Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander...

Universal Correction To The Veneziano Amplitude

Alexander Zhiboedov, Harvard U

Strings 2017, Tel Aviv, Israel

with S. Caron-Huot, Z. Komargodski, A. Sever, 1607’(talk by Zohar at Strings2016)

with A. Sever, (to appear)

Homework from Strings2014

Problem 72 (Juan):

What is the general theory of weakly coupled, interacting, higher spin particles?

Homework from Strings2014

Problem 72 (Juan):

(related homework from Nima about weakly coupled completion of gravity amplitudes)

What is the general theory of weakly coupled, interacting, higher spin particles?

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 ⌘ meromorphicfss�

f�ss✓

m2�, J

What is WIHS?

A(s, t) =Two-to-two scattering amplitude

• weakly coupled ⌘ meromorphicfss�

f�ss✓

m2�, J

• unitarity A(s, t)|s'm2�' f2

ss�

PJ(1 +2t

m2��4m2

s)

s�m2�

positive

cos ✓

What is WIHS?

A(s, t) =Two-to-two scattering amplitude

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

• weakly coupled ⌘ meromorphicfss�

f�ss✓

m2�, J

• unitarity A(s, t)|s'm2�' f2

ss�

PJ(1 +2t

m2��4m2

s)

s�m2�

positive

cos ✓

What is WIHS?

A(s, t) =Two-to-two scattering amplitude

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

• weakly coupled ⌘ meromorphicfss�

f�ss✓

m2�, J

• unitarity A(s, t)|s'm2�' f2

ss�

PJ(1 +2t

m2��4m2

s)

s�m2�

positive

cos ✓

A(s, t) = A(t, s)• crossing

• soft high energy limit (causality for HS)

lims!1

A(s, t0) < sJ0

J0

t

s

What is WIHS?

[talk Caron-Huot]

• soft high energy limit (causality for HS)

lims!1

A(s, t0) < sJ0

J0

t

s

clash with unitarity! A(s, t) ⇠ sJ0

What is WIHS?

[talk Caron-Huot]

• soft high energy limit (causality for HS)

lims!1

A(s, t0) < sJ0

J0

t

s

• no accumulation point in the spectrum#{of particles mi < E} < 1

clash with unitarity! A(s, t) ⇠ sJ0

What is WIHS?

[talk Caron-Huot]

• soft high energy limit (causality for HS)

lims!1

A(s, t0) < sJ0

J0

t

s

• no accumulation point in the spectrum#{of particles mi < E} < 1

clash with unitarity! A(s, t) ⇠ sJ0

What is WIHS?

fundamental strings, large N confining gauge theories, …

[Veneziano][Andreev, Siegel]

[Veneziano, Yankielowicz, Onofri]

�(�s)�(�t)

�(�t� s)

Solutions:

[talk Caron-Huot]

• soft high energy limit (causality for HS)

lims!1

A(s, t0) < sJ0

J0

t

s

• no accumulation point in the spectrum#{of particles mi < E} < 1

clash with unitarity! A(s, t) ⇠ sJ0

What is WIHS?

fundamental strings, large N confining gauge theories, …

[Veneziano][Andreev, Siegel]

[Veneziano, Yankielowicz, Onofri]

�(�s)�(�t)

�(�t� s)

Solutions:very non-generic

[talk Caron-Huot]

t

j(t)

ST

(mass)

(spin)

lims!1

A(s, t) ⇠ sj(t)

spectrumscattering

The Regge Trajectory j(t)

�(�s)�(�t)

�(�t� s)

t

j(t)

STYM

(mass)

(spin)

lims!1

A(s, t) ⇠ sj(t)

spectrumscattering

The Regge Trajectory j(t)

�(�s)�(�t)

�(�t� s)

t

j(t)

STYM

(mass)

(spin)

lims!1

A(s, t) ⇠ sj(t)

spectrum

non-universal

?

scattering

The Regge Trajectory j(t)

�(�s)�(�t)

�(�t� s)

t

j(t)

STYM

(mass)

(spin)

lims!1

A(s, t) ⇠ sj(t)

universal

spectrum

non-universal

?

scattering

The Regge Trajectory j(t)

�(�s)�(�t)

�(�t� s)

s

t

u

scattering

scattering

scattering

spectrum

spectrumspectrum

Mandelstam Plane

Universal/Imaginary Angles (analytic methods)

Non-universal/Real Angles (numerical methods)

WIHS amplitudes are universal at imaginary scattering angles

s, t > 0

s > 0, � s < t < 0

Results

High Energy Asymptotic

High Energy Asymptotic

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

High Energy Asymptotic

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

lim logA(s, t) = ↵0[(s+ t) log(s+ t)� s log(s)� t log(t)]

s, t ! 1s/t fixed

High Energy Asymptotic

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

limit of the Veneziano amplitude (Zohar’s talk at Strings2016)

⇠ E2logElim logA(s, t) = ↵0

[(s+ t) log(s+ t)� s log(s)� t log(t)]s, t ! 1s/t fixed

High Energy Asymptotic

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

limit of the Veneziano amplitude (Zohar’s talk at Strings2016)

⇠ E2logElim logA(s, t) = ↵0

[(s+ t) log(s+ t)� s log(s)� t log(t)]s, t ! 1s/t fixed

�16p⇡

3↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+ . . .

⇠ E1/2logE

High Energy Asymptotic

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

elliptic integral of the first kind EllipticK[x]

limit of the Veneziano amplitude (Zohar’s talk at Strings2016)

⇠ E2logElim logA(s, t) = ↵0

[(s+ t) log(s+ t)� s log(s)� t log(t)]s, t ! 1s/t fixed

�16p⇡

3↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+ . . .

⇠ E1/2logE

High Energy Asymptotic

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

elliptic integral of the first kind EllipticK[x]

limit of the Veneziano amplitude (Zohar’s talk at Strings2016)

⇠ E2logE

correction due to the slowdown of the string (massive endpoints)/spectrum non-degeneracy

lim logA(s, t) = ↵0[(s+ t) log(s+ t)� s log(s)� t log(t)]

s, t ! 1s/t fixed

�16p⇡

3↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+ . . .

⇠ E1/2logE

High Energy Asymptotic

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

elliptic integral of the first kind EllipticK[x]

limit of the Veneziano amplitude (Zohar’s talk at Strings2016)

⇠ E2logE

correction due to the slowdown of the string (massive endpoints)/spectrum non-degeneracy

corrections are O(log E)

lim logA(s, t) = ↵0[(s+ t) log(s+ t)� s log(s)� t log(t)]

s, t ! 1s/t fixed

�16p⇡

3↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+ . . .

⇠ E1/2logE

Result (leading)

lim logA(s, t) = ↵0[(s+ t) log(s+ t)� s log(s)� t log(t)]

s, t ! 1s/t fixed

Result (leading)

lim logA(s, t) = ↵0[(s+ t) log(s+ t)� s log(s)� t log(t)]

s, t ! 1s/t fixed

• amplitude is exponentially large (unitarity universality))

Result (leading)

lim logA(s, t) = ↵0[(s+ t) log(s+ t)� s log(s)� t log(t)]

s, t ! 1s/t fixed

• stringy. Infinitely many asymptotically linear Regge trajectories

object of transverse size = a string

⇠ log(s)

b

j(t) = ↵0t+ corrections

+parallel trajectories

Im A(s, b) ⇠ e�b2

↵0log s)

• amplitude is exponentially large (unitarity universality))

Result (leading)

lim logA(s, t) = ↵0[(s+ t) log(s+ t)� s log(s)� t log(t)]

s, t ! 1s/t fixed

• stringy. Infinitely many asymptotically linear Regge trajectories

object of transverse size = a string

⇠ log(s)

b

j(t) = ↵0t+ corrections

+parallel trajectories

Im A(s, b) ⇠ e�b2

↵0log s)

• amplitude is exponentially large (unitarity universality))

• insensitive to the microscopic spectrum degeneracy

Result (sub-leading)

� logA(s, t) = �16

p⇡

3

↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+ . . .

Result (sub-leading)

� logA(s, t) = �16

p⇡

3

↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+ . . .

• worldsheet: slowdown of the string endpoints

mm

[Chodos, Thorn, 74’]

j(t) = ↵0✓t� 8

p⇡

3m3/2t1/4 + ...

◆

[Sonnenschein et al.][Wilczek]

[Baker, Steinke]

Result (sub-leading)

� logA(s, t) = �16

p⇡

3

↵0m3/2

✓s t

s+ t

◆ 14K

✓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

mm

[Chodos, Thorn, 74’]

j(t) = ↵0✓t� 8

p⇡

3m3/2t1/4 + ...

◆

[Sonnenschein et al.][Wilczek]

[Baker, Steinke]

Computing the Correction

• Scattering of Strings With Massive Endpoints

• Universality (Holography & EFT of Long Strings)

• Bootstrap

Worldsheet Computation (review)

s/t fixed|s|, |t| ! 1lim A(s, t) = e�SE(s,t) [Gross, Mende]

[Gross, Mañes][Alday, Maldacena]

Worldsheet Computation (review)

s/t fixed|s|, |t| ! 1lim 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| ! 1lim 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

2⇡↵0

Zd

2z @x · @̄x� i

X

j

kj · x(�j)Flat space

Worldsheet Computation (review)

SE =1

2⇡↵0

Zd

2z @x · @̄x� i

X

j

kj · x(�j)Flat space

• general solutionx

µ0 = i

X

i

k

µi log |z � �i|2

Worldsheet Computation (review)

• Virasoro (scattering equations)X

j

ki · kj�i � �j

= 0

SE =1

2⇡↵0

Zd

2z @x · @̄x� i

X

j

kj · x(�j)Flat space

• general solutionx

µ0 = i

X

i

k

µi log |z � �i|2

Worldsheet Computation (review)

• Virasoro (scattering equations)X

j

ki · kj�i � �j

= 0

SE =1

2⇡↵0

Zd

2z @x · @̄x� i

X

j

kj · x(�j)Flat space

logA(s, t) = ↵0[(s+ t) log(s+ t)� s log s� t log t])

s, t > 0

• general solutionx

µ0 = i

X

i

k

µi log |z � �i|2

Adding The Mass

Adding The Mass

SE =1

2⇡↵0

Zd

2z @x · @̄x+m

Zd�

p|@�x|2 � i

X

j

kj · x(�j)

[Chodos, Thorn]

Adding The Mass

Modified boundary condition:

1

2⇡↵0 @⌧x+m @�@�xp

@�x · @�x= i

X

j

kj �(� � �j)

SE =1

2⇡↵0

Zd

2z @x · @̄x+m

Zd�

p|@�x|2 � i

X

j

kj · x(�j)

[Chodos, Thorn]

Adding The Mass

Modified boundary condition:

1

2⇡↵0 @⌧x+m @�@�xp

@�x · @�x= i

X

j

kj �(� � �j)

is zero for a free string! @�x0 · @�x0 = 0

SE =1

2⇡↵0

Zd

2z @x · @̄x+m

Zd�

p|@�x|2 � i

X

j

kj · x(�j)

[Chodos, Thorn]

Adding The Mass

Modified boundary condition:

1

2⇡↵0 @⌧x+m @�@�xp

@�x · @�x= i

X

j

kj �(� � �j)

is zero for a free string! @�x0 · @�x0 = 0

SE =1

2⇡↵0

Zd

2z @x · @̄x+m

Zd�

p|@�x|2 � i

X

j

kj · x(�j)

[Chodos, Thorn]

The expansion reorganizes itself in terms of : pm

x

µ = x

µ0 +

pm x

µ1 + ... S = S0 +

pmS1 +mS2 +m3/2S3

Adding The Mass

Modified boundary condition:

1

2⇡↵0 @⌧x+m @�@�xp

@�x · @�x= i

X

j

kj �(� � �j)

is zero for a free string! @�x0 · @�x0 = 0

SE =1

2⇡↵0

Zd

2z @x · @̄x+m

Zd�

p|@�x|2 � i

X

j

kj · x(�j)

[Chodos, Thorn]

The expansion reorganizes itself in terms of : pm

x

µ = x

µ0 +

pm x

µ1 + ... S = S0 +

pmS1 +mS2 +m3/2S3

The on-shell action evaluates to

Adding The Mass

Lb =p2⇡↵0

m

Zd� (@2

�x0 · @2�x0)

1/4

SE = SGM +2

3mLb + ...

Gross-Mende solution

The on-shell action evaluates to

Adding The Mass

Lb =p2⇡↵0

m

Zd� (@2

�x0 · @2�x0)

1/4

SE = SGM +2

3mLb + ...

Gross-Mende solution

The on-shell action evaluates to

reparameterization invariant

Adding The Mass

Lb =p2⇡↵0

m

Zd� (@2

�x0 · @2�x0)

1/4

SE = SGM +2

3mLb + ...

Gross-Mende solution

The on-shell action evaluates to

reparameterization invariant

Adding The Mass

For four external particles

� logA(s, t) = �16

p⇡

3

↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+O(m5/2

)

Lb =p2⇡↵0

m

Zd� (@2

�x0 · @2�x0)

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

� logA(s, t) = �16

p⇡

3

↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+O(m5/2

)

Lb =p2⇡↵0

m

Zd� (@2

�x0 · @2�x0)

1/4

SE = SGM +2

3mLb + ...

Emergent s-u Crossing Symmetry

lim logA(s, t) = ↵0[(s+ t) log(s+ t)� s log(s)� t log(t)]

s, t ! 1s/t fixed

�16p⇡

3↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+ . . .

Emergent s-u Crossing Symmetry

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

lim logA(s, t) = ↵0[(s+ t) log(s+ t)� s log(s)� t log(t)]

s, t ! 1s/t fixed

�16p⇡

3↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+ . . .

Emergent s-u Crossing Symmetry

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

What about the s-u crossing?

lim logA(s, t) = ↵0[(s+ t) log(s+ t)� s log(s)� t log(t)]

s, t ! 1s/t fixed

�16p⇡

3↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+ . . .

Emergent s-u Crossing Symmetry

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

What about the s-u crossing?logA(s, t) = Re[logA(u, t)]

u = �s� t

lim logA(s, t) = ↵0[(s+ t) log(s+ t)� s log(s)� t log(t)]

s, t ! 1s/t fixed

�16p⇡

3↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+ . . .

Emergent s-u Crossing Symmetry

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

What about the s-u crossing?logA(s, t) = Re[logA(u, t)]

u = �s� t

1

2

3

4

1

2

34???

lim logA(s, t) = ↵0[(s+ t) log(s+ t)� s log(s)� t log(t)]

s, t ! 1s/t fixed

�16p⇡

3↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+ . . .

Emergent s-u Crossing Symmetry [Komatsu]

1

2

3

4

1

2

34???

Emergent s-u Crossing Symmetry [Komatsu]

1

2

3

4

1

2

34???

Asymptotic s-u Crossing

Equivalently, the asymptotic s-u crossing is:

dDiscs logA(s, t) ⌘

logA(�s� t+ i✏, t) + logA(�s� t� i✏, t)� 2 logA(s, t) = 0

Double discontinuity is zero!

Universality

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:

ds

2 = dr

2 + f(r) dx21,d�1

• AdS in the UV limr!1

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

IR

r0

Holographic radial direction

Flavor braneUV

[Polchinski, Strassler]

Polchinski-Strassler Mechanism

For mesons we add a space-filling flavor brane

IR

r0

Holographic radial direction

Flavor braneUV

real scattering angles

[Polchinski, Strassler]

Polchinski-Strassler Mechanism

For mesons we add a space-filling flavor brane

IR

r0

Holographic radial direction

Flavor braneUV

real scattering angles

imaginary scattering angles

[Polchinski, Strassler]

At high energies the string acquires the characteristic shape

IR

r0

Holographic radial direction Flavor brane

UV

Polchinski-Strassler Mechanism[Polchinski, Strassler]

At high energies the string acquires the characteristic shape

IR

r0

Holographic radial direction Flavor brane

UV

Polchinski-Strassler Mechanism[Polchinski, Strassler]

At high energies the string acquires the characteristic shape

IR

r0

Holographic radial direction Flavor brane

UV

Polchinski-Strassler Mechanism[Polchinski, Strassler]

At high energies the string acquires the characteristic shape

IR

r0

Holographic radial direction Flavor brane

UV

string in flat space

Polchinski-Strassler Mechanism[Polchinski, Strassler]

At high energies the string acquires the characteristic shape

IR

r0

Holographic radial direction Flavor brane

UV

m

string in flat space

Polchinski-Strassler Mechanism[Polchinski, Strassler]

At high energies the string acquires the characteristic shape

IR

r0

Holographic radial direction Flavor brane

UV

m ) m m

effective description

m2↵0 � 1

string in flat space

Polchinski-Strassler Mechanism[Polchinski, Strassler]

At high energies the string acquires the characteristic shape

IR

r0

Holographic radial direction Flavor brane

UV

m ) m m

effective description

m2↵0 � 1

string in flat space

• At the holographic model reduces to the string with massive ends s, t � 1

Polchinski-Strassler Mechanism[Polchinski, Strassler]

At high energies the string acquires the characteristic shape

IR

r0

Holographic radial direction Flavor brane

UV

m ) m m

effective description

m2↵0 � 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[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) = ↵0✓t� 8

p⇡

3m3/2t1/4

◆[Hellerman, Swanson]

EFT of Long Strings

• quantum correctionsPolchinski-Strominger term

(@2x · @̄x)(@x · @̄2

x)

(@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) = ↵0✓t� 8

p⇡

3m3/2t1/4

◆[Hellerman, Swanson]

EFT of Long Strings

• quantum correctionsPolchinski-Strominger term

(@2x · @̄x)(@x · @̄2

x)

(@x · @̄x)2

• higher derivative corrections (closed strings)

[hopefully somebody in progress]

[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) = ↵0✓t� 8

p⇡

3m3/2t1/4

◆[Hellerman, Swanson]

EFT of Long Strings

• quantum correctionsPolchinski-Strominger term

(@2x · @̄x)(@x · @̄2

x)

(@x · @̄x)2

• higher derivative corrections (closed strings)

[hopefully somebody in progress]

[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) = ↵0✓t� 8

p⇡

3m3/2t1/4

◆[Hellerman, Swanson]

Bootstrap

Bootstrap (Leading Order)

For s,t large and positive a thermodynamic picture emerges

s, t ! 1s/t fixed

lim logA ((1 + i✏)s, (1 + i✏)t) [Caron-Huot, Komargodski, Sever, AZ]

1�1

J even

J odd

PJ(x)

we are here

Partial wave

Bootstrap (Leading Order)

For s,t large and positive a thermodynamic picture emerges

s, t ! 1s/t fixed

lim logA ((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

Bootstrap (Leading Order)

For s,t large and positive a thermodynamic picture emerges

s, t ! 1s/t fixed

lim logA ((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

Bootstrap (Leading Order)

For s,t large and positive a thermodynamic picture emerges

s, t ! 1s/t fixed

lim logA ((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

poles

zeros

Bootstrap (Leading Order)

For s,t large and positive a thermodynamic picture emerges

�t

s

We are here

s, t ! 1s/t fixed

lim logA ((1 + i✏)s, (1 + i✏)t)

distribution of excess zeros ⇢

logA(s, t) ' j(t) log s ) j(t) = Diss logA =

X(zeros� poles)

Bootstrap (Leading Order)

To leading order we have

logA(s, t) ' log

j(t)Z

0

dj cj(t) Pj

✓1 +

2s

t

◆, cj(t) � 0

The unique solution is

�t

s

logA(s, t) = ↵

0t

1Z

0

dx⇢(x) log

⇣1 +

s

tx

⌘

= ↵0[(s+ t) log(s+ t)� s log s� t log t]

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 (Correction)

• Unitarity and the s-t crossing are not enough

logA(s, t) = logA(t, s)

Bootstrap (Correction)

• Unitarity and the s-t crossing are not enough

logA(s, t) = logA(t, s)

Bootstrap (Correction)

• Impose the s-u crossing

dDiscs logA(s, t) = 0

s

t

u

Bootstrap (Integral Equation)

The extra condition leads to an integral equation

�⇢k(y) =

1Z

0

dx [K(y, x) +K(1� y, 1� x)] �⇢k(x)

+

(1� y)

k�1

⇡ sin⇡k

✓y

x+ y � x y

+ k log

x (1� y)

x+ y � x y

◆

�j(t) = tk

correction to the trajectory

correction to the distribution

K(y, x) =

cot⇡k

⇡

✓yP

1

x� y

� k log

x

|x� y|

◆

• The correction we found obeys the equation

�⇢k(y) =

1Z

0

dx [K(y, x) +K(1� y, 1� x)] �⇢k(x)

Bootstrap (Integral Equation)

�j(t) = tk

• The correction we found obeys the equation

• Easy to show that only k=1/4, k=3/4 are possible

�⇢k(y) =

1Z

0

dx [K(y, x) +K(1� y, 1� x)] �⇢k(x)

Bootstrap (Integral Equation)

�j(t) = tk

• The correction we found obeys the equation

• Easy to show that only k=1/4, k=3/4 are possible

The solution is unique? (in progress)

�⇢k(y) =

1Z

0

dx [K(y, x) +K(1� y, 1� x)] �⇢k(x)

Bootstrap (Integral Equation)

�j(t) = tk

Conclusions and Open questions

• WIHS is an exciting, unexplored and stringy territory[Hagedorn?]

Conclusions and Open questions

• WIHS is an exciting, unexplored and stringy territory[Hagedorn?]

• Subject to the analytic bootstrap [Systematic expansion, EFT+holography?]

[talk Alday]

Conclusions and Open questions

• WIHS is an exciting, unexplored and stringy territory[Hagedorn?]

• Non-universal regime[Numerical bootstrap, graviton, DIS?]

• Subject to the analytic bootstrap [Systematic expansion, EFT+holography?]

[talk Alday]

Conclusions and Open questions

• WIHS is an exciting, unexplored and stringy territory[Hagedorn?]

• Non-universal regime[Numerical bootstrap, graviton, DIS?]

• Bootstrap in AdS (Mellin space)[Theories with accumulation?]

• Subject to the analytic bootstrap [Systematic expansion, EFT+holography?]

[talk Alday]

Conclusions and Open questions

• WIHS is an exciting, unexplored and stringy territory[Hagedorn?]

• Non-universal regime[Numerical bootstrap, graviton, DIS?]

• Bootstrap in AdS (Mellin space)[Theories with accumulation?]

• Subject to the analytic bootstrap [Systematic expansion, EFT+holography?]

[talk Alday]

• Quantum theories [Universal?]

[talk Penedones]

Conclusions and Open questions

• WIHS is an exciting, unexplored and stringy territory[Hagedorn?]

• Non-universal regime[Numerical bootstrap, graviton, DIS?]

• Bootstrap in AdS (Mellin space)[Theories with accumulation?]

thank you!

• Subject to the analytic bootstrap [Systematic expansion, EFT+holography?]

[talk Alday]

• Quantum theories [Universal?]

[talk Penedones]

Bootstrap Method

Take your physical problem and:

Bootstrap Method

1. Solve analytically for things that ``must happen.’’

Take your physical problem and:

Bootstrap Method

1. Solve analytically for things that ``must happen.’’

Take your physical problem and:

2. Feed this knowledge into a computer. Learn things that ``never happen’’ and ``special occasions.’’

Bootstrap Method

1. Solve analytically for things that ``must happen.’’

3. Feed this knowledge into 1.

Take your physical problem and:

2. Feed this knowledge into a computer. Learn things that ``never happen’’ and ``special occasions.’’

Bootstrap Method

1. Solve analytically for things that ``must happen.’’

3. Feed this knowledge into 1.

Take your physical problem and:

2. Feed this knowledge into a computer. Learn things that ``never happen’’ and ``special occasions.’’

Bootstrap Method

1. Solve analytically for things that ``must happen.’’

3. Feed this knowledge into 1.

Take your physical problem and:

2. Feed this knowledge into a computer. Learn things that ``never happen’’ and ``special occasions.’’

Clearly, the final result is independent of the starting point. [Simmons-Duffin]

Bootstrap in Mellin space

Mack polynomials at large spin take the form

M(s, t) 'c2��⌧Q

�,⌧,dJ,m (s)

t� (⌧ + 2m)+ ...

Crossing equation takes the form

M(s, t) 'J(t)X

cJJs =

J(s)XcJJ

t ' M(t, s)

The solution islogM(s, t) =

1

cs t

⌧(J) = c log J

Worldsheet Computation

Worldsheet Computation

SE =1

2⇡↵0

Zd

2z @zx

µ@z̄xµ +

1

2

Zd�

✓e @�x

µ@�xµ +

m

2

e

◆+ i

X

j

k

µj xµ(�j)

Worldsheet Computation

SE =1

2⇡↵0

Zd

2z @zx

µ@z̄xµ +

1

2

Zd�

✓e @�x

µ@�xµ +

m

2

e

◆+ i

X

j

k

µj xµ(�j)

• introduce the boundary metric

Worldsheet Computation

SE =1

2⇡↵0

Zd

2z @zx

µ@z̄xµ +

1

2

Zd�

✓e @�x

µ@�xµ +

m

2

e

◆+ i

X

j

k

µj xµ(�j)

• introduce the boundary metric e(�)2 =m

2

@�xµ@�xµ

Worldsheet Computation

SE =1

2⇡↵0

Zd

2z @zx

µ@z̄xµ +

1

2

Zd�

✓e @�x

µ@�xµ +

m

2

e

◆+ i

X

j

k

µj xµ(�j)

• introduce the boundary metric e(�)2 =m

2

@�xµ@�xµ

• write the solution as x

µ = x

µ0 + y

µ

Gross-Mende solution

Worldsheet Computation

SE =1

2⇡↵0

Zd

2z @zx

µ@z̄xµ +

1

2

Zd�

✓e @�x

µ@�xµ +

m

2

e

◆+ i

X

j

k

µj xµ(�j)

• introduce the boundary metric e(�)2 =m

2

@�xµ@�xµ

• impose the Virasoro constraint

1

(2⇡↵

0)

2

m

2

e

2= e

2@

2�x

µ0@

2�x

µ0 �m

2[@� log e]

2+ e

2⇥2@

2�x

µ0@

2�y

µ+ @

2�y

µ@

2�y

µ⇤

• write the solution as x

µ = x

µ0 + y

µ

Gross-Mende solution

Worldsheet Computation

Consider the small m expansion

1

(2⇡↵

0)

2

m

2

e

2= e

2@

2�x

µ0@

2�x

µ0 �m

2[@� log e]

2+ e

2⇥2@

2�x

µ0@

2�y

µ+ @

2�y

µ@

2�y

µ⇤

Worldsheet Computation

Consider the small m expansion

1

(2⇡↵

0)

2

m

2

e

2= e

2@

2�x

µ0@

2�x

µ0 �m

2[@� log e]

2+ e

2⇥2@

2�x

µ0@

2�y

µ+ @

2�y

µ@

2�y

µ⇤

Worldsheet Computation

Consider the small m expansion

1

(2⇡↵

0)

2

m

2

e

2= e

2@

2�x

µ0@

2�x

µ0 �m

2[@� log e]

2+ e

2⇥2@

2�x

µ0@

2�y

µ+ @

2�y

µ@

2�y

µ⇤

e⇤(�)2 =

m

2⇡↵0p

@

2�x0 · @2

�x0e⇤ ⇠

pm

The leading solution is