Scattering amplitudes from soft theorems Yohei Ema University of - PowerPoint PPT Presentation

Scattering amplitudes from soft theorems Yohei Ema University of Tokyo PPP2017 @ YITP 2017.08.02 On-going work with S. Chigusa, H. Shimizu, Y. Tachikawa and T. Yamaura

Introduction

Scattering amplitude theory • Scattering amplitude is of central importance in particle physics. • It sometimes shows a surprising simplicity that is not obvious from the standard Feynman diagrammatic method. ex. 6pt Maximally Helicity Violated (MHV) amplitude of pure YM: h 1 2 i 4 A 6 [1 − 2 − 3 + 4 + 5 + 6 + ] = h 1 2 ih 2 3 ih 3 4 ih 4 5 ih 5 6 ih 6 1 i after summing over 220 (!) diagrams. • Scattering amplitude program tries to construct amplitudes from analytical properties, not relying (heavily) on Feynman diagrams.

Goal of this talk • Show that tree-level YM/gravity amplitudes are recursively constructible, with leading soft theorem being an input. • We also review basic ingredients of modern scattering amplitude theory.

Outline 1. Introduction 2. Review A: spinor helicity formalism 3. Review B: on-shell recursion 4. Review C: soft theorems 5. Idea (and explicit computation) 6. Summary

Outline 1. Introduction 2. Review A: spinor helicity formalism 3. Review B: on-shell recursion 4. Review C: soft theorems 5. Idea (and explicit computation) 6. Summary

Spinor helicity formalism • Consider 4-dim theory with only massless particles. • The momentum satisfies σ µ ) ˙ ab p ˙ ab ≡ p µ (¯ a [ p | b . det p = − p µ p µ = 0 ab = � | p i ˙ p ˙ • The momentum product in this language is ˙ [ p q ] ≡ ✏ ab [ p | a [ q | b . 2 p · q = h p q i [ p q ] b | p i ˙ a | q i b where and h p q i ⌘ ✏ ˙ a ˙ Amplitudes are constructed from these products. • Little group keeps momentum intact. In terms of angle/square brackets: | p i ! t | p i , | p ] ! t − 1 | p ] . • Amplitude transforms due to the external lines as: = t − 2 h i ..., { t i | i i , t − 1 � � i | i ] , h i } , ... A n ( ..., {| i i , | i ] , h i } , ... ) . A n i • Three point amplitude is determined solely from little group scaling.

Outline 1. Introduction 2. Review A: spinor helicity formalism 3. Review B: on-shell recursion 4. Review C: soft theorems 5. Idea (and explicit computation) 6. Summary

Complex momentum shift [Britto, Cachazo, Feng, 04; Britto, Cachazo, Feng, Witten, 05] • Consider the following complex momentum shift: p i ( z ) ≡ p i + zq i , where on-shell condition of p i · q i = q 2 ˆ p i ( z ) ˆ i = 0 : X and momentum conservation of . p i ( z ) ˆ q i = 0 : i Shifted amplitude is a function of z : ˆ A n ( z ) A n = ˆ (original amplitude is ). A n (0) • Poles: associated with on-shell intermediate particle (Locality). Oltshenamp . ACH d 3 Au Ar , . ¥-1 / \ I .µ - 2 fdiacpnnaitpnns ? n ; ' → • Amplitude factorizes near the poles as I Fi y . . , NL 1 A ( z ) → ˆ ˆ A R ( z I ) , ˆ ˆ P 2 A L ( z I ) I ( z ) ∝ ( z − z I ) . ˆ P 2 I ( z )

On-shell recursion [Britto, Cachazo, Feng, 04; Britto, Cachazo, Feng, Witten, 05] • From the standard complex analysis, we obtain 1 A ( z ) = − 1 1 dz dz I I ˆ ˆ ˆ ˆ X A (0) = A L ( z I ) A R ( z I ) + B ∞ . ˆ 2 π i 2 π i z z P 2 I ( z ) | z | =0 | z − z I | =0 I (b) (a) • : products of lower point on-shell amplitudes. (a) • : contribution from which vanishes when lim ˆ (b) | z | = ∞ , A ( z ) = 0 . | z | →∞ on-shell constructibility of the theory B ∞ = 0 ⇔ • Two (or more) ways to achieve on-shell constructibility: (1) Invent a good momentum shift (such as BCFW shift) ˆ ˆ A ( z ) A ( z ) (2) Modify the integrand as zf ( z ) . → [Cheung, Kampf, Novotny, Shen, Trnka, 15] z (We should know how the amplitude behaves as ) f ( z ) → 0 .

On-shell recursion [Britto, Cachazo, Feng, 04; Britto, Cachazo, Feng, Witten, 05] • From the standard complex analysis, we obtain 1 A ( z ) = − 1 1 dz dz I I ˆ ˆ ˆ ˆ X A (0) = A L ( z I ) A R ( z I ) + B ∞ . ˆ 2 π i 2 π i z z P 2 I ( z ) | z | =0 | z − z I | =0 I (b) (a) • : products of lower point on-shell amplitudes. (a) • : contribution from which vanishes when lim ˆ (b) | z | = ∞ , A ( z ) = 0 . | z | →∞ on-shell constructibility of the theory B ∞ = 0 ⇔ • Two (or more) ways to achieve on-shell constructibility: (1) Invent a good momentum shift (such as BCFW shift) ˆ ˆ A ( z ) A ( z ) (2) Modify the integrand as zf ( z ) . → [Cheung, Kampf, Novotny, Shen, Trnka, 15] z (We should know how the amplitude behaves as ) f ( z ) → 0 .

(Anti-)holomorphic shift [Cohen, Elvang, Kiermaier, 10] • We will stick to the following complex momentum shifts. X | ˆ i i = | i i � a i z | X i , | ˆ Holomorphic shift: with a i | i ] = 0 . i ] = | i ] | ˆ i i = | i i , | ˆ X Anti-holomorphic shift: with a i | i i = 0 . i ] = | i ] � a i z | X ] q i = � a i | X i [ i | ! q 2 * for holomorphic shift. i = q i · p i = 0 ** Similar relation holds for anti-holomorphic shift. • We will take or in the following. | X i = | 1 i | X ] = | 1] z = 1 /a 1 corresponds to the soft limit of the particle 1.

Large z behavior P h ... i a n [ ... ] s n • If coupling dimension is unique, amplitude is A n = g P h ... i a d [ ... ] s d . * : common due to little group scaling and mass dimension. a i , s i Let us define a ≡ a n − a d , s ≡ s n − s d . • Dimensional analysis: a + s = 4 − n − [ g ] where mass dimension of coupling [ g ] : g. • Little group scaling: X h i where helicity of i-th particle. a − s = − h i : [Cohen, Elvang, Kiermaier, 10] i ˆ Hol shift: lim X A n ( z ) → O ( z a ) with 2 a = 4 − n − [ g ] − h i . z →∞ i Anti-hol shift: ˆ X with 2 s = 4 − n − [ g ] + lim A n ( z ) → O ( z s ) h i . z →∞ i * YM: Einstein gravity: [ g ] = − n + 2 . [ g ] = 0 ,

Outline 1. Introduction 2. Review A: spinor helicity formalism 3. Review B: on-shell recursion 4. Review C: soft theorems 5. Idea (and explicit computation) 6. Summary

Leading soft theorems [Low 58; Weinberg 65; …] • Leading soft theorem: { p ✏ | 1 i , p ✏ | 1] , h 1 } , ... = 1 ✏ S (0) A n − 1 ( {| 2 i , | 2] , h 2 } , ... ) + O ( ✏ 0 ) � � A n n [1 k ] h x k ih y k i S (0) = X where for positive helicity graviton h 1 k ih x 1 ih y 1 i k =2 h x 2 i h x 4 i S (0) = and for positive helicity gluon (color-ordered). h x 1 ih 1 2 i � h x 1 ih 1 4 i • From little group scaling, it behaves under hol/anti-hol soft limit as A n ( { ✏ | 1 i , | 1] , h 1 } , ... ) = ✏ − 1 − h 1 S (0) A n − 1 ( {| 2 i , | 2] , h 2 } , ... ) + O ( ✏ − h 1 ) and A n ( {| 1 i , ✏ | 1] , h 1 } , ... ) = ✏ − 1+ h 1 S (0) A n − 1 ( {| 2 i , | 2] , h 2 } , ... ) + O ( ✏ h 1 ) .

Outline 1. Introduction 2. Review A: spinor helicity formalism 3. Review B: on-shell recursion 4. Review C: soft theorems 5. Idea (and explicit computation) 6. Summary

What we learned so far • Integrand should fall off at large z for on-shell constructibility. • Under holomorphic or anti-holomorphic shift: ˆ A 4 [1 + 2 + 3 − 4 − ] → const for pure YM theory lim z →∞ ˆ and at worst for Einstein gravity. lim M n ( z ) → z z →∞ • Under holomorphic/anti-holomorphic soft limit: A n ( { ✏ | 1 i , | 1] , h 1 } , ... ) = ✏ − 1 − h 1 S (0) A n − 1 + O ( ✏ − h 1 ) A n ( {| 1 i , ✏ | 1] , h 1 } , ... ) = ✏ − 1+ h 1 S (0) A n − 1 + O ( ✏ h 1 ) . and

Idea • Main idea: use soft theorem to take better integrand. • Consider the worst case and assume X h i = 0 h 1 > 0 . Under anti-holomorphic soft shift, ˆ ˆ large z behavior is for YM and for gravity. A 4 ( z ) → z 0 M n ( z ) → z 1 lim lim z →∞ z →∞ S (0) ˆ A 4 ( z ) = ˆ ˆ A 3 | z =1 /a 1 + O ( ✏ 1 ) Soft limit is for YM S (0) ˆ M n ( z ) = ✏ ˆ ˆ M n − 1 | z =1 /a 1 + O ( ✏ 2 ) and for gravity with ✏ ≡ 1 − a 1 z. Take integrand as I dz I dz ˆ ˆ M n ( z ) A 4 ( z ) for YM and for gravity! (1 − a 1 z ) 2 1 − a 1 z z z Integrand falls off rapidly enough at large z. Residue at is nothing but the leading soft term. z = 1 /a 1

Computation: gluon 4pt • Consider 4pt (color-ordered) YM amplitude A 4 [1 + 2 + 3 − 4 − ] . • Under anti-holomorphic soft shift, pole is only at z = 1 /a 1 . p j ( z )) 2 ∝ (1 − a 1 z ) ( p i + p j ) 2 . * (ˆ p i ( z ) + ˆ We need to consider only the soft factor (soft limit is ``exact’’): S (0) ˆ A 4 [1 + 2 + 3 − 4 − ] = ˆ A 3 [2 + 3 − 4 − ] | z =1 /a 1 3pt from little group h 3 4 i 4 ✓ ◆ h x 2 i h x 4 i = h x 1 ih 1 2 i � h x 1 ih 1 4 i h 2 3 ih 3 4 ih 4 2 i h 3 4 i 4 = h 1 2 ih 2 3 ih 3 4 ih 4 1 i , where Schouten identity is used. | 1 ih 2 4 i + | 2 ih 4 1 i + | 4 ih 1 2 i = 0 : It correctly reproduces the Parke-Taylor MHV amplitude. [Parke and Taylor 86]