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?


• weakly coupled ⌘ meromorphicfss�

f�ss✓

m2�, J

What is WIHS?



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?


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


f�ss✓

m2�, J


ss�

PJ(1 +2t

m2��4m2

s)

s�m2�

positive

cos ✓

What is WIHS?


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


f�ss✓

m2�, J


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]


lims!1

A(s, t0) < sJ0

J0

t

s

clash with unitarity! A(s, t) ⇠ sJ0

What is WIHS?

[talk Caron-Huot]


lims!1

A(s, t0) < sJ0

J0

t

s

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


What is WIHS?

[talk Caron-Huot]


lims!1

A(s, t0) < sJ0

J0

t

s



What is WIHS?

fundamental strings, large N confining gauge theories, …

[Veneziano][Andreev, Siegel]

[Veneziano, Yankielowicz, Onofri]

�(�s)�(�t)

�(�t� s)

Solutions:

[talk Caron-Huot]


lims!1

A(s, t0) < sJ0

J0

t

s



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


�(�s)�(�t)

�(�t� s)

t

j(t)

STYM

(mass)

(spin)

lims!1

A(s, t) ⇠ sj(t)

spectrum

non-universal

?

scattering


�(�s)�(�t)

�(�t� s)

t

j(t)

STYM

(mass)

(spin)

lims!1

A(s, t) ⇠ sj(t)

universal

spectrum

non-universal

?

scattering


�(�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


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



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



elliptic integral of the first kind EllipticK[x]




�16p⇡

3↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+ . . .

⇠ E1/2logE





⇠ E2logE

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


s, t ! 1s/t fixed

�16p⇡

3↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+ . . .

⇠ E1/2logE





⇠ E2logE

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

corrections are O(log E)


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)


s, t ! 1s/t fixed

Result (leading)


s, t ! 1s/t fixed

• amplitude is exponentially large (unitarity universality))

Result (leading)


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)


Result (leading)


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)


• 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

◆�+ . . .


� 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]


� 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]


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]


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]



SE =1

2⇡↵0

Zd

2z @x · @̄x� i

X

j

kj · x(�j)Flat space


SE =1

2⇡↵0

Zd

2z @x · @̄x� i

X

j


• general solutionx

µ0 = i

X

i

k

µi log |z � �i|2


• Virasoro (scattering equations)X

j

ki · kj�i � �j

= 0

SE =1

2⇡↵0

Zd

2z @x · @̄x� i

X

j



µ0 = i

X

i

k



• Virasoro (scattering equations)X

j

ki · kj�i � �j

= 0

SE =1

2⇡↵0

Zd

2z @x · @̄x� i

X

j


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

s, t > 0


µ0 = i

X

i

k


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


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


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]

The expansion reorganizes itself in terms of : pm

x

µ = x

µ0 +

pm x

µ1 + ... S = S0 +

pmS1 +mS2 +m3/2S3

Adding The Mass


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]

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


Adding The Mass

Lb =p2⇡↵0

m

Zd� (@2

�x0 · @2�x0)

1/4

SE = SGM +2

3mLb + ...



reparameterization invariant

Adding The Mass

Lb =p2⇡↵0

m

Zd� (@2

�x0 · @2�x0)

1/4

SE = SGM +2

3mLb + ...




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 + ...


non-universal O(t�1/4)



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


s, t ! 1s/t fixed

�16p⇡

3↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+ . . .


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


s, t ! 1s/t fixed

�16p⇡

3↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+ . . .



What about the s-u crossing?


s, t ! 1s/t fixed

�16p⇡

3↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+ . . .



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

u = �s� t


s, t ! 1s/t fixed

�16p⇡

3↵0m3/2

✓s t

s+ t

◆ 14K

✓s

s+ t

◆+K

✓t

s+ t

◆�+ . . .



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

u = �s� t

1

2

3

4

1

2

34???


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


Flavor braneUV

[Polchinski, Strassler]



IR

r0


Flavor braneUV

real scattering angles




IR

r0


Flavor braneUV

real scattering angles

imaginary scattering angles


At high energies the string acquires the characteristic shape

IR

r0

Holographic radial direction Flavor brane

UV

Polchinski-Strassler Mechanism[Polchinski, Strassler]


IR

r0


UV



IR

r0


UV



IR

r0


UV

string in flat space



IR

r0


UV

m




IR

r0


UV

m ) m m

effective description

m2↵0 � 1




IR

r0


UV

m ) m m


m2↵0 � 1


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



IR

r0


UV

m ) m m


m2↵0 � 1


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

• Insensitive to the details of the background


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.]


[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.]


[Dubovsky et al.]


j(t) = ↵0✓t� 8

p⇡

3m3/2t1/4


EFT of Long Strings


(@2x · @̄x)(@x · @̄2

x)

(@x · @̄x)2

• higher derivative corrections (closed strings)

[hopefully somebody in progress]

[Aharony et al.]

[Hellerman et al.]


[Dubovsky et al.]


j(t) = ↵0✓t� 8

p⇡

3m3/2t1/4


EFT of Long Strings


(@2x · @̄x)(@x · @̄2

x)

(@x · @̄x)2

• higher derivative corrections (closed strings)

[hopefully somebody in progress]

[Aharony et al.]

[Hellerman et al.]


[Dubovsky et al.]


j(t) = ↵0✓t� 8

p⇡

3m3/2t1/4


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



s, t ! 1s/t fixed


1�1

J even

J odd

PJ(x)

we are here

Partial wave

) • All residues are positive



s, t ! 1s/t fixed


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




s, t ! 1s/t fixed


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


Complex s plane at fixed real t

poles

zeros



�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)


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



�t

s


density>0



�t

s


Indeed, the massive ends correction is of this form!

density>0


• Unitarity and the s-t crossing are not enough

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


• Unitarity and the s-t crossing are not enough

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


• 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)


�j(t) = tk


• 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)


�j(t) = tk


• 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)


�j(t) = tk

Conclusions and Open questions

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



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

[talk Alday]



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


[talk Alday]




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


[talk Alday]






[talk Alday]

• Quantum theories [Universal?]

[talk Penedones]





thank you!


[talk Alday]

• Quantum theories [Universal?]

[talk Penedones]

Bootstrap Method

Take your physical problem and:

Bootstrap Method

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


Bootstrap Method



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

Bootstrap Method


3. Feed this knowledge into 1.



Bootstrap Method





Bootstrap Method





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


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)


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


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


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)


2

@�xµ@�xµ

• write the solution as x

µ = x

µ0 + y

µ



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)


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



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⇤



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⇤



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

Date post:	12-May-2018
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Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander...

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