Tales of the Unexpected: One-Loop Soft Theorems via Hidden - PowerPoint PPT Presentation

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

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

Motivation hep-th Papers with ‘Soft’ in the Title Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 3 / 31

Motivation Solution to the Information Paradox? Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 4 / 31

Motivation Origin of IR Divergences Massless QFTs have two types of IR divergences Brehmsstrahlung processes (soft, collinear) p ′ p k � − E 2 � − q 2 � � α l ∼ dσ 0 π log log µ 2 m 2 Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 5 / 31

Motivation Origin of IR Divergences Massless QFTs have two types of IR divergences Brehmsstrahlung processes (soft, collinear) p ′ p k � − E 2 � − q 2 � � α l ∼ dσ 0 π log log µ 2 m 2 Massless particles running in loops p ′ p k � − q 2 � − q 2 � � �� 1 − α ∼ dσ 0 π log log µ 2 m 2 Divergences cancel in S -matrix across amplitude loop orders Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 5 / 31

Motivation Soft Theorems Ancient... Leading tree-level soft universal in QED and gravity [Weinberg 1964] → 1 A tree δ 2 S (0) A tree n − 1 + . . . as p n → 0 n Leading tree-level soft universal in YM theories [Berends, Giele 1988] Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 6 / 31

Motivation Soft Theorems Ancient... Leading tree-level soft universal in QED and gravity [Weinberg 1964] → 1 A tree δ 2 S (0) A tree n − 1 + . . . as p n → 0 n 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

Motivation Soft Theorems Ancient... Leading tree-level soft universal in QED and gravity [Weinberg 1964] → 1 A tree δ 2 S (0) A tree n − 1 + . . . as p n → 0 n 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] → 1 � � S (0)tree A 1-loop A 1-loop + S (0)1-loop A tree + . . . n n − 1 n − 1 δ 2 Leading soft not renormalised in gravity [W 1964, B+ 1998] � log s ij � A 1-loop ∼ A tree � s ij n n ǫ i � = j Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 6 / 31

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

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] → 1 n − 1 + 1 A tree δ 2 S (0) A tree δ S (1) A tree n − 1 + . . . n Symmetry interpretations for YM and QED [Lipstein, Strominger, Lysov, Paterski] Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 7 / 31

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] → 1 n − 1 + 1 A tree δ 2 S (0) A tree δ S (1) A tree n − 1 + . . . n 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

Motivation The Question Is 1 -loop subleading soft behaviour universal? Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 8 / 31

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

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

Soft Theorems at Tree Level Definition of Soft Operators Consider holomorphic soft scaling of a positive helicity gluon | n � → δ | n � , | n ] → | n ] , p n → δp n In N = 4 we can write a tree-level soft theorem [Casali] � 1 δ 2 S (0) + 1 � δ S (1) + . . . A tree A tree = n n − 1 Universal soft operators are given by � n − 1 1 � S (0) = � n − 1 n �� n 1 � | n ] | n ] ∂ ∂ S (1) = � n − 1 n � · ∂ | n − 1] + � n 1 � · ∂ | 1] Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 10 / 31

Soft Theorems at Tree Level Soft Theorem for Stripped Amplitudes Amplitudes contain an overall momentum conservation delta function A n = A n δ (4) ( P n ) Can define various equivalent stripped amplitudes by integrating out � A ( ab ) = d | a ] d | b ] |� a b �|A n n Soft theorems require consistent solution of momentum conservation [Bern, Nohle, Davies] �� 1 � ( ab ) δ 2 S (0) + 1 � A tree( ab ) δ S (1) A tree = n − 1 n Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 11 / 31

Soft Theorems at Tree Level Soft Operators from Conformal Symmetry N = 4 has a conformal symmetry with boost generator n − 1 ∂ 2 ∂ 2 α + 1 � k α ˙ α A n = α A n = 0 ∂ | i � α [ i | ˙ ∂ | n � α ∂ [ n | ˙ δ i =1 Substituting the amplitude soft expansion gives constraint equations ∂ 2 � � S (0) A tree = 0 n − 1 ∂ | n � α ∂ [ n | ˙ α n − 1 ∂ 2 ∂ 2 � � � � � S (0) A tree S (1) A tree + = 0 n − 1 n − 1 ∂ | i � α [ i | ˙ α ∂ | n � α ∂ [ n | ˙ α i =1 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

Soft Theorems at Tree Level Amplitude/Wilson Loop Duality Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 13 / 31

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 α and ( θ i − θ i +1 ) A α = � i | α η A ( x i − x i +1 ) α ˙ α = ( p i ) α ˙ i The dual conformal boost generator is n � � ∂ ∂ ∂ � x β + θ A α � i | α β | i ] ˙ i +1 α | i ] ˙ K α ˙ α = ∂ | i � β + x i +1 α ˙ α α i ˙ ∂η A ∂ | i ] ˙ β i i =1 Tree-level superamplitudes are covariant � n � α A tree � A tree = − K α ˙ x iα ˙ α n n i =1 Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 14 / 31

Soft Theorems at Tree Level Soft Dependence in Dual Space We must determine x i as a function of p j 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] δp n x 1 x 2 x 3 Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 15 / 31

Soft Theorems at Tree Level Constraint Equations at Tree Level Leading soft   n − 1 α ) O ( δ 0 ) S (0) = �  S (0) ( K α ˙ p j  j =3 Subleading soft   n − 1 − 2 | n ] � 1 | � n 1 � + S (0) ( K α ˙ � α ) O ( δ 0 ) , S (1) � �  S (1) α ) O ( δ 1 ) + ( K α ˙ = p j  j =3 Suffice to fix soft operators S (0) and S (1) Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 16 / 31

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

Soft Theorems at One Loop Definition of Subleading Soft Anomaly One-loop amplitudes have IR divergences; we need regulator ǫ Ansatz for soft limit = 1 � � S (0)tree A 1-loop A 1-loop + S (0)1-loop A tree n n − 1 n − 1 δ 2 + 1 � � S (1)tree A 1-loop + S (1)1-loop A tree + . . . n − 1 n − 1 δ Define the subleading soft anomaly Z by S (1)1-loop = F (0) S (1)tree + 1 ǫ Z − 1 + Z 0 + 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

Soft Theorems at One Loop The Question Is Z 0 universal? Edward Hughes (QMUL) One-Loop Soft Theorems Young Theorists’ Forum 19 / 31