCdn2kSKMctLQVDPSjiRBoc9Iyx9dZ3nrgUhFBb/X44h0QzTgNKAYaWP17L07eAm9QCKcuGlSTaF3Ac/N17PLTsWZFJwXbi7KIK96z/72+gLHIeEaM6RUx3Ui3U2Q1BQzkpa8WJEI4REakI6RHIVEdZPJASk8NE4fBkKaxzWcuL8nEhQqNQ590xkiPVSzWWb+m2VOhB7FzH4dnHUTyqNYE46n64OYQS1gxgj2qSRYs7ERCEtqLoB4iAwjbUiWDBp3FsS8aFYr7nHFuT0p165ySEWwDw7AEXDBKaiBG1AHDYBBCp7BC3i1nqw36936mLYWrHxmF/wp6/MH3sOd0Q==</latexit>

Sµ = � 12m

eµnrsPn Jrs .<latexit sha1_base64="35dnpsdXX0Cp5iMkPSLdFGQOXz4=">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</latexit>

Arbitrary S :

Large S limit :

[21] = h21i+ xh2qihq1im

=) [21] = h2|S✓

I +x|qihq|

m

◆�S|1iS

<latexit sha1_base64="+2sdG2bHCp0BuAtHatk5N5GnEbQ=">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</latexit>

✓I +

x|qihq|m

◆�S! exp

✓q · Sm

◆

<latexit sha1_base64="rWfKQcyd+46Lit/Q3RqDKVtlHhg=">AAACenicdVFNb9QwEHXCR8vylcKRy6grpFaVVgkUKKqEKnGBWxFsW6leVo53smvVibP2pHRJ80M5cuNHcMC7CahUMJLtp/dm3mjGaamVozj+FoQ3bt66vbZ+p3f33v0HD6ONR0fOVFbiUBpt7EkqHGpV4JAUaTwpLYo81Xicnr1d6sfnaJ0yxSdalDjKxbRQmZKCPDWOaq4xo633OzyzQtYXl3PgVhRTjcB1+84vmzpvuFXTGW1/rrmZGIKPDUCP7/N9f68UYa358ofCi7I1htZ3zmVb5Z2gs4Jx1I8H8SrgCngRJ69fJpB0TJ91cTiOfvCJkVWOBUktnDtN4pJGtbCkpMamxyuHpZBnYoqnHhYiRzeqV0tq4KlnJpAZ609BsGKvVtQid26Rpz4zFzRz17Ul+U9tyZTiq7nWn7K9Ua2KsiIsZNs+qzSQgeU/wERZlKQXHghplZ8A5Ez4XZH/rZ5fze/54f/g6NkgeT6IP+z2D950S1pnT9gm22IJe8UO2Dt2yIZMsu/BWhAFG8HPcDPcDnfa1DDoah6zvyLc/QUCzL+i</latexit>

limn!•

⇣1 +

xn

⌘n= ex

<latexit sha1_base64="/BrM8exMQaLpme9vQfA4IqxDHlU=">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</latexit>

Minimal coupling for large spin

3-point interaction :

Spin interaction for the minimal coupling :

Hspin = exp✓

q · S

m

◆

<latexit sha1_base64="YrVGv0W8SYI8nF+YNoDCS2G8azw=">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</latexit>

#⇤(P2) · #(P1) =h12iS[12]S

m2S vs Amin(1S, 2S, 3+h) = khmxh h12i2S

m2S<latexit sha1_base64="0KgnN1HRBi9JnxI7zpyjqkiwqVw=">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</latexit>

The other spin dependence :

in the large S limit.

P =12(P1 + P2) , q = P1 � P2 .

<latexit sha1_base64="S1DJxTktVJobunEGYWZg4JOaBSM=">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</latexit>

#⇤(P2) · #(P1) ⇡ #⇤(P)

"I +

i2

~pm

⇥~Sm

!·~q#

#(P)<latexit sha1_base64="T5AEsxYYjUkjcTiJJTv70tm4okU=">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</latexit>

Gravitational minimal coupling

A+3 =

m2

Mplx2

✓h12i

m

◆2S.

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

On-shell minimal 3-point coupling :

c1 = 0<latexit sha1_base64="g0v9tRr6Vr5Qhu/8gLYuMi1ZmqQ=">AAACBHicdVDLSgMxFL1TX7W+qi7dBIvgqsz4dqEU3LisYFuhHUomTdvQTDIkGaEO3bp3q7/gTtz6H/6Bn2GmHaEVPRA4nHNv7r0niDjTxnU/ndzc/MLiUn65sLK6tr5R3NyqaxkrQmtEcqnuAqwpZ4LWDDOc3kWK4jDgtBEMrlK/cU+VZlLcmmFE/RD3BOsygo2VGqTtoQvktoslt+yOgabIseudn3jIy5QSZKi2i1+tjiRxSIUhHGvd9NzI+AlWhhFOR4VWrGmEyQD3aNNSgUOq/WS87gjtWaWDulLZJwwaq9MdCQ61HoaBrQyx6evfXir+5TVj0z3zEyai2FBBJoO6MUdGovR21GGKEsOHlmCimN0VkT5WmBibUGFmTPp5hB/kyEbzcz/6n9QPyt5h2b05KlUus5DysAO7sA8enEIFrqEKNSAwgCd4hhfn0Xl13pz3SWnOyXq2YQbOxzdVxZfT</latexit>

Universal gravitational potential :

ds2 = e2F(dt2 + ~A · d~x)2 � e�2Fd~x2<latexit sha1_base64="OSwAP/U2EiRtz04AzGQ4xc4qdgY=">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</latexit>

V = mF � 12~S · ~B

<latexit sha1_base64="m8K6RcC9OazeVE0hfMbzQX6hl28=">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</latexit>

[Johansson,Ochirov 1906.12292]Double copy relation for massive particles

One particle effective action

Wµn = hABeAµ DeB

n

Ds, hABeA

µeBn = gµn .


S =Z

ds

✓�m

pu2 � 1

2SµnWµn + LSI

◆


Non-minimal spin-interaction terms:

Action for a spinning particle [Levi 1807.01699 and references therein]

Eµn ⌘ Rµrnsurus , Bµn ⌘ 12

ersgµRrsdnugud .

<latexit sha1_base64="xSPWaHT4vb3pwybB38mEbAvLzFI=">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</latexit>

LSI =•

Ân=1

(�1)n

(2n)!C2n

m2n�1 Dµ2n · · · Dµ3

Eµ1µ2pu2

Sµ1 · · · Sµ2n

+•

Ân=1

(�1)n

(2n + 1)!C2n+1m2n Dµ2n+1 · · · Dµ3

Bµ1µ2pu2

Sµ1 · · · Sµ2n+1 .<latexit sha1_base64="efB/hOaYMshIfe8Sktqk+1ZEqsA=">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</latexit>

C and c

LSI =•

Ân=1

(�1)n

(2n)!C2n

m2n�1 Dµ2n · · · Dµ3

Eµ1µ2pu2

Sµ1 · · · Sµ2n

+•

Ân=1

(�1)n

(2n + 1)!C2n+1m2n Dµ2n+1 · · · Dµ3

Bµ1µ2pu2

Sµ1 · · · Sµ2n+1 .<latexit sha1_base64="efB/hOaYMshIfe8Sktqk+1ZEqsA=">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</latexit>

In the large S limit, the generating functions satisfy

h(x) ⌘•

Ân=0

cn(x/S)n , f (x) ⌘•

Ân=0

Cnn!

xn =) f (x) = exh(x) .<latexit sha1_base64="MrJwT2WjekbvOcyS4IXxu/+GAn0=">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</latexit>

A+h3 = khmxh

✓h12i2S

m2S + c1xh12i2S�1h13ih32i

m2S+1 + · · ·◆

.<latexit sha1_base64="X9SCNOPZrnTuPgmgDfSmLDBqhfU=">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</latexit>

Wilson coefficients

BH = minimally coupled particle !

[Vines, 1709.06016]Kerr BH solution to all orders in spin

hµn =

✓uµun cos(a · ∂) + u(µen)

rsguras∂g sin(a · ∂)(a · ∂)

◆4GM

r<latexit sha1_base64="+T7PlsLVugkXEDCkS7GWGaE9hek=">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</latexit>

In terms of on-shell 3-point coupling,

C2n = C2n+1 = 1<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Kerr BH :

Kerr BH : c0 = 1 , cn = 0 (n � 1)<latexit sha1_base64="YAc3EcPdWNzcoSXj+nFlQ4ugY8M=">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</latexit>

aµ = Sµ/m<latexit sha1_base64="QPeO9ZF0dlwr/X+yNzC8E26krsM=">AAACD3icdVDLSgMxFM34rPU16tJNsAiuasb6qAuh4MZlRfuAtpZMmrahSWZIMoVa+hHu3eovuBO3foJ/4GeYmY5gRQ+EHM65N/fm+CFn2iD04czNLywuLWdWsqtr6xub7tZ2VQeRIrRCAh6ouo815UzSimGG03qoKBY+pzV/cBn7tSFVmgXy1oxC2hK4J1mXEWys1HZdfNcUEbyAN8l9KNpuDuVPkHd+iiDKowQJKXoFD3qpkgMpym33s9kJSCSoNIRjrRseCk1rjJVhhNNJthlpGmIywD3asFRiQXVrnGw+gftW6cBuoOyRBibqz44xFlqPhG8rBTZ9/duLxb+8RmS6xdaYyTAyVJLpoG7EoQlgHAPsMEWJ4SNLMFHM7gpJHytMjA0rOzMmfjzE98HERvP9f/g/qR7lvUIeXR/nSsU0pAzYBXvgAHjgDJTAFSiDCiBgCB7BE3h2HpwX59V5m5bOOWnPDpiB8/4F9LSb6Q==</latexit>

1PM : kinematics

P2 = (Ea,~p �~q/2)P1 = (Ea,~p +~q/2)

<latexit sha1_base64="f7AE5H3UnHw0A3AOujwXXr+NkFQ=">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</latexit>

on-shell 3pton-shell 3pt

HCL

HCL (holomorphic classical limit) : [Guevara 1706]

q = (0,~q) = (0,~ ) + (0,~ ⇤) , ~ 2 = 0 = (~ ⇤)2 , 2~ ·~ ⇤ = ~q2 .<latexit sha1_base64="cW4gqCAeeEb/ZhemxN+adjTLsqc=">AAACm3icdZHfTtswFMadMFgXGCtwOU2yViGVraqc8v8ChMTNNO2CSRSQSFc5zmmxcOLUdiqVqG/Ci/EGPAZOyVCK4EiWjn/n+2Sfc8JUcG0IeXDchQ+LSx9rn7zllc+rX+pr6xdaZopBl0kh1VVINQieQNdwI+AqVUDjUMBleHta1C/HoDSXybmZpNCL6TDhA86osahfvx/hI9wkrWAMLB9Ntyq3AISw4Oc8+PdjK2i1vGCU0Qh7Fd6xVlLY57QWV+T28lIMWCQNroqtufzGzNXG/XqDtMkscCXZJf7hno/9kjRQGWf9+mMQSZbFkBgmqNbXPklNL6fKcCZg6gWZhpSyWzqEa5smNAbdy2dTnOJNSyI8kMqexOAZrTpyGms9iUOrjKm50a9rBXyzVpCU3slX75vBQS/nSZoZSNjz84NMYCNxsSgccQXMiIlNKFPcdoDZDVWUGbtOz47mf//4/eSi0/a32+TvTuPkuBxSDX1F31ET+WgfnaBf6Ax1EXNcp+n4Tsf95p66v90/z1LXKT0baC7c7hMgAMTQ</latexit>

! q(b) = `+ b`⇤ (b ! 0)<latexit sha1_base64="0UiOVgQQxZL4sEAOepPXagr3LmQ=">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</latexit>

P4<latexit sha1_base64="STucaW2jJTg6U4AxCTTvZoTIu+Q=">AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKexqQE8S8OIxonlAsoTZyWwyZHZmmZkV4pKLd6/6C97Eq3/iH/gZziZ70MSChqKqm+6uIOZMG9f9cgorq2vrG8XN0tb2zu5eef+gpWWiCG0SyaXqBFhTzgRtGmY47cSK4ijgtB2MrzO//UCVZlLcm0lM/QgPBQsZwcZKd41+rV+uuFV3BrRMvJxUIEejX/7uDSRJIioM4VjrrufGxk+xMoxwOi31Ek1jTMZ4SLuWChxR7aezU6foxCoDFEplSxg0U39PpDjSehIFtjPCZqQXvUz818uUGD/Khf0mvPRTJuLEUEHm68OEIyNRlgYaMEWJ4RNLMFHMfoDICCtMjM2sZKPxFoNYJq2zqndedW9rlfpVHlIRjuAYTsGDC6jDDTSgCQSG8Awv8Oo8OW/Ou/Mxby04+cwh/IHz+QOTnJbY</latexit>

P3<latexit sha1_base64="c6eKpBxwpHt34+nUIys0jRwjLJI=">AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKewaQU8S8OIxonlAsoTZyWwyZHZmmZkV4pKLd6/6C97Eq3/iH/gZziZ70MSChqKqm+6uIOZMG9f9cgorq2vrG8XN0tb2zu5eef+gpWWiCG0SyaXqBFhTzgRtGmY47cSK4ijgtB2MrzO//UCVZlLcm0lM/QgPBQsZwcZKd41+rV+uuFV3BrRMvJxUIEejX/7uDSRJIioM4VjrrufGxk+xMoxwOi31Ek1jTMZ4SLuWChxR7aezU6foxCoDFEplSxg0U39PpDjSehIFtjPCZqQXvUz818uUGD/Khf0mvPRTJuLEUEHm68OEIyNRlgYaMEWJ4RNLMFHMfoDICCtMjM2sZKPxFoNYJq2zqlerurfnlfpVHlIRjuAYTsGDC6jDDTSgCQSG8Awv8Oo8OW/Ou/Mxby04+cwh/IHz+QOSApbX</latexit>

P1<latexit sha1_base64="qDJLWWcjw5+gE15YQaeXklBGPrw=">AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKexqQE8S8OIxonlAsoTZyWwyZHZmmZkV4pKLd6/6C97Eq3/iH/gZziZ70MSChqKqm+6uIOZMG9f9cgorq2vrG8XN0tb2zu5eef+gpWWiCG0SyaXqBFhTzgRtGmY47cSK4ijgtB2MrzO//UCVZlLcm0lM/QgPBQsZwcZKd42+1y9X3Ko7A1omXk4qkKPRL3/3BpIkERWGcKx113Nj46dYGUY4nZZ6iaYxJmM8pF1LBY6o9tPZqVN0YpUBCqWyJQyaqb8nUhxpPYkC2xlhM9KLXib+62VKjB/lwn4TXvopE3FiqCDz9WHCkZEoSwMNmKLE8IklmChmP0BkhBUmxmZWstF4i0Esk9ZZ1Tuvure1Sv0qD6kIR3AMp+DBBdThBhrQBAJDeIYXeHWenDfn3fmYtxacfOYQ/sD5/AGOzpbV</latexit>

P2<latexit sha1_base64="2OBlW9aX6KrIiQ8zJComV3TcbLE=">AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKexGQU8S8OIxonlAsoTZyWwyZHZmmZkV4pKLd6/6C97Eq3/iH/gZziZ70MSChqKqm+6uIOZMG9f9cgorq2vrG8XN0tb2zu5eef+gpWWiCG0SyaXqBFhTzgRtGmY47cSK4ijgtB2MrzO//UCVZlLcm0lM/QgPBQsZwcZKd41+rV+uuFV3BrRMvJxUIEejX/7uDSRJIioM4VjrrufGxk+xMoxwOi31Ek1jTMZ4SLuWChxR7aezU6foxCoDFEplSxg0U39PpDjSehIFtjPCZqQXvUz818uUGD/Khf0mvPRTJuLEUEHm68OEIyNRlgYaMEWJ4RNLMFHMfoDICCtMjM2sZKPxFoNYJq1a1TururfnlfpVHlIRjuAYTsGDC6jDDTSgCQSG8Awv8Oo8OW/Ou/Mxby04+cwh/IHz+QOQaJbW</latexit>

A3<latexit sha1_base64="D+3QLAI4TVbAqrYDK0IPbUDEJ4c=">AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyqoCeJePEY0TwgWcLsZDYZMjuzzMwKccnFu1f9BW/i1T/xD/wMZ5M9aGJBQ1HVTXdXEHOmjet+OYWl5ZXVteJ6aWNza3unvLvX1DJRhDaI5FK1A6wpZ4I2DDOctmNFcRRw2gpG15nfeqBKMynuzTimfoQHgoWMYGOlu6veaa9ccavuFGiReDmpQI56r/zd7UuSRFQYwrHWHc+NjZ9iZRjhdFLqJprGmIzwgHYsFTii2k+np07QkVX6KJTKljBoqv6eSHGk9TgKbGeEzVDPe5n4r5cpMX6Uc/tNeOGnTMSJoYLM1ocJR0aiLA3UZ4oSw8eWYKKY/QCRIVaYGJtZyUbjzQexSJonVe+06t6eVWqXeUhFOIBDOAYPzqEGN1CHBhAYwDO8wKvz5Lw5787HrLXg5DP78AfO5w953pbI</latexit>

A3<latexit sha1_base64="D+3QLAI4TVbAqrYDK0IPbUDEJ4c=">AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyqoCeJePEY0TwgWcLsZDYZMjuzzMwKccnFu1f9BW/i1T/xD/wMZ5M9aGJBQ1HVTXdXEHOmjet+OYWl5ZXVteJ6aWNza3unvLvX1DJRhDaI5FK1A6wpZ4I2DDOctmNFcRRw2gpG15nfeqBKMynuzTimfoQHgoWMYGOlu6veaa9ccavuFGiReDmpQI56r/zd7UuSRFQYwrHWHc+NjZ9iZRjhdFLqJprGmIzwgHYsFTii2k+np07QkVX6KJTKljBoqv6eSHGk9TgKbGeEzVDPe5n4r5cpMX6Uc/tNeOGnTMSJoYLM1ocJR0aiLA3UZ4oSw8eWYKKY/QCRIVaYGJtZyUbjzQexSJonVe+06t6eVWqXeUhFOIBDOAYPzqEGN1CHBhAYwDO8wKvz5Lw5787HrLXg5DP78AfO5w953pbI</latexit>

P4 = (Eb,�~p +~q/2)P3 = (Eb,�~p �~q/2)

<latexit sha1_base64="BA09auz9whVl03JsY6EWrpFp6wc=">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</latexit>

From amplitude to potential

cosh r =P1 · P3mamb

=) xa/xb = e2r

<latexit sha1_base64="/hixHcyc9AMKHLVTo5Hp/fgDwVs=">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</latexit>

Rapidity (for velocity-expansion later)

The BH-BH amplitude

LS = (8pG)m2

am2b

(�t)

h(xa/xb)

2 h12i2Sa [34]2Sb + (xb/xa)2[12]2Sah34i2Sb

i

<latexit sha1_base64="op+nQ65VyxJ5w+actsdjEBsbgds=">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</latexit>

Large S conversion

h12i2Sa ! exp(q · S/m) , [12]2Sa ! exp(�q · S/m) .<latexit sha1_base64="yoRSdcCr2pdgyVaC1KWs50c1icE=">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</latexit>

h(x) ⌘•

Ân=0

cn(x/S)n , f (x) ⌘•

Ân=0

Cnn!

xn =) f (x) = exh(x) .<latexit sha1_base64="3qKs8us/Gt7CSrVU17l4TqKEF9Y=">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</latexit>

1PM, leading-velocity, all-spinO(G)

<latexit sha1_base64="wpstcDCdl0iAu/QjIqpmEdb54uc=">AAACC3icdVDLSgMxFM34rPVVdekmWIS6GTLWR90VXOjOCvYB7VgyadqGZiZDkhHq0E9w71Z/wZ249SP8Az/DzHQELXrgwuGce7n3Hi/kTGmEPqy5+YXFpeXcSn51bX1js7C13VAikoTWieBCtjysKGcBrWumOW2FkmLf47Tpjc4Tv3lHpWIiuNHjkLo+HgSszwjWRrrt+FgPCebx1aR0cdAtFJF9jJyzEwSRjVKkpOKUHehkShFkqHULn52eIJFPA004VqrtoFC7MZaaEU4n+U6kaIjJCA9o29AA+1S5cXr1BO4bpQf7QpoKNEzVnxMx9pUa+57pTK5Us14i/uklSojvxcx+3a+4MQvCSNOATNf3Iw61gEkwsMckJZqPDcFEMvMBJEMsMdEmvryJ5vt/+D9pHNpO2UbXR8VqJQspB3bBHigBB5yCKrgENVAHBEjwCJ7As/VgvViv1tu0dc7KZnbAL1jvXzlCm5Q=</latexit>

O(v0),O(v1)<latexit sha1_base64="mBLHlSJp85jnvr3nVYXtVYQk1nk=">AAACIXicdZDLSgMxFIYz9VbrbdSlm9AitCAlY73UXcGNOyvYC7S1ZNK0Dc1cSDKFOnTve7h3q6/gTtyJL+BjmJmOYIv+EPj5zjmcnN/2OZMKoQ8jtbS8srqWXs9sbG5t75i7e3XpBYLQGvG4J5o2lpQzl9YUU5w2fUGxY3PasEeXUb0xpkIyz71VE592HDxwWZ8RrDTqmtm2g9WQYB5eT/PjO1Q4gvPEKnTNHCqeIuviDEFURLFiU7ZKFrQSkgOJql3zq93zSOBQVxGOpWxZyFedEAvFCKfTTDuQ1MdkhAe0pa2LHSo7YXzLFB5q0oN9T+jnKhjT3xMhdqScOLbujP4pF2sR/LMWER/fewv7Vb/cCZnrB4q6ZLa+H3CoPBjFBXtMUKL4RBtMBNMXQDLEAhOlQ83oaH7uh/+b+nHRKhXRzUmuUk5CSoMDkAV5YIFzUAFXoApqgIAH8ASewYvxaLwab8b7rDVlJDP7YE7G5zdCoaNp</latexit>

Generating function for Wilson coefficients :

f (x) ⌘•

Ân=0

Cnn!

xn , f±(x) =12( f (x)± f (�x)) , g+(x) =

f�(x)x

.<latexit sha1_base64="kRdkSlicXdWvFRK2z/CJs65ggFQ=">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</latexit>

Velocity and "spin-length" vectors :

~vi ⌘~pimi

, ~ai ⌘~Simi

.<latexit sha1_base64="BSIUuQtTWGoVh/L9b3ZTJ3OPiWc=">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</latexit>

Spins carry derivatives in classical potential :

ai ⌘ �i~ai ·r , bij ⌘ ~vi ⇥~aj ·r .<latexit sha1_base64="B38NhqWLWqZRzjI9j/ksXghPujc=">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</latexit>

1PM, leading velocity, all-spin

V0 = � [ f+(aa) f+(ab) + f�(aa) f�(ab)]

✓Gmamb

r

◆.

<latexit sha1_base64="7FLeJG6iZYnWUopPe0Nt7VzFiOE=">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</latexit>

General :

BH-BH : V0 = � cosh [(~aa +~ab)⇥r]

✓Gmamb

r

◆.

<latexit sha1_base64="DZ3fUhom1gvZH1muK14PjcvK9i8=">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</latexit>

V1 =12(baa � bbb) cosh [(~aa +~ab)⇥r]

✓Gmamb

r

◆

+ 2(~va �~vb) · sinh [(~aa +~ab)⇥r]

✓Gmamb

r

◆.

<latexit sha1_base64="/mXs2IF3T5boPKlf+tLe2BluBDI=">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</latexit>

V1 =12

⇣baa � bbb

⌘[ f+(aa) f+(ab) + f�(aa) f�(ab)]

✓Gmamb

r

◆

� 2⇣

baa � bba

⌘g+(aa) f+(ab)

✓Gmamb

r

◆

� 2⇣

bab � bbb

⌘f+(aa)g+(ab)

✓Gmamb

r

◆.

<latexit sha1_base64="gjydfYwOW5q26vu+HuxG0e3NIeo=">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</latexit>

1PMSpin-orbit and spin-spin terms up to S4

VS4aS0

b= � 3G

8m2ar5

(~Sa · ~Sa)

2 � 10(~Sa · ~Sa)(~Sa ·~n)2 +353(~Sa ·~n)4

�

<latexit sha1_base64="irk+Jxb7GpZKmRTzgc47AZnVQIc=">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</latexit>

VS2aS2

b=� 3G

4mambr5

h~S2

a~S2b + 2(~Sa · ~Sb)

2 � 5n(~Sa ·~n)2~S2

b + ~S2a(~Sb ·~n)2

+4(~Sa · ~Sb)(~Sa ·~n)(~Sb ·~n)� 7(~Sa ·~n)2(~Sb ·~n)2oi

<latexit sha1_base64="AXeSs63KjqqmPFBcGuKqgxa3IBg=">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</latexit>

VS3aS1

b= � 3G

2m2ar5

h~S2

a(~Sa · ~Sb)� 5n(~Sa · ~Sb)(~Sa ·~n)2 + ~S2

a(~Sa ·~n)(~Sb ·~n)o

+353(~Sa ·~n)3(~Sb ·~n)

�

<latexit sha1_base64="NZSzv2Oefx29rr+m2guk1mDvP+I=">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</latexit>

All are consistent with known results.

[Porto 05][Holstein, Ross 08][Levi 10][Levi, Steinhoff 15][]

=<latexit sha1_base64="1c4dBfrJusIzw7HmUM5seKPqISU=">AAAB/nicbVDLSgNBEOyNrxhfUY9eBoPgKeyqoBch4CXHBMwDkiXMTmaTIbMzy8ysEJeAd6/6C97Eq7/iH/gZziZ70MSChqKqm+6uIOZMG9f9cgpr6xubW8Xt0s7u3v5B+fCorWWiCG0RyaXqBlhTzgRtGWY47caK4ijgtBNM7jK/80CVZlLcm2lM/QiPBAsZwcZKzdtBueJW3TnQKvFyUoEcjUH5uz+UJImoMIRjrXueGxs/xcowwums1E80jTGZ4BHtWSpwRLWfzg+doTOrDFEolS1h0Fz9PZHiSOtpFNjOCJuxXvYy8V8vU2L8KJf2m/DGT5mIE0MFWawPE46MRFkWaMgUJYZPLcFEMfsBImOsMDE2sZKNxlsOYpW0L6reZdVtXlVq9TykIpzAKZyDB9dQgzo0oAUEKDzDC7w6T86b8+58LFoLTj5zDH/gfP4AQb6WKA==</latexit>

+<latexit sha1_base64="0/Sbv9MIeWo7LzpZQwhJfeRbs0o=">AAAB/nicbVDLSgNBEOyNrxhfUY9eBoMgCGFXBT0GvOSYgHlAsoTZyWwyZHZmmZkV4hLw7lV/wZt49Vf8Az/D2WQPmljQUFR1090VxJxp47pfTmFtfWNzq7hd2tnd2z8oHx61tUwUoS0iuVTdAGvKmaAtwwyn3VhRHAWcdoLJXeZ3HqjSTIp7M42pH+GRYCEj2FipeTEoV9yqOwdaJV5OKpCjMSh/94eSJBEVhnCsdc9zY+OnWBlGOJ2V+ommMSYTPKI9SwWOqPbT+aEzdGaVIQqlsiUMmqu/J1IcaT2NAtsZYTPWy14m/utlSowf5dJ+E976KRNxYqggi/VhwpGRKMsCDZmixPCpJZgoZj9AZIwVJsYmVrLReMtBrJL2ZdW7qrrN60qtnodUhBM4hXPw4AZqUIcGtIAAhWd4gVfnyXlz3p2PRWvByWeO4Q+czx8k6pYW</latexit>



Each diagram is gauge-dependent

Gravity Compton amplitudes

Gravity Compton amplitudes

Massive particle of spin up to 2

Naive extrapolation to S > 2 fails, reflecting the fact that such particles cannot be elementary.

[Arkani-Hamed,Huang,Huang, 17]P1

<latexit sha1_base64="qDJLWWcjw5+gE15YQaeXklBGPrw=">AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKexqQE8S8OIxonlAsoTZyWwyZHZmmZkV4pKLd6/6C97Eq3/iH/gZziZ70MSChqKqm+6uIOZMG9f9cgorq2vrG8XN0tb2zu5eef+gpWWiCG0SyaXqBFhTzgRtGmY47cSK4ijgtB2MrzO//UCVZlLcm0lM/QgPBQsZwcZKd42+1y9X3Ko7A1omXk4qkKPRL3/3BpIkERWGcKx113Nj46dYGUY4nZZ6iaYxJmM8pF1LBY6o9tPZqVN0YpUBCqWyJQyaqb8nUhxpPYkC2xlhM9KLXib+62VKjB/lwn4TXvopE3FiqCDz9WHCkZEoSwMNmKLE8IklmChmP0BkhBUmxmZWstF4i0Esk9ZZ1Tuvure1Sv0qD6kIR3AMp+DBBdThBhrQBAJDeIYXeHWenDfn3fmYtxacfOYQ/sD5/AGOzpbV</latexit>


k3<latexit sha1_base64="/TfBX4Mz7n+jzWZOhLbpWkTrdTE=">AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKewaQU8S8OIxonlAsoTZyWwyZHZmmZkV4pKLd6/6C97Eq3/iH/gZziZ70MSChqKqm+6uIOZMG9f9cgorq2vrG8XN0tb2zu5eef+gpWWiCG0SyaXqBFhTzgRtGmY47cSK4ijgtB2MrzO//UCVZlLcm0lM/QgPBQsZwcZKd+N+rV+uuFV3BrRMvJxUIEejX/7uDSRJIioM4VjrrufGxk+xMoxwOi31Ek1jTMZ4SLuWChxR7aezU6foxCoDFEplSxg0U39PpDjSehIFtjPCZqQXvUz818uUGD/Khf0mvPRTJuLEUEHm68OEIyNRlgYaMEWJ4RNLMFHMfoDICCtMjM2sZKPxFoNYJq2zqlerurfnlfpVHlIRjuAYTsGDC6jDDTSgCQSG8Awv8Oo8OW/Ou/Mxby04+cwh/IHz+QO9dpby</latexit>

k4<latexit sha1_base64="0fuZda5LoUeIXH0osILqKQoKiRk=">AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKexqQE8S8OIxonlAsoTZyWwyZHZmmZkV4pKLd6/6C97Eq3/iH/gZziZ70MSChqKqm+6uIOZMG9f9cgorq2vrG8XN0tb2zu5eef+gpWWiCG0SyaXqBFhTzgRtGmY47cSK4ijgtB2MrzO//UCVZlLcm0lM/QgPBQsZwcZKd+N+rV+uuFV3BrRMvJxUIEejX/7uDSRJIioM4VjrrufGxk+xMoxwOi31Ek1jTMZ4SLuWChxR7aezU6foxCoDFEplSxg0U39PpDjSehIFtjPCZqQXvUz818uUGD/Khf0mvPRTJuLEUEHm68OEIyNRlgYaMEWJ4RNLMFHMfoDICCtMjM2sZKPxFoNYJq2zqndedW9rlfpVHlIRjuAYTsGDC6jDDTSgCQSG8Awv8Oo8OW/Ou/Mxby04+cwh/IHz+QO/EJbz</latexit>

M(1S, 2S, 3�1, 4+1) = (8pG)h3|p1|4]4�2S(h32i[14] + h31i[24])2S

(s � m2)(u � m2)t<latexit sha1_base64="fJf8GwDt1qRGAa3wJUnHIhRdWNY=">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</latexit>

Gravity Compton amplitudes for S > 2Remove spurious poles.

Match residues at physical (s,t,u) poles.

Polynomial ambiguities remain.

1-loop (2PM) gravity

[Hostein,Ross 08][Neill,Rothstein 13]

[Donoghue 96][Goldberger, Rothstein 04]

V(r) = �Gmambr

✓1 + Ac

GMr

+ AQGhr2 + · · ·

◆

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[Johansson, Ochirov][Maybee, O'Connell, Vines][Plefka, Shi, Steinhoff, Wang][Bjerrum-Bohr, Cristofoli, Damgaard, Vanhove][di Vecchia, Koemans Collado, Russo]

[Bautista, Guevara][Foffa, Porto, Rothstein, Sturani][Foffa, Mastrolia, Sturani, Sturm, Torres Bobadilla]

[Antonelli, Buonanno, Steinhoff, van De Meent, Vines][Bern, Cheung, Roiban, Shen, Solon, Zeng][Guevara, Ochirov, Vines]

[Buonanno, Steinhoff, Vines][Kosower, Maybee, O'Connell][Cheung, Rothstein, Solon][Bjerrum-Bohr, Damgaard, Festuccia, Plante, Vanhove]

(2018-2019)

Leading singularity [Arkani-Hamed,Cachazo,Kaplan 08]

“Cut” all propagators running in the loop.

Cut :

Ex) Massless box integral

1p2 � m2 ! d(p2 � m2)

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LS(box) =Z d4`

(2p)4

4

’i=1

d(`2i ) =

1st

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Leading Singularity and amplitudes

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On-shellCompton

On-shell 3pt

On-shell 3pt

At 1-loop (2PM), the triangle LS suffices!

[Guevara, 1706.02314]

LS provides a short-cut to Feynman diagram computations.

Efficiently computed by HCL

[Cachazo,Guevara, 1705.10262]

[Neill,Rothstein 13]

2PM, partial results

Spin-independent

Spin-orbit

Spin-spin

[Hostein,Ross 08]

2PM, partial results

Cubic in spin

Quartic in spin

NEW !

2PM, missing pieces

Separating the iteration : (1PM)2 vs (2PM)

EFT matching with spin will be useful.

Polynomial ambiguity : terms are affected.Sn�4<latexit sha1_base64="uSBwUENWGTA1nZd+SjHcmz24pSE=">AAACB3icdZDLSgMxFIbP1Futt6pLN8EiuCozWm8bKbhxWdFesB1LJk3b0ExmSDJCHfoA7t3qK7gTtz6Gb+BjmGlHqEV/CPx85xxOzu+FnClt259WZm5+YXEpu5xbWV1b38hvbtVUEElCqyTggWx4WFHOBK1qpjlthJJi3+O07g0uknr9nkrFAnGjhyF1fdwTrMsI1gbdXt/FotWjqDRq5wt20R4LTZkj2zk7dpCTkgKkqrTzX61OQCKfCk04Vqrp2KF2Yyw1I5yOcq1I0RCTAe7RprEC+1S58fjHI7RnSAd1A2me0GhMpydi7Cs19D3T6WPdV7O1BP5ZS0iIH4KZ/bp76sZMhJGmgkzWdyOOdICSUFCHSUo0HxqDiWTmAkT6WGKiTXQ5E83P/eh/UzsoOodF+6pUKJ+nIWVhB3ZhHxw4gTJcQgWqQEDAEzzDi/VovVpv1vukNWOlM9vwS9bHNxW8meQ=</latexit>

[Cheung, Rothstein, Solon 18]

Summary

Black holes are minimally coupled to gravity.

Wilson coefficients for non-BH can be incorporated.

Dissipative effects require separate treatment.

Compton amplitudes for S > 2 have been constructed.

More work is needed to complete the 2PM result.

Massive S-matrix from minimal coupling to black holes (and ......Massive S-matrix from minimal...

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