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arxiv:1511.06716 Tales of the Unexpected: One-Loop Soft Theorems via Hidden Symmetries Andreas Brandhuber Edward Hughes Bill Spence Gabriele Travaglini Queen Mary University of London Young Theorists Forum, 15th January 2016 Edward

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Tales of the Unexpected: One-Loop Soft Theorems via Hidden Symmetries

Andreas Brandhuber Edward Hughes Bill Spence Gabriele Travaglini

Queen Mary University of London

Young Theorists’ Forum, 15th January 2016

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 1 / 31

arxiv:1511.06716

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SLIDE 2

Outline

  • I. Motivation
  • II. Soft Theorems at Tree Level
  • III. Soft Theorems at One Loop
  • IV. Applications

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 2 / 31

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SLIDE 3

Motivation

hep-th Papers with ‘Soft’ in the Title

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 3 / 31

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SLIDE 4

Motivation

Solution to the Information Paradox?

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 4 / 31

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SLIDE 5

Motivation

Origin of IR Divergences

Massless QFTs have two types of IR divergences Brehmsstrahlung processes (soft, collinear)

p p′ k

∼ dσ0 α π log −E2

l

µ2

  • log

−q2 m2

  • Edward Hughes (QMUL)

One-Loop Soft Theorems Young Theorists’ Forum 5 / 31

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SLIDE 6

Motivation

Origin of IR Divergences

Massless QFTs have two types of IR divergences Brehmsstrahlung processes (soft, collinear)

p p′ k

∼ dσ0 α π log −E2

l

µ2

  • log

−q2 m2

  • Massless particles running in loops

p p′ k

∼ dσ0

  • 1 − α

π log −q2 µ2

  • log

−q2 m2

  • Divergences cancel in S-matrix across amplitude loop orders

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 5 / 31

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Motivation

Soft Theorems Ancient...

Leading tree-level soft universal in QED and gravity [Weinberg 1964] Atree

n

→ 1 δ2 S(0)Atree

n−1 + . . . as pn → 0

Leading tree-level soft universal in YM theories [Berends, Giele 1988]

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 6 / 31

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Motivation

Soft Theorems Ancient...

Leading tree-level soft universal in QED and gravity [Weinberg 1964] Atree

n

→ 1 δ2 S(0)Atree

n−1 + . . . as pn → 0

Leading tree-level soft universal in YM theories [Berends, Giele 1988] Subleading tree-level soft universal in QED [Low et al. 1968]

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 6 / 31

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SLIDE 9

Motivation

Soft Theorems Ancient...

Leading tree-level soft universal in QED and gravity [Weinberg 1964] Atree

n

→ 1 δ2 S(0)Atree

n−1 + . . . as pn → 0

Leading tree-level soft universal in YM theories [Berends, Giele 1988] Subleading tree-level soft universal in QED [Low et al. 1968] Leading loop corrections universal in QCD [Bern et al. 1998] A1-loop

n

→ 1 δ2

  • S(0)treeA1-loop

n−1

+ S(0)1-loopAtree

n−1

  • + . . .

Leading soft not renormalised in gravity [W 1964, B+ 1998] A1-loop

n

∼ Atree

n

  • i=j
  • sij

log sij ǫ

  • Edward Hughes (QMUL)

One-Loop Soft Theorems Young Theorists’ Forum 6 / 31

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SLIDE 10

Motivation

...and Modern

Subleading tree-level soft behaviour universal in gravity [White] Ward identity for Virasoro symmetry at null infinity [Cachazo et al.] Valid in arbitrary dimension via scattering equations [Schwab et al.]

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 7 / 31

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Motivation

...and Modern

Subleading tree-level soft behaviour universal in gravity [White] Ward identity for Virasoro symmetry at null infinity [Cachazo et al.] Valid in arbitrary dimension via scattering equations [Schwab et al.] Subleading tree-level soft behaviour universal in YM theory [Casali] Atree

n

→ 1 δ2 S(0)Atree

n−1 + 1

δ S(1)Atree

n−1 + . . .

Symmetry interpretations for YM and QED [Lipstein, Strominger, Lysov, Paterski]

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 7 / 31

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Motivation

...and Modern

Subleading tree-level soft behaviour universal in gravity [White] Ward identity for Virasoro symmetry at null infinity [Cachazo et al.] Valid in arbitrary dimension via scattering equations [Schwab et al.] Subleading tree-level soft behaviour universal in YM theory [Casali] Atree

n

→ 1 δ2 S(0)Atree

n−1 + 1

δ S(1)Atree

n−1 + . . .

Symmetry interpretations for YM and QED [Lipstein, Strominger, Lysov, Paterski] Limited knowledge of subleading behaviour at 1-loop [Bern, Dixon, Nohle, Neill, Stewart, Larkoski, Broedel, de Leeuw, Plefka, Rosso]

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 7 / 31

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SLIDE 13

Motivation

The Question

Is 1-loop subleading soft behaviour universal?

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 8 / 31

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SLIDE 14

Motivation

The Question

Is 1-loop subleading soft behaviour universal?

(in planar N = 4 SYM theory)

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 8 / 31

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SLIDE 15

Soft Theorems at Tree Level

Outline

  • I. Motivation
  • II. Soft Theorems at Tree Level
  • III. Soft Theorems at One Loop
  • IV. Applications

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 9 / 31

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Soft Theorems at Tree Level

Definition of Soft Operators

Consider holomorphic soft scaling of a positive helicity gluon |n → δ|n , |n] → |n] , pn → δpn In N = 4 we can write a tree-level soft theorem [Casali] Atree

n

= 1 δ2 S(0) + 1 δ S(1) + . . .

  • Atree

n−1

Universal soft operators are given by

S(0) = n − 1 1 n − 1 nn 1 S(1) = |n] n − 1 n · ∂ ∂|n − 1] + |n] n 1 · ∂ ∂|1]

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 10 / 31

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Soft Theorems at Tree Level

Soft Theorem for Stripped Amplitudes

Amplitudes contain an overall momentum conservation delta function An = Anδ(4)(Pn) Can define various equivalent stripped amplitudes by integrating out A(ab)

n

=

  • d|a]d|b]|a b|An

Soft theorems require consistent solution of momentum conservation [Bern, Nohle, Davies] Atree(ab)

n

= 1 δ2 S(0) + 1 δ S(1)

  • Atree

n−1

(ab)

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 11 / 31

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Soft Theorems at Tree Level

Soft Operators from Conformal Symmetry

N = 4 has a conformal symmetry with boost generator kα ˙

αAn = n−1

  • i=1

∂2 ∂|iα[i| ˙

α + 1

δ ∂2 ∂|nα∂[n| ˙

α An = 0

Substituting the amplitude soft expansion gives constraint equations

∂2 ∂|nα∂[n| ˙

α

  • S(0)Atree

n−1

  • = 0

n−1

  • i=1

∂2 ∂|iα[i| ˙

α

  • S(0)Atree

n−1

  • +

∂2 ∂|nα∂[n| ˙

α

  • S(1)Atree

n−1

  • = 0

which fix soft operators S(0) and S(1) [Larkoski] Difficult to generalize to 1-loop since anomaly unknown

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 12 / 31

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Soft Theorems at Tree Level

Amplitude/Wilson Loop Duality

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 13 / 31

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Soft Theorems at Tree Level

Dual Superconformal Symmetry

Wilson loop has a conformal symmetry. . . . . . which is a new hidden symmetry of the amplitude! Define dual coordinates (xi − xi+1)α ˙

α = (pi)α ˙ α and (θi − θi+1)A α = i|αηA i

The dual conformal boost generator is Kα ˙

α = n

  • i=1
  • xβ

i ˙ αi|α

∂ ∂|iβ + xi+1α ˙

β|i] ˙ α

∂ ∂|i] ˙

β

+ θA

i+1α|i] ˙ α

∂ ∂ηA

i

  • Tree-level superamplitudes are covariant

Kα ˙

αAtree n

= − n

  • i=1

xiα ˙

α

  • Atree

n

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 14 / 31

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SLIDE 21

Soft Theorems at Tree Level

Soft Dependence in Dual Space

We must determine xi as a function of pj Ambiguities! Which base point? Which way round polygon? Related to ambiguity in defining stripped amplitude Choose minimal δ dependence compatible with eliminating |1] and |2]

δpn x1 x2 x3

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 15 / 31

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Soft Theorems at Tree Level

Constraint Equations at Tree Level

Leading soft (Kα ˙

α)O(δ0) S(0) =

 

n−1

  • j=3

pj   S(0) Subleading soft −2|n]1| n 1 + S(0) (Kα ˙

α)O(δ1) +

  • (Kα ˙

α)O(δ0) , S(1)

=  

n−1

  • j=3

pj   S(1) Suffice to fix soft operators S(0) and S(1)

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 16 / 31

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Soft Theorems at One Loop

Outline

  • I. Motivation
  • II. Soft Theorems at Tree Level
  • III. Soft Theorems at One Loop
  • IV. Applications

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 17 / 31

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Soft Theorems at One Loop

Definition of Subleading Soft Anomaly

One-loop amplitudes have IR divergences; we need regulator ǫ Ansatz for soft limit A1-loop

n

= 1 δ2

  • S(0)treeA1-loop

n−1

+ S(0)1-loopAtree

n−1

  • + 1

δ

  • S(1)treeA1-loop

n−1

+ S(1)1-loopAtree

n−1

  • + . . .

Define the subleading soft anomaly Z by S(1)1-loop = F (0)S(1)tree + 1 ǫ Z−1 + Z0 + O(ǫ) IR divergent piece Z−1 known and universal [Bern, Nohle, Davies]

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 18 / 31

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Soft Theorems at One Loop

The Question

Is Z0 universal?

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 19 / 31

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SLIDE 26

Soft Theorems at One Loop

IR Divergences and Dual Conformal Anomaly

1-loop IR divergences in amplitude correspond to. . . . . . cusp divergences in Wilson loop This suggests simple dual conformal anomaly

Kα ˙

αA1-loop n

= 2 ǫ cΓ n

  • i=1

xiα ˙

α

  • −[i i − 1]i − 1 i

−ǫ

  • Atree

n

− n

  • i=1

xiα ˙

α

  • A1-loop

n

Proved via unitarity [Brandhuber, Spence, Travaglini] Use this to derive constraint equations for soft theorems at 1-loop

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 20 / 31

pn x1 x2 p1 p2

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Soft Theorems at One Loop

Soft Anomaly from Dual Conformal Anomaly

Constraint equation for Z0

  • (Kα ˙

α)O(δ0) − n−1

  • j=3

|j]j|

  • Z0

= 2

  • Z−1

n−1

  • j=3

|j]j| − |n − 1]n − 1| [n − 2 n] n − 1 n[n − 1 n − 2] + |1]1| [2 n] [2 1]1 n

  • +
  • |n]1|

n 1 + 2 |n]n − 1| n − 1 n − |1]n| n − 1 1[n n − 1] n − 1 nn 1[1 n − 1]

  • log
  • −

(n − 1 1) (n − 1 n)(n 1)

  • − 2

|n]n − 1| n − 1 n log(−(n − 1 1)) + 2|n]n| n − 1 1 n − 1 nn 1 log(−(n 1))

Suggests the subleading log δ terms

Z0|log δ =

  • (n 1)

(n − 1 1) + (n − 2 n) (n − 2 n − 1) − (n − 2 1)(n − 1 n) (n − 2 n − 1)(n − 1 1)

  • S(0) log(−(n − 1 n))

Must verify this with explicit calculations

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 21 / 31

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Soft Theorems at One Loop

MHV Amplitudes at One Loop

Box-function form [Bern et al. 1994] AMHV,1-loop

n

= AMHV,tree

n channels

F 2me + cyclic

  • Edward Hughes (QMUL)

One-Loop Soft Theorems Young Theorists’ Forum 22 / 31

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Soft Theorems at One Loop

MHV Amplitudes at One Loop

Box-function form [Bern et al. 1994] AMHV,1-loop

n

= AMHV,tree

n channels

F 2me + cyclic

  • Dual-conformal-adapted form

A1-loop

n

Atree

n

= 1 2

  • i
  • j∈{i−2,i−1,i,i+1,i+2}
  • −Li2(1 − uij) + log x2

ij log uij

  • +
  • i

log(x2

ii−2) log

  x2

i+1i−2

x2

i+1i−1

  • x2

ii−2

  , Convenient to use momentum twistor variables

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 22 / 31

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Soft Theorems at One Loop

Complete Soft Anomaly in MHV Sector

Compact formula involving bulk terms n − 1 1 n − 1 n

n−4

  • j=4

log

  • y2

n−1j

y2

1j

  • n − 2 n − 1 j − 1 jn − 2 n − 1 n 1

n − 2 n − 1 1 j − 1n − 2 n − 1 1 j Universal boundary terms, including the log δ piece (n 1) + (n 2) (1 2) − sn−1,1,2(n 1) (n − 1 1)(1 2)

  • log(−(n 1))

Feed-down contributions, including a log δ piece Agreement with prediction from dual conformal anomaly!

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 23 / 31

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Soft Theorems at One Loop

Low-Point Verification

Numerically verified against soft behaviour of box-function form Perfect agreement at 5, 6, 7-points Anomaly from box functions is hard to simplify analytically # points # terms Zdual # terms Zboxes 5 6 22 6 20 205 7 38 1233

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 24 / 31

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Soft Theorems at One Loop

NMHV Amplitudes

Tree-level NMHV amplitudes equal MHV × ratio functions ANMHV,tree

n

= AMHV,tree

n

  • j,k

R1jk Ratio functions are dual superconformal invariant [Drummond et al.] Each R1jk satisfies the tree-level soft theorem [Bianchi et al.]

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 25 / 31

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SLIDE 33

Soft Theorems at One Loop

NMHV Amplitudes

Tree-level NMHV amplitudes equal MHV × ratio functions ANMHV,tree

n

= AMHV,tree

n

  • j,k

R1jk Ratio functions are dual superconformal invariant [Drummond et al.] Each R1jk satisfies the tree-level soft theorem [Bianchi et al.] Dual superconformal invariant V functions enter at 1-loop ANMHV,1-loop

n

= AMHV,tree

n

  • i,j,k

Rtree

ijk V 1-loop ijk

+ AMHV,1-loop

n

  • j,k

Rtree

1jk

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 25 / 31

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Soft Theorems at One Loop

log δ Anomaly in NMHV Sector

Surprising and non-trivial cancellations 6-point anomaly has non-trivial log δ piece 1 2 n − 1 1 n − 1 nR524 2 3 4 54 5 6 1 1 2 4 53 4 5 1 log(−(6 1)) 7-point anomaly has non-trivial log δ piece

1 2 n − 1 1 n − 1 n

  • R624

2 3 4 65 6 7 1 3 4 6 11 2 5 6 + R625 2 4 5 65 6 7 1 4 5 6 11 2 5 6

  • log(−(7 1))

Evidence for universality within NMHV sector But no agreement with prediction from dual conformal anomaly

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 26 / 31

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SLIDE 35

Soft Theorems at One Loop

The Question

Is 1-loop subleading soft behaviour universal?

(in planar N = 4 SYM theory)

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 27 / 31

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SLIDE 36

Soft Theorems at One Loop

The Question

Is 1-loop subleading soft behaviour universal?

(in planar N = 4 SYM theory)

Within but not between helicity sectors!

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 27 / 31

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SLIDE 37

Soft Theorems at One Loop

The Question

Is 1-loop subleading soft behaviour universal?

(in planar N = 4 SYM theory)

Within but not between helicity sectors!

(we conjecture)

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 27 / 31

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Applications

Outline

  • I. Motivation
  • II. Soft Theorems at Tree Level
  • III. Soft Theorems at One Loop
  • IV. Applications

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 28 / 31

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Applications

A Cornucopia of Applications

Better resummation in QCD [Bonocore et al.] Improved soft-collinear bootstrap [Bourjaily et al.] Flat space holography [Adamo, Casali, Skinner, Lipstein, . . . ] Theory construction [Cheung et al.] Black hole entropy [Hawking, Perry, Strominger]

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 29 / 31

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Applications

Future Work

More sectors, higher loops! Soft anomaly from one-loop scattering equations [Geyer et al.] Relation to soft-collinear effective theory [Larkoski et al.] Subleading soft friendly amplitudes [Bern, Dixon, Smirnov] Symmetry intepretation of quantum corrections [Strominger et al.] Subleading collinear theorems [Stieberger, Kosower] Subleading soft behaviour of form factors [Bork et al.] Color-kinematics duality → gravity [White, Oxburgh, Feng et al.]

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 30 / 31

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Applications

www.collabor8research.org

Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 31 / 31