Bootstrap for Large N Confining Gauge Theories Alexander Zhiboedov, - - PowerPoint PPT Presentation

Bootstrap for Large N Confining Gauge Theories

Alexander Zhiboedov, Harvard U IGST 2017, Paris, France

with S. Caron-Huot, Z. Komargodski, A. Sever

Introduction

In this talk I discuss theories of weakly interacting higher spin particles in flat space (d>2).

Introduction

In this talk I discuss theories of weakly interacting higher spin particles in flat space (d>2). These are characterized by: spectrum of particles mi three-point couplings fijk i j k

Introduction

In this talk I discuss theories of weakly interacting higher spin particles in flat space (d>2). This data is combined into 2-2 scattering amplitude

A(s, t)

These are characterized by: spectrum of particles mi three-point couplings fijk i j k

What is WIHS?

What is WIHS?

A(s, t) =

Two-to-two scattering amplitude

What is WIHS?

A(s, t) =

Two-to-two scattering amplitude

• weakly coupled ≡

meromorphic

fssφ

fφss

θ

m2

φ, J

What is WIHS?

A(s, t) =

Two-to-two scattering amplitude

• weakly coupled ≡

meromorphic

fssφ

fφss

θ

m2

φ, J

• unitarity

A(s, t)|s'm2

φ ' f 2

ssφ

PJ(1 +

2t m2

φ4m2 s )

s m2

φ positive

cos θ

What is WIHS?

A(s, t) =

Two-to-two scattering amplitude

• weakly coupled ≡

meromorphic

fssφ

fφss

θ

m2

φ, J

• unitarity

A(s, t)|s'm2

φ ' f 2

ssφ

PJ(1 +

2t m2

φ4m2 s )

s m2

φ positive

cos θ

A(s, t) = A(t, s)

• crossing

Examples

• QFT

A(s, t) = 1 s − m2 + 1 t − m2

Examples

• fundamental strings

[Veneziano]

A(s, t) = Γ(−s)Γ(−t) Γ(−s − t)

• QFT

A(s, t) = 1 s − m2 + 1 t − m2

Examples

• fundamental strings

[Veneziano]

A(s, t) = Γ(−s)Γ(−t) Γ(−s − t)

• QFT

A(s, t) = 1 s − m2 + 1 t − m2

• confining gauge theories/large N QCD

[Kronfeld]

Examples

• fundamental strings

[Veneziano]

A(s, t) = Γ(−s)Γ(−t) Γ(−s − t)

• QFT

A(s, t) = 1 s − m2 + 1 t − m2

• confining gauge theories/large N QCD

[Kronfeld]

• something else???

• soft high energy limit

lim

s→∞ A(s, t0) < sJ0

J0 t s

What is WIHS?

• soft high energy limit

lim

s→∞ A(s, t0) < sJ0

J0 t s

What is WIHS?

clash with unitarity!

A(s, t) ∼ sJ0 ⇒

• soft high energy limit

lim

s→∞ A(s, t0) < sJ0

J0 t s

• no accumulation point in the spectrum

#{of particles mi < E} < ∞

What is WIHS?

clash with unitarity!

A(s, t) ∼ sJ0 ⇒

• soft high energy limit

lim

s→∞ A(s, t0) < sJ0

J0 t s

• no accumulation point in the spectrum

#{of particles mi < E} < ∞

What is WIHS?

clash with unitarity!

A(s, t) ∼ sJ0 ⇒

• interacting higher spin ≡ exchange of a particle with spin > 2

(softness is causality)

J=2 - Shapiro time delay

Examples

• QFT

A(s, t) = 1 s − m2 + 1 t − m2

high energy is not soft

lim

s→∞ A(s, t0) < sJ0

Examples

• QFT

A(s, t) = 1 s − m2 + 1 t − m2

high energy is not soft

lim

s→∞ A(s, t0) < sJ0

• fundamental strings

Very non-generic:

• linear Regge trajectories
• spectrum degeneracy

[Veneziano]

A(s, t) = Γ(−s)Γ(−t) Γ(−s − t)

lim

s→∞ A(s, t) ∼ sα0t

Examples

• QFT

A(s, t) = 1 s − m2 + 1 t − m2

high energy is not soft

lim

s→∞ A(s, t0) < sJ0

• confining gauge theories
• Regge trajectories are not linear
• spectrum is not degenerate
• fundamental strings

Very non-generic:

• linear Regge trajectories
• spectrum degeneracy

[Veneziano]

A(s, t) = Γ(−s)Γ(−t) Γ(−s − t)

lim

s→∞ A(s, t) ∼ sα0t

Goal

Goal

In principle, this is a well-defined math problem and such functions are subject to a classification.

Goal

In principle, this is a well-defined math problem and such functions are subject to a classification. In practice, almost nothing is known.

Goal

In principle, this is a well-defined math problem and such functions are subject to a classification. In practice, almost nothing is known. Can we bootstrap it?

Goal

In principle, this is a well-defined math problem and such functions are subject to a classification. In practice, almost nothing is known. Can we bootstrap it?

• 1. Universal properties

Goal

In principle, this is a well-defined math problem and such functions are subject to a classification. In practice, almost nothing is known. Can we bootstrap it?

• 1. Universal properties

[El-Showk, Paulos, Poland, Rychkov, Simmons-Duffin, Vichi]

• 2. Special points

Hint

What fundamental strings and YM strings have in common?

Hint

What fundamental strings and YM strings have in common? What do we mean by strings?

t j(t)

ST

(mass) (spin)

lim

s→∞ A(s, t) ∼ sj(t)

spectrum scattering

The Regge Trajectory j(t)

Γ(−s)Γ(−t) Γ(−t − s)

t j(t)

ST Y M

(mass) (spin)

lim

s→∞ A(s, t) ∼ sj(t)

spectrum scattering

The Regge Trajectory j(t)

Γ(−s)Γ(−t) Γ(−t − s)

t j(t)

ST Y M

(mass) (spin)

lim

s→∞ A(s, t) ∼ sj(t)

spectrum

non-universal

?

scattering

The Regge Trajectory j(t)

Γ(−s)Γ(−t) Γ(−t − s)

t j(t)

ST Y M

(mass) (spin)

lim

s→∞ A(s, t) ∼ sj(t)

universal

spectrum

non-universal

?

scattering

The Regge Trajectory j(t)

Γ(−s)Γ(−t) Γ(−t − s)

s

t u

s c a t t e r i n g scattering s c a t t e r i n g spectrum s p e c t r u m s p e c t r u m

Mandelstam Plane

Universal/Imaginary Angles Non-universal/Real Angles Key idea: scattering at imaginary angles is universal

s, t > 0

s > 0, − s < t < 0

s + t + u = 0

s = E2

cm

t = −E2

cm

2 (1 − cos θ)

Part I

Asymptotic Uniqueness of the Veneziano amplitude

Result Part I

The high energy limit of WIHS amplitudes at imaginary scattering angles takes the form lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞ s/t fixed

Result Part I

The high energy limit of WIHS amplitudes at imaginary scattering angles takes the form lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞ s/t fixed

∼ E2

Result Part I

The high energy limit of WIHS amplitudes at imaginary scattering angles takes the form

limit of the Veneziano amplitude

lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞ s/t fixed

∼ E2

Result

lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞ s/t fixed

Result

lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞ s/t fixed

• amplitude is exponentially large (unitarity universality)

⇒

Result

lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞ s/t fixed

• stringy. Infinitely many asymptotically linear Regge trajectories
• bject of transverse size

= a string

∼ log(s)

b

j(t) = α0t + corrections +parallel trajectories

Im A(s, b) ∼ e−

b2 α0 log s

⇒

• amplitude is exponentially large (unitarity universality)

⇒

Result

lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞ s/t fixed

• stringy. Infinitely many asymptotically linear Regge trajectories
• bject of transverse size

= a string

∼ log(s)

b

j(t) = α0t + corrections +parallel trajectories

Im A(s, b) ∼ e−

b2 α0 log s

⇒

• amplitude is exponentially large (unitarity universality)

⇒

• insensitive to the microscopic spectrum degeneracy

Bootstrap (Leading Order)

For s,t large and positive a thermodynamic picture emerges

s, t → ∞ s/t fixed

lim log A ((1 + i✏)s, (1 + i✏)t)

[Caron-Huot, Komargodski, Sever, AZ] 1

−1

J even J odd

PJ(x)

we are here

Partial wave

Complex s plane at fixed real t

Bootstrap (Leading Order)

For s,t large and positive a thermodynamic picture emerges

s, t → ∞ s/t fixed

lim log A ((1 + i✏)s, (1 + i✏)t)

[Caron-Huot, Komargodski, Sever, AZ] 1

−1

J even J odd

PJ(x)

we are here

Partial wave

⇒ • All residues are positive

Complex s plane at fixed real t

Bootstrap (Leading Order)

For s,t large and positive a thermodynamic picture emerges

s, t → ∞ s/t fixed

lim log A ((1 + i✏)s, (1 + i✏)t)

[Caron-Huot, Komargodski, Sever, AZ] 1

−1

J even J odd

PJ(x)

we are here

Partial wave

⇒ • At least one zero between every two poles

• There could be more zeros

⇒ • All residues are positive

Complex s plane at fixed real t

Bootstrap (Leading Order)

For s,t large and positive a thermodynamic picture emerges

s, t → ∞ s/t fixed

lim log A ((1 + i✏)s, (1 + i✏)t)

[Caron-Huot, Komargodski, Sever, AZ] 1

−1

J even J odd

PJ(x)

we are here

Partial wave

⇒ • At least one zero between every two poles

• There could be more zeros

⇒ • All residues are positive

log A(s, t) ' j(t) log s ⇒ j(t) = Diss log A = X (zeros − poles) Complex s plane at fixed real t

poles zeros

Bootstrap (Leading Order)

For s,t large and positive a thermodynamic picture emerges

−t

s

We are here

s, t → ∞ s/t fixed

lim log A ((1 + i✏)s, (1 + i✏)t)

distribution of excess zeros ρ

Bootstrap (Leading Order)

log A = Z d2z ρ(t; z, ¯ z) log(z − s) The amplitude takes the form

Bootstrap (Leading Order)

log A = Z d2z ρ(t; z, ¯ z) log(z − s) The amplitude takes the form

dimensional analysis at large s,t

ρ(t; z, ¯ z) = j(t) t2 ρ(z/t, ¯ z/t)

In the large t limit we can write for the distribution

Bootstrap (Leading Order)

log A = Z d2z ρ(t; z, ¯ z) log(z − s)

⇒

log A(s, t) = j(t) Z d2z ρ(z, ¯ z) log(1 − s tz )

normalization unitarity

ρ(z, ¯ z) ≥ 0 Z d2z ρ(z, ¯ z) = 1

The amplitude takes the form

dimensional analysis at large s,t

ρ(t; z, ¯ z) = j(t) t2 ρ(z/t, ¯ z/t)

In the large t limit we can write for the distribution

Bootstrap (Leading Order)

Unitarity

Unitarity

The distribution of the zeros comes from a sum of Legendre polynomials with positive coefficients

j(t)

X

n=0

C2

nPn (1 + 2β)

Unitarity

• 1
• 0.015

0.015

together with the Regge limit (3pt couplings cannot be too small) implies finite support of the excess zeros

β = s t

• β + 1

2

• ≤ 1

The distribution of the zeros comes from a sum of Legendre polynomials with positive coefficients

j(t)

X

n=0

C2

nPn (1 + 2β)

Bootstrap (Leading Order)

⇒

log A(s, t) = j(t) Z d2z ρ(z, ¯ z) log(1 − s tz )

normalization unitarity

ρ(z, ¯ z) ≥ 0 Z d2z ρ(z, ¯ z) = 1

Assume

j(t) = tk

Bootstrap (Leading Order)

2d “electric field”

ρ(z, ¯ z)

2d “electric potential” for a positive distribution of charge

V (β) = log A(s, t) j(t) = Z d2z ρ(z, ¯ z) log(1 − β z ) =

E(β) = t ∂sV (s/t) = Z d2z ρ(z, ¯ z) β − z =

s/t

Bootstrap (Leading Order)

2d “electric field”

ρ(z, ¯ z)

2d “electric potential” for a positive distribution of charge E(β) analytic where ρ = 0 crossing

+

⇒

E(β) = 1 β 2F1(k, k, k + 1; −1/β) = 1 β − k2 k + 1 1 β2 + . . .

M1

V (β) = log A(s, t) j(t) = Z d2z ρ(z, ¯ z) log(1 − β z ) =

E(β) = t ∂sV (s/t) = Z d2z ρ(z, ¯ z) β − z =

s/t

Bootstrap (Leading Order)

2d “electric field”

ρ(z, ¯ z)

2d “electric potential” for a positive distribution of charge E(β) analytic where ρ = 0 crossing

+

⇒

E(β) = 1 β 2F1(k, k, k + 1; −1/β) = 1 β − k2 k + 1 1 β2 + . . .

M1

V (β) = log A(s, t) j(t) = Z d2z ρ(z, ¯ z) log(1 − β z ) =

E(β) = t ∂sV (s/t) = Z d2z ρ(z, ¯ z) β − z =

s/t

k ≤ 1

lim

β→∞ E(β) = 1

β + . . . ⇒ lim

β→0 E(β) = −k βk−1 log β + . . .

k ≤ 1 ⇒

crossing

β

Bootstrap (Leading Order)

mathematical identity

∂2

θ log

X

n

C2

n Pn(cos θ) ≥ 0

M1 ≥ 1 2

⇒

k ≥ 1

⇒

k = 1

⇒

Bootstrap (Leading Order)

mathematical identity

∂2

θ log

X

n

C2

n Pn(cos θ) ≥ 0

M1 ≥ 1 2

⇒

k ≥ 1

⇒

k = 1

⇒

⇒

for ρ(x, x) = 1

The unique solution is

−t

s

log A(s, t) = α0t

1

Z dxρ(x) log ⇣ 1 + s tx ⌘ = α0 [(s + t) log(s + t) − s log s − t log t]

−1 < x < 0

= classical string theory

Part II

Universal Correction to the Veneziano Amplitude

Result Part II

The high energy limit of WIHS amplitudes at imaginary scattering angles takes the form

lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞

s/t fixed

∼ E2

Result Part II

The high energy limit of WIHS amplitudes at imaginary scattering angles takes the form

lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞

s/t fixed

−16√π 3 α0m3/2 ✓ s t s + t ◆ 1

4 

K ✓ s s + t ◆ + K ✓ t s + t ◆ + . . .

∼ E2

Result Part II

The high energy limit of WIHS amplitudes at imaginary scattering angles takes the form

lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞

s/t fixed

−16√π 3 α0m3/2 ✓ s t s + t ◆ 1

4 

K ✓ s s + t ◆ + K ✓ t s + t ◆ + . . .

∼ E

1 2

∼ E2

Result Part II

The high energy limit of WIHS amplitudes at imaginary scattering angles takes the form

elliptic integral of the first kind EllipticK[x]

lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞

s/t fixed

−16√π 3 α0m3/2 ✓ s t s + t ◆ 1

4 

K ✓ s s + t ◆ + K ✓ t s + t ◆ + . . .

∼ E

1 2

∼ E2

Result Part II

The high energy limit of WIHS amplitudes at imaginary scattering angles takes the form

elliptic integral of the first kind EllipticK[x] correction due to the slowdown of the string (massive endpoints)/spectrum non-degeneracy

lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞

s/t fixed

−16√π 3 α0m3/2 ✓ s t s + t ◆ 1

4 

K ✓ s s + t ◆ + K ✓ t s + t ◆ + . . .

∼ E

1 2

∼ E2

Result Part II

The high energy limit of WIHS amplitudes at imaginary scattering angles takes the form

elliptic integral of the first kind EllipticK[x] correction due to the slowdown of the string (massive endpoints)/spectrum non-degeneracy corrections are O(1)

lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞

s/t fixed

−16√π 3 α0m3/2 ✓ s t s + t ◆ 1

4 

K ✓ s s + t ◆ + K ✓ t s + t ◆ + . . .

∼ E

1 2

∼ E2

Result

δ log A(s, t) = −16√π 3 α0m3/2 ✓ s t s + t ◆ 1

4 

K ✓ s s + t ◆ + K ✓ t s + t ◆ + . . .

Result

δ log A(s, t) = −16√π 3 α0m3/2 ✓ s t s + t ◆ 1

4 

K ✓ s s + t ◆ + K ✓ t s + t ◆ + . . .

• worldsheet: slowdown of the string endpoints

m m

[Chodos, Thorn, 74’]

j(t) = α0 ✓ t − 8√π 3 m3/2t1/4 + ... ◆

[Sonnenschein et al.] [Wilczek] [Baker, Steinke]

Result

δ log A(s, t) = −16√π 3 α0m3/2 ✓ s t s + t ◆ 1

4 

K ✓ s s + t ◆ + K ✓ t s + t ◆ + . . .

• bootstrap: removal of the spectrum degeneracy

jsub−leading(t) 6= jleading(t) + integer

• worldsheet: slowdown of the string endpoints

m m

[Chodos, Thorn, 74’]

j(t) = α0 ✓ t − 8√π 3 m3/2t1/4 + ... ◆

[Sonnenschein et al.] [Wilczek] [Baker, Steinke]

• Scattering of Strings With Massive Endpoints
• Universality (Holography & EFT of Long Strings)
• Bootstrap

Worldsheet Computation (review)

s/t fixed |s|, |t| → ∞

lim A(s, t) = e−SE(s,t)

[Gross, Mende] [Gross, Mañes] [Alday, Maldacena]

Worldsheet Computation (review)

s/t fixed |s|, |t| → ∞

lim A(s, t) = e−SE(s,t)

• real scattering angles (amplitude is small)

SE 1

[Gross, Mende] [Gross, Mañes] [Alday, Maldacena]

Worldsheet Computation (review)

s/t fixed |s|, |t| → ∞

lim A(s, t) = e−SE(s,t)

• real scattering angles (amplitude is small)

SE 1

• imaginary scattering angles (amplitude is large) SE 1

[Gross, Mende] [Gross, Mañes] [Alday, Maldacena]

Worldsheet Computation (review)

Worldsheet Computation (review)

SE = 1 2πα0 Z d2z ∂x · ¯ ∂x − i X

j

kj · x(σj)

Flat space

Worldsheet Computation (review)

SE = 1 2πα0 Z d2z ∂x · ¯ ∂x − i X

j

kj · x(σj)

Flat space

• general solution

xµ

0 = i

X

i

kµ

i log |z − σi|2

Worldsheet Computation (review)

• Virasoro (scattering equations)

X

j

ki · kj σi − σj = 0

SE = 1 2πα0 Z d2z ∂x · ¯ ∂x − i X

j

kj · x(σj)

Flat space

• general solution

xµ

0 = i

X

i

kµ

i log |z − σi|2

Worldsheet Computation (review)

• Virasoro (scattering equations)

X

j

ki · kj σi − σj = 0

SE = 1 2πα0 Z d2z ∂x · ¯ ∂x − i X

j

kj · x(σj)

Flat space log A(s, t) = α0 [(s + t) log(s + t) − s log s − t log t] ⇒

s, t > 0

• general solution

xµ

0 = i

X

i

kµ

i log |z − σi|2

Adding The Mass

Adding The Mass

SE = 1 2πα0 Z d2z ∂x · ¯ ∂x + m Z dσ p |∂σx|2 − i X

j

kj · x(σj)

[Chodos, Thorn]

Adding The Mass

Modified boundary condition: 1 2πα0 ∂τx + m ∂σ ∂σx √∂σx · ∂σx = i X

j

kj δ(σ − σj)

SE = 1 2πα0 Z d2z ∂x · ¯ ∂x + m Z dσ p |∂σx|2 − i X

j

kj · x(σj)

[Chodos, Thorn]

Adding The Mass

Modified boundary condition: 1 2πα0 ∂τx + m ∂σ ∂σx √∂σx · ∂σx = i X

j

kj δ(σ − σj) is zero for a free string!

∂σx0 · ∂σx0 = 0

SE = 1 2πα0 Z d2z ∂x · ¯ ∂x + m Z dσ p |∂σx|2 − i X

j

kj · x(σj)

[Chodos, Thorn]

Adding The Mass

Modified boundary condition: 1 2πα0 ∂τx + m ∂σ ∂σx √∂σx · ∂σx = i X

j

kj δ(σ − σj) is zero for a free string!

∂σx0 · ∂σx0 = 0

SE = 1 2πα0 Z d2z ∂x · ¯ ∂x + m Z dσ p |∂σx|2 − i X

j

kj · x(σj)

[Chodos, Thorn]

The expansion reorganizes itself in terms of :

√m

xµ = xµ

0 + √m xµ 1 + ...

S = S0 + √mS1 + mS2 + m3/2S3

Adding The Mass

Modified boundary condition: 1 2πα0 ∂τx + m ∂σ ∂σx √∂σx · ∂σx = i X

j

kj δ(σ − σj) is zero for a free string!

∂σx0 · ∂σx0 = 0

SE = 1 2πα0 Z d2z ∂x · ¯ ∂x + m Z dσ p |∂σx|2 − i X

j

kj · x(σj)

[Chodos, Thorn]

The expansion reorganizes itself in terms of :

√m

xµ = xµ

0 + √m xµ 1 + ...

S = S0 + √mS1 + mS2 + m3/2S3

The on-shell action evaluates to

Adding The Mass

Lb = √ 2πα0 m Z dσ (∂2

σx0 · ∂2 σx0)1/4

SE = SGM + 2 3mLb + ...

Gross-Mende solution

The on-shell action evaluates to

Adding The Mass

Lb = √ 2πα0 m Z dσ (∂2

σx0 · ∂2 σx0)1/4

SE = SGM + 2 3mLb + ...

Gross-Mende solution

The on-shell action evaluates to

reparameterization invariant

Adding The Mass

Lb = √ 2πα0 m Z dσ (∂2

σx0 · ∂2 σx0)1/4

SE = SGM + 2 3mLb + ...

Gross-Mende solution

The on-shell action evaluates to

reparameterization invariant

Adding The Mass

For four external particles

δ log A(s, t) = −16√π 3 α0m3/2 ✓ s t s + t ◆ 1

4 

K ✓ s s + t ◆ + K ✓ t s + t ◆ + O(m5/2)

Lb = √ 2πα0 m Z dσ (∂2

σx0 · ∂2 σx0)1/4

SE = SGM + 2 3mLb + ...

Gross-Mende solution

non-universal O(t−1/4)

The on-shell action evaluates to

reparameterization invariant

Adding The Mass

For four external particles

δ log A(s, t) = −16√π 3 α0m3/2 ✓ s t s + t ◆ 1

4 

K ✓ s s + t ◆ + K ✓ t s + t ◆ + O(m5/2)

Lb = √ 2πα0 m Z dσ (∂2

σx0 · ∂2 σx0)1/4

SE = SGM + 2 3mLb + ...

Emergent s-u Crossing Symmetry

lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞

s/t fixed

−16√π 3 α0m3/2 ✓ s t s + t ◆ 1

4 

K ✓ s s + t ◆ + K ✓ t s + t ◆ + . . .

Emergent s-u Crossing Symmetry

The s-t crossing is manifest: log A(s, t) = log A(t, s)

lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞

s/t fixed

−16√π 3 α0m3/2 ✓ s t s + t ◆ 1

4 

K ✓ s s + t ◆ + K ✓ t s + t ◆ + . . .

Emergent s-u Crossing Symmetry

The s-t crossing is manifest: log A(s, t) = log A(t, s) What about the s-u crossing?

lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞

s/t fixed

−16√π 3 α0m3/2 ✓ s t s + t ◆ 1

4 

K ✓ s s + t ◆ + K ✓ t s + t ◆ + . . .

Emergent s-u Crossing Symmetry

The s-t crossing is manifest: log A(s, t) = log A(t, s) What about the s-u crossing? log A(s, t) = Re[log A(u, t)] u = −s − t

lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞

s/t fixed

−16√π 3 α0m3/2 ✓ s t s + t ◆ 1

4 

K ✓ s s + t ◆ + K ✓ t s + t ◆ + . . .

Emergent s-u Crossing Symmetry

The s-t crossing is manifest: log A(s, t) = log A(t, s) What about the s-u crossing? log A(s, t) = Re[log A(u, t)] u = −s − t 1 2 3 4 1 2 3 4 ???

lim log A(s, t) = α0 [(s + t) log(s + t) − s log(s) − t log(t)]

s, t → ∞

s/t fixed

−16√π 3 α0m3/2 ✓ s t s + t ◆ 1

4 

K ✓ s s + t ◆ + K ✓ t s + t ◆ + . . .

Emergent s-u Crossing Symmetry

[Komatsu]

1 2

3

4 1

2

3 4 ???

Emergent s-u Crossing Symmetry

[Komatsu]

1 2

3

4 1

2

3 4 ???

Asymptotic s-u Crossing

Equivalently, the asymptotic s-u crossing is: dDiscs log A(s, t) ≡ log A(−s − t + i✏, t) + log A(−s − t − i✏, t) − 2 log A(s, t) = 0 Double discontinuity is zero!

Why is the correction universal?

Why is the correction universal?

Why is the massive ends model physical?

Holographic Argument

Holographic dual of a confining gauge theory: ds2 = dr2 + f(r) dx2

1,d−1

• AdS in the UV

lim

r→∞ f(r) = e2r

• Cutoff in the IR

f(0) = 1

[Sonnenschein] [Erdmenger et al.]

Holographic radial direction

r x

Polchinski-Strassler Mechanism

For mesons we add a space-filling flavor brane

[Polchinski, Strassler]

IR r0 Holographic radial direction Flavor brane UV

Polchinski-Strassler Mechanism

For mesons we add a space-filling flavor brane

real scattering angles

[Polchinski, Strassler]

IR r0 Holographic radial direction Flavor brane UV

Polchinski-Strassler Mechanism

For mesons we add a space-filling flavor brane

real scattering angles imaginary scattering angles

[Polchinski, Strassler]

IR r0 Holographic radial direction Flavor brane UV

IR r0 Holographic radial direction Flavor brane UV

Polchinski-Strassler Mechanism

For mesons we add a space-filling flavor brane

[Polchinski, Strassler]

IR r0 Holographic radial direction Flavor brane UV

Polchinski-Strassler Mechanism

For mesons we add a space-filling flavor brane

[Polchinski, Strassler]

IR r0 Holographic radial direction Flavor brane UV

Polchinski-Strassler Mechanism

For mesons we add a space-filling flavor brane

[Polchinski, Strassler]

IR r0 Holographic radial direction Flavor brane UV

string in flat space

Polchinski-Strassler Mechanism

For mesons we add a space-filling flavor brane

[Polchinski, Strassler]

IR r0 Holographic radial direction Flavor brane UV

m

string in flat space

Polchinski-Strassler Mechanism

For mesons we add a space-filling flavor brane

[Polchinski, Strassler]

IR r0 Holographic radial direction Flavor brane UV

m

⇒

m m

effective description

m2α0 1

string in flat space

Polchinski-Strassler Mechanism

For mesons we add a space-filling flavor brane

[Polchinski, Strassler]

IR r0 Holographic radial direction Flavor brane UV

m

⇒

m m

effective description

m2α0 1

string in flat space

• At the holographic model reduces to the string with massive ends

s, t 1

Polchinski-Strassler Mechanism

For mesons we add a space-filling flavor brane

[Polchinski, Strassler]

IR r0 Holographic radial direction Flavor brane UV

m

⇒

m m

effective description

m2α0 1

string in flat space

• At the holographic model reduces to the string with massive ends

s, t 1

• Insensitive to the details of the background

Polchinski-Strassler Mechanism

For mesons we add a space-filling flavor brane

[Polchinski, Strassler]

EFT of Long Strings

[Aharony et al.] [Hellerman et al.] [Polchinski, Strominger] [Dubovsky et al.]

EFT of Long Strings

[Aharony et al.] [Hellerman et al.] [Polchinski, Strominger] [Dubovsky et al.]

• boundary corrections (open strings)

Unique in the effective theory of open strings

j(t) = α0 ✓ t − 8√π 3 m3/2t1/4 ◆

[Hellerman, Swanson]

EFT of Long Strings

• quantum corrections

Polchinski-Strominger term

(∂2x · ¯ ∂x)(∂x · ¯ ∂2x) (∂x · ¯ ∂x)2

[Aharony et al.] [Hellerman et al.] [Polchinski, Strominger] [Dubovsky et al.]

• boundary corrections (open strings)

Unique in the effective theory of open strings

j(t) = α0 ✓ t − 8√π 3 m3/2t1/4 ◆

[Hellerman, Swanson]

EFT of Long Strings

• quantum corrections

Polchinski-Strominger term

(∂2x · ¯ ∂x)(∂x · ¯ ∂2x) (∂x · ¯ ∂x)2

• higher derivative corrections (closed strings)

[Aharony et al.] [Hellerman et al.] [Polchinski, Strominger] [Dubovsky et al.]

• boundary corrections (open strings)

Unique in the effective theory of open strings

j(t) = α0 ✓ t − 8√π 3 m3/2t1/4 ◆

[Hellerman, Swanson]

EFT of Long Strings

• quantum corrections

Polchinski-Strominger term

(∂2x · ¯ ∂x)(∂x · ¯ ∂2x) (∂x · ¯ ∂x)2

• higher derivative corrections (closed strings)

[Aharony et al.] [Hellerman et al.] [Polchinski, Strominger] [Dubovsky et al.]

• boundary corrections (open strings)

Unique in the effective theory of open strings

j(t) = α0 ✓ t − 8√π 3 m3/2t1/4 ◆

[Hellerman, Swanson]

Bootstrap (Correction)

• Spectrum Non-degeneracy/Support of excess zeros

−t

s

non-degeneracy zeros

Bootstrap (Correction)

• Spectrum Non-degeneracy/Support of excess zeros

−t

s

non-degeneracy zeros density>0

Bootstrap (Correction)

• Spectrum Non-degeneracy/Support of excess zeros

−t

s

non-degeneracy zeros

Indeed, the massive ends correction is of this form!

density>0

Bootstrap (Assumptions)

• Unitarity and the s-t crossing are not enough

log A(s, t) = log A(t, s)

Bootstrap (Assumptions)

• Unitarity and the s-t crossing are not enough

log A(s, t) = log A(t, s)

Bootstrap (Assumptions)

• Impose the s-u crossing

dDiscs log A(s, t) = 0

s

t

u

• Unitarity and the s-t crossing are not enough

log A(s, t) = log A(t, s)

Bootstrap (Assumptions)

• Impose the s-u crossing

dDiscs log A(s, t) = 0

s

t

u

• boundary type

δ log A(s, t) = f(s, t) + f(t, s)

corrects the trajectory only in one channel

Bootstrap (Integral Equation)

The extra condition leads to an integral equation δρk(y) =

1

Z dx [K(y, x) + K(1 − y, 1 − x)] δρk(x)

+(1 − y)k−1 π sin πk ✓ y x + y − x y + k log x (1 − y) x + y − x y ◆

δj(t) = tk

correction to the trajectory correction to the distribution

K(y, x) = cot πk π ✓ y P 1 x − y − k log x |x − y| ◆

δρk(y) =

1

Z dx [K(y, x) + K(1 − y, 1 − x)] δρk(x)

Bootstrap (Integral Equation)

δj(t) = tk

• The correction we found obeys the equation

δρk(y) =

1

Z dx [K(y, x) + K(1 − y, 1 − x)] δρk(x)

Bootstrap (Integral Equation)

δj(t) = tk

• The correction we found obeys the equation

δρk(y) =

1

Z dx [K(y, x) + K(1 − y, 1 − x)] δρk(x)

Bootstrap (Integral Equation)

δj(t) = tk

• One can show that the solution is of the form

ρs(x) = xk−1

∞

X

n=0

xn(an log x + bn) .

• The correction we found obeys the equation
• Easy to show that only k=1/4, k=3/4 are possible

δρk(y) =

1

Z dx [K(y, x) + K(1 − y, 1 − x)] δρk(x)

Bootstrap (Integral Equation)

δj(t) = tk

• One can show that the solution is of the form

ρs(x) = xk−1

∞

X

n=0

xn(an log x + bn) .

• The correction we found obeys the equation
• Easy to show that only k=1/4, k=3/4 are possible

δρk(y) =

1

Z dx [K(y, x) + K(1 − y, 1 − x)] δρk(x)

Bootstrap (Integral Equation)

δj(t) = tk

• One can show that the solution is of the form

ρs(x) = xk−1

∞

X

n=0

xn(an log x + bn) .

• To prove the uniqueness we do numerics

χ2 =

nmax

X

n=0

(aLHS

n

− aRHS

n

)2 + (bLHS

n

− bRHS

n

)2

Minimize

Truncation

Conclusions and Open Questions

Conclusions and Open Questions

• Relax some assumptions

Conclusions and Open Questions

• Relax some assumptions
• Removal of the degeneracy and Hagedorn

Conclusions and Open Questions

• Relax some assumptions
• Non-universal regime

[Numerical bootstrap, graviton, DIS?]

• Removal of the degeneracy and Hagedorn

Conclusions and Open Questions

• Relax some assumptions
• Non-universal regime

[Numerical bootstrap, graviton, DIS?]

• Bootstrap in AdS (Mellin space)

[Theories with accumulation?]

• Removal of the degeneracy and Hagedorn

Conclusions and Open Questions

• Relax some assumptions
• Non-universal regime

[Numerical bootstrap, graviton, DIS?]

• Bootstrap in AdS (Mellin space)

[Theories with accumulation?]

• Removal of the degeneracy and Hagedorn
• Quantum theories

[Universal?]

Conclusions and Open Questions

• Relax some assumptions
• Non-universal regime

[Numerical bootstrap, graviton, DIS?]

• Bootstrap in AdS (Mellin space)

[Theories with accumulation?] thank you!

• Removal of the degeneracy and Hagedorn
• Quantum theories

[Universal?]