universal correction to the veneziano amplitude correction to the veneziano amplitude alexander...
TRANSCRIPT
![Page 1: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/1.jpg)
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)
![Page 2: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/2.jpg)
Homework from Strings2014
Problem 72 (Juan):
What is the general theory of weakly coupled, interacting, higher spin particles?
![Page 3: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/3.jpg)
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?
![Page 4: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/4.jpg)
What is WIHS?
![Page 5: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/5.jpg)
What is WIHS?
A(s, t) =Two-to-two scattering amplitude
![Page 6: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/6.jpg)
What is WIHS?
A(s, t) =Two-to-two scattering amplitude
• weakly coupled ⌘ meromorphicfss�
f�ss✓
m2�, J
![Page 7: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/7.jpg)
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 ✓
![Page 8: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/8.jpg)
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 ✓
![Page 9: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/9.jpg)
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
![Page 10: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/10.jpg)
• soft high energy limit (causality for HS)
lims!1
A(s, t0) < sJ0
J0
t
s
What is WIHS?
[talk Caron-Huot]
![Page 11: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/11.jpg)
• 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]
![Page 12: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/12.jpg)
• 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]
![Page 13: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/13.jpg)
• 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]
![Page 14: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/14.jpg)
• 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]
![Page 15: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/15.jpg)
t
j(t)
ST
(mass)
(spin)
lims!1
A(s, t) ⇠ sj(t)
spectrumscattering
The Regge Trajectory j(t)
�(�s)�(�t)
�(�t� s)
![Page 16: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/16.jpg)
t
j(t)
STYM
(mass)
(spin)
lims!1
A(s, t) ⇠ sj(t)
spectrumscattering
The Regge Trajectory j(t)
�(�s)�(�t)
�(�t� s)
![Page 17: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/17.jpg)
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)
![Page 18: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/18.jpg)
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)
![Page 19: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/19.jpg)
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
![Page 20: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/20.jpg)
Results
![Page 21: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/21.jpg)
High Energy Asymptotic
![Page 22: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/22.jpg)
High Energy Asymptotic
The high energy limit of WIHS amplitudes at imaginary scattering angles takes the form
![Page 23: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/23.jpg)
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
![Page 24: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/24.jpg)
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
![Page 25: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/25.jpg)
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
![Page 26: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/26.jpg)
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
![Page 27: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/27.jpg)
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
![Page 28: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/28.jpg)
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
![Page 29: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/29.jpg)
Result (leading)
lim logA(s, t) = ↵0[(s+ t) log(s+ t)� s log(s)� t log(t)]
s, t ! 1s/t fixed
![Page 30: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/30.jpg)
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))
![Page 31: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/31.jpg)
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))
![Page 32: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/32.jpg)
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
![Page 33: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/33.jpg)
Result (sub-leading)
� logA(s, t) = �16
p⇡
3
↵0m3/2
✓s t
s+ t
◆ 14K
✓s
s+ t
◆+K
✓t
s+ t
◆�+ . . .
![Page 34: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/34.jpg)
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]
![Page 35: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/35.jpg)
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]
![Page 36: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/36.jpg)
Computing the Correction
![Page 37: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/37.jpg)
• Scattering of Strings With Massive Endpoints
• Universality (Holography & EFT of Long Strings)
• Bootstrap
![Page 38: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/38.jpg)
Worldsheet Computation (review)
s/t fixed|s|, |t| ! 1lim A(s, t) = e�SE(s,t) [Gross, Mende]
[Gross, Mañes][Alday, Maldacena]
![Page 39: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/39.jpg)
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]
![Page 40: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/40.jpg)
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]
![Page 41: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/41.jpg)
Worldsheet Computation (review)
![Page 42: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/42.jpg)
Worldsheet Computation (review)
SE =1
2⇡↵0
Zd
2z @x · @̄x� i
X
j
kj · x(�j)Flat space
![Page 43: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/43.jpg)
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
![Page 44: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/44.jpg)
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
![Page 45: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/45.jpg)
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
![Page 46: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/46.jpg)
Adding The Mass
![Page 47: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/47.jpg)
Adding The Mass
SE =1
2⇡↵0
Zd
2z @x · @̄x+m
Zd�
p|@�x|2 � i
X
j
kj · x(�j)
[Chodos, Thorn]
![Page 48: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/48.jpg)
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]
![Page 49: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/49.jpg)
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]
![Page 50: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/50.jpg)
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
![Page 51: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/51.jpg)
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
![Page 52: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/52.jpg)
The on-shell action evaluates to
Adding The Mass
Lb =p2⇡↵0
m
Zd� (@2
�x0 · @2�x0)
1/4
SE = SGM +2
3mLb + ...
![Page 53: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/53.jpg)
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 + ...
![Page 54: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/54.jpg)
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 + ...
![Page 55: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/55.jpg)
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 + ...
![Page 56: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/56.jpg)
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 + ...
![Page 57: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/57.jpg)
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
◆�+ . . .
![Page 58: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/58.jpg)
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
◆�+ . . .
![Page 59: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/59.jpg)
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
◆�+ . . .
![Page 60: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/60.jpg)
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
◆�+ . . .
![Page 61: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/61.jpg)
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
◆�+ . . .
![Page 62: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/62.jpg)
Emergent s-u Crossing Symmetry [Komatsu]
1
2
3
4
1
2
34???
![Page 63: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/63.jpg)
Emergent s-u Crossing Symmetry [Komatsu]
1
2
3
4
1
2
34???
![Page 64: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/64.jpg)
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!
![Page 65: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/65.jpg)
Universality
![Page 66: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/66.jpg)
Why is the correction universal?
![Page 67: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/67.jpg)
Why is the correction universal?
Why is the massive ends model physical?
![Page 68: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/68.jpg)
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
![Page 69: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/69.jpg)
Polchinski-Strassler Mechanism
For mesons we add a space-filling flavor brane
IR
r0
Holographic radial direction
Flavor braneUV
[Polchinski, Strassler]
![Page 70: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/70.jpg)
Polchinski-Strassler Mechanism
For mesons we add a space-filling flavor brane
IR
r0
Holographic radial direction
Flavor braneUV
real scattering angles
[Polchinski, Strassler]
![Page 71: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/71.jpg)
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]
![Page 72: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/72.jpg)
At high energies the string acquires the characteristic shape
IR
r0
Holographic radial direction Flavor brane
UV
Polchinski-Strassler Mechanism[Polchinski, Strassler]
![Page 73: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/73.jpg)
At high energies the string acquires the characteristic shape
IR
r0
Holographic radial direction Flavor brane
UV
Polchinski-Strassler Mechanism[Polchinski, Strassler]
![Page 74: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/74.jpg)
At high energies the string acquires the characteristic shape
IR
r0
Holographic radial direction Flavor brane
UV
Polchinski-Strassler Mechanism[Polchinski, Strassler]
![Page 75: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/75.jpg)
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]
![Page 76: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/76.jpg)
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]
![Page 77: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/77.jpg)
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]
![Page 78: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/78.jpg)
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]
![Page 79: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/79.jpg)
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]
![Page 80: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/80.jpg)
EFT of Long Strings [Aharony et al.]
[Hellerman et al.]
[Polchinski, Strominger]
[Dubovsky et al.]
![Page 81: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/81.jpg)
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]
![Page 82: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/82.jpg)
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]
![Page 83: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/83.jpg)
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]
![Page 84: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/84.jpg)
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]
![Page 85: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/85.jpg)
Bootstrap
![Page 86: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/86.jpg)
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
![Page 87: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/87.jpg)
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
![Page 88: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/88.jpg)
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
![Page 89: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/89.jpg)
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
![Page 90: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/90.jpg)
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)
![Page 91: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/91.jpg)
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]
![Page 92: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/92.jpg)
Bootstrap (Correction)
• Spectrum Non-degeneracy/Support of excess zeros
�t
s
non-degeneracy zeros
![Page 93: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/93.jpg)
Bootstrap (Correction)
• Spectrum Non-degeneracy/Support of excess zeros
�t
s
non-degeneracy zeros
density>0
![Page 94: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/94.jpg)
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
![Page 95: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/95.jpg)
Bootstrap (Correction)
![Page 96: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/96.jpg)
• Unitarity and the s-t crossing are not enough
logA(s, t) = logA(t, s)
Bootstrap (Correction)
![Page 97: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/97.jpg)
• 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
![Page 98: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/98.jpg)
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|
◆
![Page 99: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/99.jpg)
• 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
![Page 100: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/100.jpg)
• 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
![Page 101: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/101.jpg)
• 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
![Page 102: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/102.jpg)
Conclusions and Open questions
• WIHS is an exciting, unexplored and stringy territory[Hagedorn?]
![Page 103: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/103.jpg)
Conclusions and Open questions
• WIHS is an exciting, unexplored and stringy territory[Hagedorn?]
• Subject to the analytic bootstrap [Systematic expansion, EFT+holography?]
[talk Alday]
![Page 104: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/104.jpg)
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]
![Page 105: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/105.jpg)
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]
![Page 106: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/106.jpg)
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]
![Page 107: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/107.jpg)
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]
![Page 108: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/108.jpg)
Bootstrap Method
Take your physical problem and:
![Page 109: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/109.jpg)
Bootstrap Method
1. Solve analytically for things that ``must happen.’’
Take your physical problem and:
![Page 110: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/110.jpg)
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.’’
![Page 111: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/111.jpg)
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.’’
![Page 112: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/112.jpg)
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.’’
![Page 113: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/113.jpg)
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]
![Page 114: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/114.jpg)
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
![Page 115: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/115.jpg)
Worldsheet Computation
![Page 116: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/116.jpg)
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)
![Page 117: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/117.jpg)
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
![Page 118: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/118.jpg)
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
![Page 119: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/119.jpg)
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
![Page 120: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/120.jpg)
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
![Page 121: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/121.jpg)
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⇤
![Page 122: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/122.jpg)
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⇤
![Page 123: Universal Correction To The Veneziano Amplitude Correction To The Veneziano Amplitude Alexander Zhiboedov, Harvard U Strings 2017, Tel Aviv, Israel with S. Caron-Huot, Z. Komargodski,](https://reader036.vdocuments.site/reader036/viewer/2022062600/5af6e0717f8b9a4d4d9100ab/html5/thumbnails/123.jpg)
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