Towards the S-matrix of massless QFTs on the Riemann sphere Piotr - - PowerPoint PPT Presentation

Towards the S-matrix of massless QFTs on the Riemann sphere

Piotr Tourkine, University of Cambridge

1

IGST 2017, ENS Paris

In collaboration with

• Eduardo Casali, Yannick Herfray,
• Yvonne Geyer, Lionel Mason, Ricardo Monteiro.

• Motivations
• Review: CHY formulae & Twistor strings
• Loops
• Null strings

Outline

2

Scattering equations formalism

[arXiv:1307.2199] Scattering of Massless Particles in Arbitrary Dimensions

• F. Cachazo, S. He, E. Y. Yuan

[arXiv:1311.2564] Ambitwistor strings and the scattering equations

• L. Mason, D. Skinner

3

Scattering equations formalism

[arXiv:1307.2199] Scattering of Massless Particles in Arbitrary Dimensions

• F. Cachazo, S. He, E. Y. Yuan

[arXiv:1311.2564] Ambitwistor strings and the scattering equations

• L. Mason, D. Skinner

4

Witten; Roiban Spradlin Volovich

5

« »

Quote from David Skinner, at Amplitudes 2015 (Zürich)

The scattering equations

6

Fairlie, Roberts ‘72, Gross Mende, ‘87

n

• i=1

ki = 0 P : CP1 → Cd P(z) =

• ki

z − zi dz z, zi ∈ CP1

z1 z2 zn

… Let and n null momenta ki ∈ Rd, k2

i = 0

Scattering equations: ResP2

• zi =
• i,j

ki · kj zi − zj = 0

The scattering equations

7

Fairlie, Roberts ‘72, Gross Mende, ‘87

ResP2

• zi =
• i,j

ki · kj zi − zj = 0

The scattering equations

• Solve for zi in terms of ki . kj
• SL(2,C) invariance
• (n-3)! solutions

8

Arise in large α’ limit of string theory Gross, Mende ‘87

ResP2

• zi =
• i,j

ki · kj zi − zj = 0

Cachazo-He-Yuan formulae (2013)

scattering data: 
 kinematics, polarisations, 
 colour structure, …

• n
• i=1

dzi ¯

• sc. eqns.
• F(ki, j, zi)

tree-level amplitude = F = IL × IR ⟶ I kinI kin ∼ gravity IL/R ∈ {I kin, I colour} = ⇒ I colourI kin ∼ gauge theory I colourI colour ∼ scalar

Cachazo-He-Yuan formulae (2013)

• n
• i=1

dzi ¯

• sc. eqns.
• F(ki, j, zi)

tree-level amplitude = I kinI kin ∼ gravity IL/R ∈ {I kin, I colour} = ⇒ I colourI kin ∼ gauge theory I colourI colour ∼ scalar just for completeness : I kin = Pf(M) I colour = Tr(T a1 . . . T an) (z1 − z2) . . . (zn − z1) + permutations

n-point field theory amplitude

Cachazo-He-Yuan formulae (2013)

=

• n
• i=1

dzi ¯

• sc. eqns.
• F(ki, j, zi)

tree-level amplitude =

• solutions

F Jac

The scattering equations

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z1

z2

…

zn

…

z1

z2 zn

…

zk zk+1

⟶ K = k1 + · · · + kk, K 2 → 0 1 K 2

• Global residue theorem
• Many works on avoiding having to solve

Baadsgaard, Bjerrum-Bohr, Bourjaily, Damgaard, Feng Dolan, Goddard 2014

Review, continued

13

Cachazo He Yuan 2015

Ambitwistor strings

• A chiral string in ambitwistor space

X X'

x x'

SA =

• Σ

P · ¯ ∂X − e 2P2

Mason, Skinner 2013

Ambitwistor strings

• A chiral string in ambitwistor space

X X'

x x'

SA =

• Σ

P · ¯ ∂X − e 2P2

Mason, Skinner 2013 Ambitwistor space = space of compexified null geodesics

x x'

space-time

X X'

Ambitwistor strings

• A chiral string in ambitwistor space

X X'

x x'

• Spectrum: type II supergravity

SA =

• Σ

P · ¯ ∂X − e 2P2

α = 0 of string theory.

Mason, Skinner 2013

Ambitwistor strings

• A chiral string in ambitwistor space

X X'

x x'

• Spectrum: type II supergravity

SA =

• Σ

P · ¯ ∂X − e 2P2

α = 0 of string theory.

NO STRINGY MODES

Mason, Skinner 2013

Ambitwistor strings

18

+ + ...

• A chiral string in ambitwistor space
• Can be extended to loops

X X'

x x'

• Spectrum: type II supergravity

Adamo, Casali, Skinner ’14, Ohmori ‘15

SA =

• Σ

P · ¯ ∂X − e 2P2

α = 0 of string theory.

NO STRINGY MODES

Mason, Skinner 2013

• Deep link to colour kinematics
• Extended to curved space
• Computation of form factors
• ℐ models; explain the BMS origin of soft theorems
• Ambitwistor models for a zoo of theories

Review, continued

19

Adamo, Casali, Skinner ’14; Geyer Lipstein Mason ‘14 Casali, Geyer, Mason, Monteiro, Roehrig Adamo, Casali, Skinner ‘14

He, Liu ’16; Brandhuber, Hughes, Panerai, Spence, Travaglini ’16; Bork Onishchenko ‘17

Ambitwistor strings : loops !

• New zero modes for P (= loop momenta)
• New scattering equations fixing the moduli of the

surface in terms of the loop momenta

• Total number of eqns : 3g-3+n

20

One-loop scattering equations

21

· ki +

• j=i

ki · kj

• 1

1 (zij, ) = 0 , i = 2, . . . , n − 1 2 + 2

n

• i=1

· ki

θ

1

θ1 (z0i, ) + n

• i=j

ki · kj

θ

1

θ1 (z0i, ) θ

1

θ1 (z0j, ) = 0

Adamo, Casali, Skinner 2014

Integrands are easy to write, string theory-like computation q = exp(2iπτ)

One-loop scattering equations

• Difficulty: hard to solve.
• number of solutions ?
• modular invariance ? after all this is just field

theory.

• Only solution in special regime

22

Adamo, Casali, Skinner 2014

Casali, Tourkine 2015

Outline

• Motivations.
• Review: CHY formulae & Twistor strings
• Loops reloaded
• Null strings

23

Geyer Mason Monteiro Tourkine PRL 2015, JHEP 2016, PRD 2016

From loops to trees

24

q = exp(2iπτ)

From loops to trees

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1-loop Ambitwistor string amplitudes have

• ne additional scattering equation

dτδ(P2)

From loops to trees

26

q = exp(2iπτ)

Integration by parts: sets q=0 dq q ¯

• 1

P2

• = − 1

P2

• q=0

= ... = − 1 2 ¯ δ(f (z)) := ¯ ∂ 1 f (z) q = exp(2iπτ)

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dτδ(P2) →

From loops to trees

From loops to trees

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q = exp(2iπτ)

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tree-level amplitude with two more points 1-loop amplitude = = + − 1 2 ×

• dD 1

2 × loop momentum Integration by parts

From loops to trees

• (n-1)!-2(n-2)! solutions
• n-point one-loop amplitudes in (super)Yang-Mills,

(super)gravity

• Maps to ‘Q-cut’ representation

30

Baadsgaard, Bjerrum-Bohr, Bourjaily, Caron-Huot, Damgaard, Feng

From loops to trees

Two loops

31

=

Geyer Monteiro Mason Tourkine, PRD 2016

1 2

1

1 2

2

1 2 Prescription for max SUSY, 
 gravity and gauge theories,
 at four points “ “

32

New formulation of the perturbative expansion based on this procedure to all loops

+ + + +…

• 1. whole new formalism
• 2. compact formulae
• 3. long term perspectives; resummation ?

the dream

the dream vs reality

• Beyond 2 loops there are new unexpected issues
• These models are really too unusual after all
• So we looked for a string theory origin

33

Siegel ‘15

• Motivations.
• Review: CHY formulae & Twistor strings
• Loops
• Null strings

Outline

34

Casali, Tourkine, JHEP 2016 Casali, Herfray, Tourkine work in progress Bandos, 1404.1299 Bagchi, Chakrabortty, Parekh 1507.04361

A clue: the scattering equations

35

• Also appear in the high energy limit of string

theory !

Paradox

36

T = (2πα)1

M2 M2 ~ T J J

Paradox

37

tension to zero

T = (2πα)1

M2 ~ T J M2 J Theory of higher spins

Paradox

38

tension to zero

T = (2πα)1

M2 ~ T J M2 J Theory of higher spins vs Ambitwistor strings =
 just normal QFTs

LST action

39

Lindstrom Sundborg Theodoris ’91

Nambu-Goto

LST action

40

Virasoro constraints

↳ Hamiltonian action

H = λ(P2 + T 2X 2) + µP · X

LST action

41

LST action

42

Integrate P out:

Casali, Tourkine, 2016

Gauge λ=0 and special quantization reduces to ambitwistor string

SA =

• Σ

P · ¯ ∂X − e 2P2

SLST =

• V α∂αX · V β∂βX

LST action

43

equations of motion :

Bagchi, Chakrabortty, Parekh 1507.04361 Lindstrom Sundborg Theodoris ’91

SLST =

• V α∂αX · V β∂βX

LST action

44

equations of motion : Open question : for Polyakov’s strings, gives integration

• ver the moduli space. What does give here ?

Dhαβ DV α

SLST =

• V α∂αX · V β∂βX

Classical and quantum

tension to zero

???

Null = tensionless

1+1+1+1+1+… = -1/2

In string theory, this Casimir effect 
 gives the tachyon.

1+1+1+1+1+… = -1/2

In string theory, this Casimir effect 
 gives the tachyon.

Classical and quantum

48

Quantum effect in the angular momentum suppresses wild fluctuations

+ +

L0 = +2

^

Classical and quantum

• Constraints : P2 and P.X’
• Constraints form a GCA2 (=BMS3) algebra

49

     [Ln, Lm] = (n − m)Ln+m + d

6 m(m2 − 1)δm+n ,

d = ηµ

µ

[Ln, Mm] = (n − m)Mn+m , [Mn, Mm] = 0 ,

Bagchi, Gopakumar, Mandal, Miwa 0912.1090

Classical and quantum

50

     Lm|phys = 0, m > 0 (L0 2)|phys = 0, Mm|phys = 0, m 0 L0 =

• nxnp−n → L0 =
• n : xnp−n :

L0 → L0 − 2

A mysterious quantization ambiguity

“Weyl ordering” Normal ordering

• J. Gamboa, C. Ramirez, M. Ruiz-Altaba, 1989

A mysterious quantization ambiguity

“Weyl ordering” Normal ordering Critical dimension : 10 (26)

• Bosonic: does not work (zero norm

states, ghosts, etc)

• Heterotic: same but 1/2
• Type II supergravity — the only that

100% works

No critical dimension Sort of higher spin theory

Ambitwistor quantization

• J. Gamboa, C. Ramirez, M. Ruiz-Altaba, 1989

A mysterious quantization ambiguity

“Weyl ordering” Normal ordering Critical dimension : 10 (26)

• Bosonic: does not work (zero norm

states, ghosts, etc)

• Heterotic: same but 1/2
• Type II supergravity — the only that

100% works

No critical dimension Sort of higher spin theory

Ambitwistor quantization ???

• J. Gamboa, C. Ramirez, M. Ruiz-Altaba, 1989

Same thing again; a curious fact about strings

A curious fact

55

A curious fact

56

conformal gauge: λ=1/2T, μ=0 S = 1 T

• (∂tX)2 − (∂σX)2dtdσ

X = XL(z) + XR(¯ z) ∼ x0 + tpO +

• αnz−n + ˜

αn(¯ z)−n αn|0 = 0, ˜ αn|0 = 0, n > 0

A curious fact

57

αn|0 = 0, ˜ αn|0 = 0, n > 0 Spectrum: m2 critical bosonic string

A curious fact

58

t → ˜ z, σ → z Complexification: SH =

• P ˜

∂X − µP∂X − λ(P2 − T 2(∂X)2)

A curious fact

59

Ambitwistor gauge: λ=0, μ=0 t → ˜ z, σ → z Complexification: SH =

• P ˜

∂X − µP∂X − λ(P2 − T 2(∂X)2) pn|0 = 0, xn|0 = 0, n > 0

Chiral CFT !

X(z) =

• xnzn,

P(z) =

• pnzn

A curious fact

60

pn|0 = 0, xn|0 = 0, n > 0 Spectrum: m2 Gravity + 2 states, m2 = ± T

Huang, Siegel, Yuan, 1603.02588 Hwang, Marnelius, Saltsidis hep-th/9804003

A curious fact

αn|0 = 0, ˜ αn|0 = 0, n > 0 pn|0 = 0, xn|0 = 0, n > 0 vs ????

A curious fact

αn = 1 2 √ T pn − in √ Txn ˜ αn = 1 2 √ T p−n − in √ Tx−n αn|0 = 0, ˜ αn|0 = 0, n > 0 pn|0 = 0, xn|0 = 0, n > 0 vs

A curious fact

αn = 1 2 √ T pn − in √ Txn ˜ αn = 1 2 √ T p−n − in √ Tx−n αn|0 = 0, ˜ αn|0 = 0, n > 0 vs αn|0A = 0, ˜ α−n|0A = 0, n > 0 pn|0A = 0, xn|0A = 0, n > 0

A curious fact

64

αn|0 = 0, ˜ αn|0 = 0, n > 0 αn|0A = 0, ˜ α−n|0A = 0, n > 0 L0 − ¯ L0 → L0 − ¯ L0 − 2 L0 + ¯ L0 − 2 → L0 + ¯ L0 Enough to truncate the spectrum

A curious fact

65

αn|0 = 0, ˜ αn|0 = 0, n > 0 αn|0A = 0, ˜ α−n|0A = 0, n > 0 L0 − ¯ L0 → L0 − ¯ L0 − 2 L0 + ¯ L0 − 2 → L0 + ¯ L0 Enough to truncate the spectrum Fact observed in the 80-90’s, reobserved later by Siegel, and Siegel + collaborators, see refs in [Casali, Tourkine 2016]

Results

• Extended CHY to one and partially two loops
• The ambitwistor string is the Null string in a special

singular gauge

• Explains why the heterotic or bosonic versions

failed

• Introduced a tensile deformation of the ambitwistor
• string. Connected to

66

R + α”R2” + α2”R3”

Hohm Siegel Zwiebach ’13

Strings Null Strings

classical quantum

Usual strings DFT ? Ambitwistor string “HS-like”

α → ∞ α → ∞

67

Casali, Tourkine, JHEP 2016

Hohm Siegel Zwiebach

Results

Perspectives

• Understand ambitwistor loops from first principles
• Solve / avoid having to solve loop sc. eqns ?
• Explain origin of twistor models. Could the other quantization

be applied in other contexts ?

• Why is there this other quantisation at all actually ?
• Relation with worldline ?