large spin analytic bootstrap via mellin...large spin analytic bootstrap via mellin parijat dey...
TRANSCRIPT
![Page 1: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/1.jpg)
Large spin analytic bootstrap via Mellin
Parijat Dey
CHEP, Indian Institute of Science,
Based on
JHEP 1801(2018) 152 with Kausik Ghosh and Aninda Sinha
Bangalore, India
JHEP1802(2018)153 with Apratim Kaviraj
OIST, OkinawaMarch 20, 2018
![Page 2: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/2.jpg)
Sampling of the results
Anomalous dimension of large spin double trace operators:All orders in inverse spin J
h=d/2, d= dimension
![Page 3: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/3.jpg)
Flashing the results for epsilon expansion
![Page 4: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/4.jpg)
Plan of talk
![Page 5: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/5.jpg)
Plan of talk
● Review of large spin bootstrap
![Page 6: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/6.jpg)
Plan of talk
● Review of large spin bootstrap
● Bootstrap in Mellin space
![Page 7: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/7.jpg)
Plan of talk
● Review of large spin bootstrap
● Bootstrap in Mellin space
● Epsilon expansion at large spin
![Page 8: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/8.jpg)
Plan of talk
● Review of large spin bootstrap
● Bootstrap in Mellin space
● Summary
● Epsilon expansion at large spin
![Page 9: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/9.jpg)
Quick Review of conformal bootstrap
![Page 10: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/10.jpg)
s-channel
Quick Review of conformal bootstrap
![Page 11: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/11.jpg)
t-channel
s-channel
Quick Review of conformal bootstrap
![Page 12: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/12.jpg)
If OPE is associative then these two expansion must give the same result
t-channel
s-channel
Quick Review of conformal bootstrap
![Page 13: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/13.jpg)
If OPE is associative then these two expansion must give the same result
Obtain all You know everything..!!
t-channel
s-channel
Quick Review of conformal bootstrap
![Page 14: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/14.jpg)
If OPE is associative then these two expansion must give the same result
This is the conventional bootstrap program
Obtain all You know everything..!!
t-channel
s-channel
Quick Review of conformal bootstrap
![Page 15: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/15.jpg)
Bootstrap equation
![Page 16: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/16.jpg)
Difficult to solve this equation analytically and get information
Bootstrap equation
![Page 17: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/17.jpg)
Difficult to solve this equation analytically and get informationfor general operators
Bootstrap equation
![Page 18: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/18.jpg)
Difficult to solve this equation analytically and get informationfor general operators
Simplifies in a certain limit...large spin limit and for double traceoperators
Bootstrap equation
Fitzpatrick-Kaplan-Poland-Simmons-Duffin, Komargodski-Zhiboedov
![Page 19: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/19.jpg)
Difficult to solve this equation analytically and get informationfor general operators
Simplifies in a certain limit...large spin limit and for double traceoperators
Bootstrap equation
Fitzpatrick-Kaplan-Poland-Simmons-Duffin, Komargodski-Zhiboedov
![Page 20: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/20.jpg)
Difficult to solve this equation analytically and get informationfor general operators
Simplifies in a certain limit...large spin limit and for double traceoperators
Bootstrap equation
Fitzpatrick-Kaplan-Poland-Simmons-Duffin, Komargodski-Zhiboedov
![Page 21: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/21.jpg)
Large spin bootstrap
Leading term
![Page 22: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/22.jpg)
The r.h.s. diverges as v ~ 0.
Large spin bootstrap
Leading term
![Page 23: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/23.jpg)
The r.h.s. diverges as v ~ 0.
Each term on the l.h.s. goes as
Large spin bootstrap
Leading term
![Page 24: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/24.jpg)
The r.h.s. diverges as v ~ 0.
Each term on the l.h.s. goes as
Need to sum over infinite large spin double trace operators
Large spin bootstrap
Leading term
![Page 25: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/25.jpg)
The r.h.s. diverges as v ~ 0.
Each term on the l.h.s. goes as
Need to sum over infinite large spin double trace operators on the left to reproduce the divergence on the r.h.s.
Large spin bootstrap
Leading term
![Page 26: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/26.jpg)
= +
![Page 27: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/27.jpg)
Double trace operators
Large spin bootstrap
![Page 28: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/28.jpg)
The Unknowns
Anomalous dimension
OPE coefficient
Double trace operators
Large spin bootstrap
![Page 29: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/29.jpg)
The Unknowns
Anomalous dimension
OPE coefficient
for simplicity
Double trace operators
Large spin bootstrap
![Page 30: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/30.jpg)
Subleading term of the bootstrap equation
Large spin bootstrap
![Page 31: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/31.jpg)
Subleading term of the bootstrap equation
Large spin bootstrap
Small v behavior on the rhs : power of v is controlled by the twist
![Page 32: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/32.jpg)
Subleading term of the bootstrap equation
Large spin bootstrap
Small v behavior on the rhs : power of v is controlled by the twist
We focus on the minimal twist operators on the rhs
![Page 33: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/33.jpg)
Subleading term of the bootstrap equation
twist spin OPE coeff
Large spin bootstrap
Small v behavior on the rhs : power of v is controlled by the twist
We focus on the minimal twist operators on the rhs
![Page 34: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/34.jpg)
Subleading term of the bootstrap equation
twist spin OPE coeff
Large spin bootstrap
Small v behavior on the rhs : power of v is controlled by the twist
We focus on the minimal twist operators on the rhs
Expanding the lhs in small u we get
![Page 35: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/35.jpg)
Subleading term of the bootstrap equation
twist spin OPE coeff
Large spin bootstrap
Small v behavior on the rhs : power of v is controlled by the twist
We focus on the minimal twist operators on the rhs
Expanding the lhs in small u we get
![Page 36: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/36.jpg)
Subleading term of the bootstrap equation
twist spin OPE coeff
Large spin bootstrap
Small v behavior on the rhs : power of v is controlled by the twist
We focus on the minimal twist operators on the rhs
Expanding the lhs in small u we get
![Page 37: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/37.jpg)
Subleading term of the bootstrap equation
twist spin OPE coeff
Large spin bootstrap
Small v behavior on the rhs : power of v is controlled by the twist
We focus on the minimal twist operators on the rhs
Expanding the lhs in small u we get
![Page 38: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/38.jpg)
Coefficient of Log u term on the lhs involves the anomalous dimension
Subleading term of the bootstrap equation
![Page 39: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/39.jpg)
Coefficient of Log u term on the lhs involves the anomalous dimension
Subleading term of the bootstrap equation
Assume the following expansion of the anomalous dimension in large spin limit
![Page 40: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/40.jpg)
Coefficient of Log u term on the lhs involves the anomalous dimension
Known Mean field OPE coefficient
Subleading term of the bootstrap equation
Assume the following expansion of the anomalous dimension in large spin limit
![Page 41: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/41.jpg)
Coefficient of Log u term on the lhs involves the anomalous dimension
Known Mean field OPE coefficient
Subleading term of the bootstrap equation
Assume the following expansion of the anomalous dimension in large spin limit
Large spin sum on the lhs can be done at the leading order by converting
![Page 42: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/42.jpg)
Summing over
Subleading term of the bootstrap equation
![Page 43: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/43.jpg)
Summing over
Subleading term of the bootstrap equation
We have from the lhs
![Page 44: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/44.jpg)
Summing over
Focus on the logu term from the rhs
Subleading term of the bootstrap equation
We have from the lhs
![Page 45: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/45.jpg)
Summing over
Focus on the logu term from the rhs
Compare the log u term to get the anomalous dimension
Subleading term of the bootstrap equation
We have from the lhs
![Page 46: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/46.jpg)
Summing over
Focus on the logu term from the rhs
Compare the log u term to get the anomalous dimension
Subleading term of the bootstrap equation
We have from the lhs
Similar analysis for non log terms gives the OPE coefficients
![Page 47: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/47.jpg)
Alday-Zhiboedov (2015)
Large spin bootstrap
Subleading order corrections can be done following
![Page 48: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/48.jpg)
Alday-Zhiboedov (2015)
Change of variable
Large spin bootstrap
Subleading order corrections can be done following
![Page 49: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/49.jpg)
Alday-Zhiboedov (2015)
Anomalous dimension: asymptotic expansion in inverse J
Change of variable
Large spin bootstrap
Subleading order corrections can be done following
![Page 50: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/50.jpg)
Alday-Zhiboedov (2015)
Anomalous dimension: asymptotic expansion in inverse J
Involves two recursion relations that one needs to solve
Change of variable
Large spin bootstrap
Subleading order corrections can be done following
![Page 51: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/51.jpg)
Alday-Zhiboedov (2015)
Anomalous dimension: asymptotic expansion in inverse J
Involves two recursion relations that one needs to solve
Simplifies in Mellin space...gives an all order expression !
Change of variable
Large spin bootstrap
Subleading order corrections can be done following
![Page 52: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/52.jpg)
Conformal bootstrap in Mellin space
![Page 53: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/53.jpg)
Mellin transform
Inverse Mellin transform
![Page 54: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/54.jpg)
Mellin transform
Inverse Mellin transform
![Page 55: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/55.jpg)
Mellin transform
Inverse Mellin transform
Scaling behavior of CFT correlation function
![Page 56: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/56.jpg)
Mellin transform
Inverse Mellin transform
Scaling behavior of CFT correlation function
Mellin representation captures conformal symmetry automatically
![Page 57: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/57.jpg)
The Mellin amplitude
Mack (2009), Penedones, Costa-Goncalves- Penedones , Fitzpatrick-Kaplan- Penedones-Raju, van Rees ....
![Page 58: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/58.jpg)
The Mellin amplitude
Mack (2009), Penedones, Costa-Goncalves- Penedones , Fitzpatrick-Kaplan- Penedones-Raju, van Rees ....
![Page 59: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/59.jpg)
Mellin transform of the 4-pt function
The Mellin amplitude
Mack (2009), Penedones, Costa-Goncalves- Penedones , Fitzpatrick-Kaplan- Penedones-Raju, van Rees ....
![Page 60: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/60.jpg)
Mellin transform of the 4-pt function
The Mellin amplitude
Mack (2009), Penedones, Costa-Goncalves- Penedones , Fitzpatrick-Kaplan- Penedones-Raju, van Rees ....
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Mellin transform of the 4-pt function
Mellin amplitude
The Mellin amplitude
Mack (2009), Penedones, Costa-Goncalves- Penedones , Fitzpatrick-Kaplan- Penedones-Raju, van Rees ....
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Properties of Mellin amplitude
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Properties of Mellin amplitude
● Mellin amplitude is a meromorphic functin of (s,t)
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Properties of Mellin amplitude
● Mellin amplitude is a meromorphic functin of (s,t)
● Poles correspond to the primary operators exchanged in
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Properties of Mellin amplitude
● Mellin amplitude is a meromorphic functin of (s,t)
● Poles correspond to the primary operators exchanged in the intermediate states
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Properties of Mellin amplitude
● Mellin amplitude is a meromorphic functin of (s,t)
● Poles correspond to the primary operators exchanged in the intermediate states
● Residues tell us about the OPE coefficients
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Properties of Mellin amplitude
● Mellin amplitude is a meromorphic functin of (s,t)
● Poles correspond to the primary operators exchanged in the intermediate states
● Residues tell us about the OPE coefficients
● Channel dualities are manifest as s, t exchange
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Mellin transform of s channel conformal block
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Mellin transform of s channel conformal block
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Mellin transform of s channel conformal block
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Mellin transform of s channel conformal block
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Expand in small anm dim
Mellin transform of s channel conformal block
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Expand in small anm dim
Mellin transform of s channel conformal block
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Mellin transform of s channel conformal block
Residue at double pole
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Mellin transform of s channel conformal block
Residue at double pole
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Continuous Hahn polynomial
Mellin transform of s channel conformal block
Residue at double pole
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Continuous Hahn polynomial
Mellin transform of s channel conformal block
Residue at double pole
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Continuous Hahn polynomial
These are orthogonal polynomials
Mellin transform of s channel conformal block
Residue at double pole
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Continuous Hahn polynomial
These are orthogonal polynomials
Mellin transform of s channel conformal block
Residue at double pole
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Continuous Hahn polynomial
These are orthogonal polynomials
Mellin transform of s channel conformal block
Play a key role in repackaging the equations in Mellin space
Residue at double pole
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Residue at
Mellin transform of s channel conformal block
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Residue at
Mellin transform of s channel conformal block
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Residue at
Mellin transform of s channel conformal block
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Residue at
where
Mellin transform of s channel conformal block
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Strategy in Mellin space
Mellin transform of the t channel can be obtained from the s channel by replacing
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Idea now is to look at the Mellin transform of the t channel
Strategy in Mellin space
Mellin transform of the t channel can be obtained from the s channel by replacing
![Page 87: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/87.jpg)
Idea now is to look at the Mellin transform of the t channel conformal block and expand the t-dependence in terms of
Strategy in Mellin space
Mellin transform of the t channel can be obtained from the s channel by replacing
![Page 88: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/88.jpg)
Idea now is to look at the Mellin transform of the t channel conformal block and expand the t-dependence in terms of the continuous Hahn polynomials
Strategy in Mellin space
Mellin transform of the t channel can be obtained from the s channel by replacing
![Page 89: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/89.jpg)
Idea now is to look at the Mellin transform of the t channel conformal block and expand the t-dependence in terms of the continuous Hahn polynomials
Strategy in Mellin space
We will decompose the bootstrap equation into the basis of
Mellin transform of the t channel can be obtained from the s channel by replacing
![Page 90: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/90.jpg)
Idea now is to look at the Mellin transform of the t channel conformal block and expand the t-dependence in terms of the continuous Hahn polynomials
Strategy in Mellin space
We will decompose the bootstrap equation into the basis of continuous Hahn polynomials
Mellin transform of the t channel can be obtained from the s channel by replacing
![Page 91: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/91.jpg)
Idea now is to look at the Mellin transform of the t channel conformal block and expand the t-dependence in terms of the continuous Hahn polynomials
Boils down to simple equations in Mellin space
Strategy in Mellin space
We will decompose the bootstrap equation into the basis of continuous Hahn polynomials
Mellin transform of the t channel can be obtained from the s channel by replacing
![Page 92: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/92.jpg)
Mack polynomial, polynomial in s, t
Mellin transform of t channel conformal block
Minimal twist operator
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Mack polynomial, polynomial in s, t
Mellin transform of t channel conformal block
Minimal twist operator
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Mack polynomial, polynomial in s, t
Evaluate the residue at
Mellin transform of t channel conformal block
Minimal twist operator
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Focus on the log u term
Mellin transform of t channel conformal block
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Focus on the log u term
Re express the t dependence in terms of Q(t) polynomials.
Mellin transform of t channel conformal block
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t integral has pole at
Mellin transform of t channel conformal block
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Bootstrap equation in Mellin space
Algebraic eq !!
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Bootstrap equation in Mellin space
Algebraic eq !!
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Bootstrap equation in Mellin space
Algebraic eq !!
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Bootstrap equation in Mellin space
Algebraic eq !!
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Bootstrap equation in Mellin space
Mack polynomial
Algebraic eq !!
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Bootstrap equation in Mellin space
Mack polynomial
Need the large spin behavior of continuous Hahn polynoimal.
Algebraic eq !!
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Asymptotics of continuous Hahn polynomial
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In Mellin space approach the large spin behavior of continuous Hahn
Asymptotics of continuous Hahn polynomial
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In Mellin space approach the large spin behavior of continuous Hahnpolynomial is the key ingredient
Asymptotics of continuous Hahn polynomial
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In Mellin space approach the large spin behavior of continuous Hahnpolynomial is the key ingredient
These polynomials can be derived from Wilson polynomials in
Asymptotics of continuous Hahn polynomial
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In Mellin space approach the large spin behavior of continuous Hahnpolynomial is the key ingredient
These polynomials can be derived from Wilson polynomials in a particular limit
Asymptotics of continuous Hahn polynomial
![Page 109: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/109.jpg)
In Mellin space approach the large spin behavior of continuous Hahnpolynomial is the key ingredient
These polynomials can be derived from Wilson polynomials in a particular limit
Wilson in 1991 worked out the large argument asymptotics of the Wilson polynomial
James Wilson 1991
Asymptotics of continuous Hahn polynomial
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In Mellin space approach the large spin behavior of continuous Hahnpolynomial is the key ingredient
These polynomials can be derived from Wilson polynomials in a particular limit
Wilson in 1991 worked out the large argument asymptotics of the Wilson polynomial
Use these results to derive the large spin behavior of continuous
James Wilson 1991
Asymptotics of continuous Hahn polynomial
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In Mellin space approach the large spin behavior of continuous Hahnpolynomial is the key ingredient
These polynomials can be derived from Wilson polynomials in a particular limit
Wilson in 1991 worked out the large argument asymptotics of the Wilson polynomial
Use these results to derive the large spin behavior of continuous Hahn polynomial
James Wilson 1991
Asymptotics of continuous Hahn polynomial
![Page 112: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/112.jpg)
Asymptotics of continuous Hahn polynomial
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Asymptotics of continuous Hahn polynomial
J. L.Fields
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Asymptotics of continuous Hahn polynomial
PD, K. Ghosh, A. Sinha
![Page 115: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/115.jpg)
Asymptotics of continuous Hahn polynomial
Given in terms of generalised Bernoulli polynomial
PD, K. Ghosh, A. Sinha
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Asymptotics of continuous Hahn polynomial
Given in terms of generalised Bernoulli polynomial
Only even powers of J will appear
PD, K. Ghosh, A. Sinha
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In the large spin limit in s channel
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In the large spin limit in s channel
From the t channel we have the large spin behavior of the
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In the large spin limit in s channel
From the t channel we have the large spin behavior of the continuous Hahn polynomial
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In the large spin limit
Comparing powers of J from both sides of the bootstrap equation
in s channel
From the t channel we have the large spin behavior of the continuous Hahn polynomial
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In the large spin limit
Comparing powers of J from both sides of the bootstrap equation will fix the anomalous dimension at all orders in inverse J
in s channel
From the t channel we have the large spin behavior of the continuous Hahn polynomial
![Page 122: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/122.jpg)
Result: Anomalous dimension
Expression for anomalous dimension to all orders in 1/J
![Page 123: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/123.jpg)
Result: Anomalous dimension
Expression for anomalous dimension to all orders in 1/J
e.g. i = 0
![Page 124: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/124.jpg)
Similar expression for OPE coefficient comes from the non log term
Result: OPE coefficient
Given in terms of generalised Bernoulli polynomial and Mack polynomial
![Page 125: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/125.jpg)
Epsilon expansion in the large spin limit
![Page 126: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/126.jpg)
Critical exponents via Epsilon expansion...
![Page 127: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/127.jpg)
● based on Feynman diagrams
Critical exponents via Epsilon expansion...
![Page 128: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/128.jpg)
● based on Feynman diagrams
● Locate fixed point for which the beta function vanishes
Critical exponents via Epsilon expansion...
![Page 129: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/129.jpg)
● based on Feynman diagrams
● Locate fixed point for which the beta function vanishes
● Use Callan-Symanzik equation to determine the anomalous dimension
Critical exponents via Epsilon expansion...
![Page 130: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/130.jpg)
● based on Feynman diagrams
● Locate fixed point for which the beta function vanishes
● Use Callan-Symanzik equation to determine the anomalous dimension
Critical exponents via Epsilon expansion...
![Page 131: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/131.jpg)
Critical exponents via Epsilon expansion...
![Page 132: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/132.jpg)
Critical exponents via Epsilon expansion...
![Page 133: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/133.jpg)
Critical exponents via Epsilon expansion...
![Page 134: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/134.jpg)
Critical exponents via Epsilon expansion...
Known up to epsilon^5
![Page 135: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/135.jpg)
Critical exponents via Epsilon expansion...
Known up to epsilon^5 Kleinert et al
![Page 136: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/136.jpg)
Critical exponents via Epsilon expansion...
Known up to epsilon^5 Kleinert et al
d=3, N=1
![Page 137: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/137.jpg)
Computing Δ for ϕ2 till 2 loops, in Wilson Fisher fixed point CFT
![Page 138: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/138.jpg)
Computing Δ for ϕ2 till 2 loops, in Wilson Fisher fixed point CFT
Computing till 3 loops involves many more diagrams
![Page 139: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/139.jpg)
Computing Δ for ϕ2 till 2 loops, in Wilson Fisher fixed point CFT
Computing till 3 loops involves many more diagrams
● OPE coefficients are even more difficult to compute:
![Page 140: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/140.jpg)
Computing Δ for ϕ2 till 2 loops, in Wilson Fisher fixed point CFT
Computing till 3 loops involves many more diagrams
● OPE coefficients are even more difficult to compute: involves three point functions
![Page 141: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/141.jpg)
Can bootstrap help?
Results from large spin bootstrap can be used to determine the operator dimension of in the large spin limit
![Page 142: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/142.jpg)
Can bootstrap help?
Results from large spin bootstrap can be used to determine the operator dimension of in the large spin limit
![Page 143: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/143.jpg)
Can bootstrap help?
Results from large spin bootstrap can be used to determine the operator dimension of in the large spin limit
Remember
![Page 144: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/144.jpg)
Can bootstrap help?
Results from large spin bootstrap can be used to determine the operator dimension of in the large spin limit
Remember
![Page 145: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/145.jpg)
Can bootstrap help?
Results from large spin bootstrap can be used to determine the operator dimension of in the large spin limit
Remember
There are multiple operators with the same minimal twist
![Page 146: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/146.jpg)
Can bootstrap help?
Results from large spin bootstrap can be used to determine the operator dimension of in the large spin limit
Remember
There are multiple operators with the same minimal twist
![Page 147: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/147.jpg)
Can bootstrap help?
Results from large spin bootstrap can be used to determine the operator dimension of in the large spin limit
Remember
There are multiple operators with the same minimal twist
![Page 148: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/148.jpg)
Can bootstrap help?
Results from large spin bootstrap can be used to determine the operator dimension of in the large spin limit
Remember
There are multiple operators with the same minimal twist
Need to sum over
![Page 149: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/149.jpg)
Strategy for epsilon expansion
Step 1. Expand the dimension and OPE in epsilon, OPE coeff can be obtained from Mellin Bootstrap Gopakumar-Kaviraj-Sen- Sinha
![Page 150: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/150.jpg)
Strategy for epsilon expansion
Step 1. Expand the dimension and OPE in epsilon, OPE coeff can be obtained from Mellin Bootstrap Gopakumar-Kaviraj-Sen- Sinha
![Page 151: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/151.jpg)
Strategy for epsilon expansion
Step 1. Expand the dimension and OPE in epsilon, OPE coeff can be obtained from Mellin Bootstrap Gopakumar-Kaviraj-Sen- Sinha
![Page 152: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/152.jpg)
Strategy for epsilon expansion
Step 1. Expand the dimension and OPE in epsilon, OPE coeff can be obtained from Mellin Bootstrap Gopakumar-Kaviraj-Sen- Sinha
![Page 153: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/153.jpg)
Strategy for epsilon expansion
Step 1. Expand the dimension and OPE in epsilon, OPE coeff can be obtained from Mellin Bootstrap Gopakumar-Kaviraj-Sen- Sinha
![Page 154: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/154.jpg)
Strategy for epsilon expansion
Step 1. Expand the dimension and OPE in epsilon, OPE coeff can be obtained from Mellin Bootstrap Gopakumar-Kaviraj-Sen- Sinha
![Page 155: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/155.jpg)
Strategy for epsilon expansion
![Page 156: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/156.jpg)
Step 2: Use the known results as input
Kleinert et al.Gracey,Manashov et al.
Gopakumar-Kaviraj-Sen- Sinha
Strategy for epsilon expansion
![Page 157: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/157.jpg)
Step 2: Use the known results as input
Step 3: Do the sum over
Compute the anomalous dimension of higher spin operators at e^5 in the large spin limit
Kleinert et al.Gracey,Manashov et al.
Gopakumar-Kaviraj-Sen- Sinha
Strategy for epsilon expansion
![Page 158: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/158.jpg)
Sampling of new result: Large spin anm. dimensionPD, A. KavirajJHEP1802(2018)153
![Page 159: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/159.jpg)
Subsequent order of anomalous dimension can also be computed using
![Page 160: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/160.jpg)
Subsequent order of anomalous dimension can also be computed using
This gives the anomalous dimension at the order
![Page 161: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/161.jpg)
Subsequent order of anomalous dimension can also be computed using
This gives the anomalous dimension at the order
Similarly, the OPE coefficients can also be computed using
![Page 162: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/162.jpg)
Subsequent order of anomalous dimension can also be computed using
This gives the anomalous dimension at the order
Similarly, the OPE coefficients can also be computed using the large spin OPE results
![Page 163: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/163.jpg)
Summary
![Page 164: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/164.jpg)
Summary
Mellin space techniques can be used to obtain the
![Page 165: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/165.jpg)
Summary
Mellin space techniques can be used to obtain the CFT data for large spin double trace operators to all orders in
![Page 166: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/166.jpg)
Summary
Mellin space techniques can be used to obtain the CFT data for large spin double trace operators to all orders in inverse conformal spin
![Page 167: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/167.jpg)
Summary
Mellin space techniques can be used to obtain the CFT data for large spin double trace operators to all orders in inverse conformal spin
Epsilon expansion in the large spin limit
![Page 168: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/168.jpg)
Summary
Mellin space techniques can be used to obtain the CFT data for large spin double trace operators to all orders in inverse conformal spin
Can be extended for higher twist operators
Epsilon expansion in the large spin limit
![Page 169: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/169.jpg)
Summary
Mellin space techniques can be used to obtain the CFT data for large spin double trace operators to all orders in inverse conformal spin
Can be extended for higher twist operators
Mixed correlator, external operators with spin, supersymmetry
Epsilon expansion in the large spin limit
![Page 170: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/170.jpg)
Summary
Mellin space techniques can be used to obtain the CFT data for large spin double trace operators to all orders in inverse conformal spin
Relation between the usual bootstrap and Mellin bootstrap
Can be extended for higher twist operators
Mixed correlator, external operators with spin, supersymmetry
Epsilon expansion in the large spin limit
![Page 171: Large spin analytic bootstrap via Mellin...Large spin analytic bootstrap via Mellin Parijat Dey CHEP, Indian Institute of Science, Based on JHEP 1801(2018) 152 with Kausik Ghosh and](https://reader033.vdocuments.mx/reader033/viewer/2022041805/5e53e8abdc58ba5c27430093/html5/thumbnails/171.jpg)
Thank you