March 2020

1

Accessing the QCD Conformal Window with Perturbation Theory and Beyond

Marco Serone, SISSA, Trieste

based on 2003.01742, in collaboration with Lorenzo Di Pietro (Univ. of Trieste)

Understanding the phases of gauge theories is one of the most interesting problems in high energy and condensed matter physics

2

Depending on the number of flavours and colors, QCD can be UV-free or not, and flow in the IR to a gapped phase (confining and/or

symmetry breaking) or to a gapless (Conformal Field Theory) phase

UV

IR Gapped or gapless

Introduction and Motivation

We focus on non-abelian gauge theories in d=4 dimensions with fermions in the fundamental representation of , which we will denote as QCDSU(nc)

3

Conformal PhaseGapped Phase

0<latexit sha1_base64="+kYPchScNyik1n1ulpTEzE1fdeA=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqHgqePHYgv2ANpTNdtKu3WzC7kYoob/AiwdFvPqTvPlv3LY5aOuDgcd7M8zMCxLBtXHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFcMWi0WsugHVKLjEluFGYDdRSKNAYCeY3M39zhMqzWP5YKYJ+hEdSR5yRo2Vmu6gXHGr7gJknXg5qUCOxqD81R/GLI1QGiao1j3PTYyfUWU4Ezgr9VONCWUTOsKepZJGqP1sceiMXFhlSMJY2ZKGLNTfExmNtJ5Gge2MqBnrVW8u/uf1UhPe+BmXSWpQsuWiMBXExGT+NRlyhcyIqSWUKW5vJWxMFWXGZlOyIXirL6+T9lXVc6te87pSv83jKMIZnMMleFCDOtxDA1rAAOEZXuHNeXRenHfnY9lacPKZU/gD5/MHdrGMrA==</latexit><latexit sha1_base64="+kYPchScNyik1n1ulpTEzE1fdeA=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqHgqePHYgv2ANpTNdtKu3WzC7kYoob/AiwdFvPqTvPlv3LY5aOuDgcd7M8zMCxLBtXHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFcMWi0WsugHVKLjEluFGYDdRSKNAYCeY3M39zhMqzWP5YKYJ+hEdSR5yRo2Vmu6gXHGr7gJknXg5qUCOxqD81R/GLI1QGiao1j3PTYyfUWU4Ezgr9VONCWUTOsKepZJGqP1sceiMXFhlSMJY2ZKGLNTfExmNtJ5Gge2MqBnrVW8u/uf1UhPe+BmXSWpQsuWiMBXExGT+NRlyhcyIqSWUKW5vJWxMFWXGZlOyIXirL6+T9lXVc6te87pSv83jKMIZnMMleFCDOtxDA1rAAOEZXuHNeXRenHfnY9lacPKZU/gD5/MHdrGMrA==</latexit><latexit sha1_base64="+kYPchScNyik1n1ulpTEzE1fdeA=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqHgqePHYgv2ANpTNdtKu3WzC7kYoob/AiwdFvPqTvPlv3LY5aOuDgcd7M8zMCxLBtXHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFcMWi0WsugHVKLjEluFGYDdRSKNAYCeY3M39zhMqzWP5YKYJ+hEdSR5yRo2Vmu6gXHGr7gJknXg5qUCOxqD81R/GLI1QGiao1j3PTYyfUWU4Ezgr9VONCWUTOsKepZJGqP1sceiMXFhlSMJY2ZKGLNTfExmNtJ5Gge2MqBnrVW8u/uf1UhPe+BmXSWpQsuWiMBXExGT+NRlyhcyIqSWUKW5vJWxMFWXGZlOyIXirL6+T9lXVc6te87pSv83jKMIZnMMleFCDOtxDA1rAAOEZXuHNeXRenHfnY9lacPKZU/gD5/MHdrGMrA==</latexit><latexit sha1_base64="+kYPchScNyik1n1ulpTEzE1fdeA=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqHgqePHYgv2ANpTNdtKu3WzC7kYoob/AiwdFvPqTvPlv3LY5aOuDgcd7M8zMCxLBtXHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFcMWi0WsugHVKLjEluFGYDdRSKNAYCeY3M39zhMqzWP5YKYJ+hEdSR5yRo2Vmu6gXHGr7gJknXg5qUCOxqD81R/GLI1QGiao1j3PTYyfUWU4Ezgr9VONCWUTOsKepZJGqP1sceiMXFhlSMJY2ZKGLNTfExmNtJ5Gge2MqBnrVW8u/uf1UhPe+BmXSWpQsuWiMBXExGT+NRlyhcyIqSWUKW5vJWxMFWXGZlOyIXirL6+T9lXVc6te87pSv83jKMIZnMMleFCDOtxDA1rAAOEZXuHNeXRenHfnY9lacPKZU/gD5/MHdrGMrA==</latexit>

UV Freedom Lost

Conformal Window

Strongly coupled Weakly coupled

n+f

nfn⇤f

At fixed nc (e.g. nc = 3)

n+f =

33

2n⇤f =?

4

Our aim is to start from the upper edge of the conformal window and go down as much as possible using perturbation theory

A way to detect a conformal behaviour is to look for fixed points of the beta-function of the theory

We will be using Borel resummation techniques and other tools.

A large part of the talk will be a review of the main properties of the methods used, in particular about

asymptotic series and their possible Borel resummation

Plan

Conclusions

5

Asymptotic series and Borel resummations

Go back to the QCD conformal window

Large order behaviour: instantons and renormalonsNature of the series of RG functions in

4d non-abelian gauge theories

Results

Asymptotic Series and Borel Resummations

6

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f(�) =1X

n=0

cn(�� �0)n

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R = limn!1

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cn+1

����<latexit sha1_base64="/c5/VhBE5ycW1UcrXH25XuKJ/08=">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</latexit><latexit sha1_base64="/c5/VhBE5ycW1UcrXH25XuKJ/08=">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</latexit><latexit sha1_base64="/c5/VhBE5ycW1UcrXH25XuKJ/08=">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</latexit><latexit sha1_base64="/c5/VhBE5ycW1UcrXH25XuKJ/08=">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</latexit>

Radius of convergence<latexit sha1_base64="hY7QWdXUIBcFpyPqqtdybSNdOn0=">AAAB/nicbVDLSgMxFM3UV62vUXHlJlgEV2WmG11JwY3LKvYB7VAymTttaCYZkoxQhoK/4saFIm79Dnf+jWk7C209EDiccy6594QpZ9p43rdTWlvf2Nwqb1d2dvf2D9zDo7aWmaLQopJL1Q2JBs4EtAwzHLqpApKEHDrh+Gbmdx5BaSbFg5mkECRkKFjMKDFWGrgn9yRimcYyxlQKmxyCoDBwq17NmwOvEr8gVVSgOXC/+pGkWQLCUE607vleaoKcKMMoh2mln2lICR2TIfQsFSQBHeTz9af43CoRjqWyTxg8V39P5CTRepKENpkQM9LL3kz8z+tlJr4KcibSzNirFh/FGcdG4lkXOGIKqOETSwhVzO6K6YgoQo1trGJL8JdPXiXtes33av5dvdq4Luooo1N0hi6Qjy5RA92iJmohinL0jF7Rm/PkvDjvzsciWnKKmWP0B87nDwullYM=</latexit><latexit sha1_base64="hY7QWdXUIBcFpyPqqtdybSNdOn0=">AAAB/nicbVDLSgMxFM3UV62vUXHlJlgEV2WmG11JwY3LKvYB7VAymTttaCYZkoxQhoK/4saFIm79Dnf+jWk7C209EDiccy6594QpZ9p43rdTWlvf2Nwqb1d2dvf2D9zDo7aWmaLQopJL1Q2JBs4EtAwzHLqpApKEHDrh+Gbmdx5BaSbFg5mkECRkKFjMKDFWGrgn9yRimcYyxlQKmxyCoDBwq17NmwOvEr8gVVSgOXC/+pGkWQLCUE607vleaoKcKMMoh2mln2lICR2TIfQsFSQBHeTz9af43CoRjqWyTxg8V39P5CTRepKENpkQM9LL3kz8z+tlJr4KcibSzNirFh/FGcdG4lkXOGIKqOETSwhVzO6K6YgoQo1trGJL8JdPXiXtes33av5dvdq4Luooo1N0hi6Qjy5RA92iJmohinL0jF7Rm/PkvDjvzsciWnKKmWP0B87nDwullYM=</latexit><latexit sha1_base64="hY7QWdXUIBcFpyPqqtdybSNdOn0=">AAAB/nicbVDLSgMxFM3UV62vUXHlJlgEV2WmG11JwY3LKvYB7VAymTttaCYZkoxQhoK/4saFIm79Dnf+jWk7C209EDiccy6594QpZ9p43rdTWlvf2Nwqb1d2dvf2D9zDo7aWmaLQopJL1Q2JBs4EtAwzHLqpApKEHDrh+Gbmdx5BaSbFg5mkECRkKFjMKDFWGrgn9yRimcYyxlQKmxyCoDBwq17NmwOvEr8gVVSgOXC/+pGkWQLCUE607vleaoKcKMMoh2mln2lICR2TIfQsFSQBHeTz9af43CoRjqWyTxg8V39P5CTRepKENpkQM9LL3kz8z+tlJr4KcibSzNirFh/FGcdG4lkXOGIKqOETSwhVzO6K6YgoQo1trGJL8JdPXiXtes33av5dvdq4Luooo1N0hi6Qjy5RA92iJmohinL0jF7Rm/PkvDjvzsciWnKKmWP0B87nDwullYM=</latexit><latexit sha1_base64="hY7QWdXUIBcFpyPqqtdybSNdOn0=">AAAB/nicbVDLSgMxFM3UV62vUXHlJlgEV2WmG11JwY3LKvYB7VAymTttaCYZkoxQhoK/4saFIm79Dnf+jWk7C209EDiccy6594QpZ9p43rdTWlvf2Nwqb1d2dvf2D9zDo7aWmaLQopJL1Q2JBs4EtAwzHLqpApKEHDrh+Gbmdx5BaSbFg5mkECRkKFjMKDFWGrgn9yRimcYyxlQKmxyCoDBwq17NmwOvEr8gVVSgOXC/+pGkWQLCUE607vleaoKcKMMoh2mln2lICR2TIfQsFSQBHeTz9af43CoRjqWyTxg8V39P5CTRepKENpkQM9LL3kz8z+tlJr4KcibSzNirFh/FGcdG4lkXOGIKqOETSwhVzO6K6YgoQo1trGJL8JdPXiXtes33av5dvdq4Luooo1N0hi6Qjy5RA92iJmohinL0jF7Rm/PkvDjvzsciWnKKmWP0B87nDwullYM=</latexit>

If �0 regular point of f<latexit sha1_base64="BPRrKaTyKzwjdJoUEYc62rzTNCU=">AAACGHicbVA9SwNBEJ3zM8avU0ubxUSwindptJKAjXYRzAfkQtjb20uW7O0eu3tCCPkZNv4VGwtFbNP5b9wkV2jig4HHezPMzAtTzrTxvG9nbX1jc2u7sFPc3ds/OHSPjptaZorQBpFcqnaINeVM0IZhhtN2qihOQk5b4fB25reeqNJMikczSmk3wX3BYkawsVLPvQyEZCKiwqD7GJUDbkcj3PPKSNF+xrFCqfUNktaMywj13JJX8eZAq8TPSQly1HvuNIgkyRK7gXCsdcf3UtMdY2UY4XRSDDJNU0yGuE87lgqcUN0dzx+boHOrRCiWypY9Yq7+nhjjROtREtrOBJuBXvZm4n9eJzPxdXfMRJoZKshiUZxxZCSapYQipigxfGQJJorZWxEZYIWJsVkWbQj+8surpFmt+F7Ff6iWajd5HAU4hTO4AB+uoAZ3UIcGEHiGV3iHD+fFeXM+na9F65qTz5zAHzjTH3wOnhg=</latexit><latexit sha1_base64="BPRrKaTyKzwjdJoUEYc62rzTNCU=">AAACGHicbVA9SwNBEJ3zM8avU0ubxUSwindptJKAjXYRzAfkQtjb20uW7O0eu3tCCPkZNv4VGwtFbNP5b9wkV2jig4HHezPMzAtTzrTxvG9nbX1jc2u7sFPc3ds/OHSPjptaZorQBpFcqnaINeVM0IZhhtN2qihOQk5b4fB25reeqNJMikczSmk3wX3BYkawsVLPvQyEZCKiwqD7GJUDbkcj3PPKSNF+xrFCqfUNktaMywj13JJX8eZAq8TPSQly1HvuNIgkyRK7gXCsdcf3UtMdY2UY4XRSDDJNU0yGuE87lgqcUN0dzx+boHOrRCiWypY9Yq7+nhjjROtREtrOBJuBXvZm4n9eJzPxdXfMRJoZKshiUZxxZCSapYQipigxfGQJJorZWxEZYIWJsVkWbQj+8surpFmt+F7Ff6iWajd5HAU4hTO4AB+uoAZ3UIcGEHiGV3iHD+fFeXM+na9F65qTz5zAHzjTH3wOnhg=</latexit><latexit sha1_base64="BPRrKaTyKzwjdJoUEYc62rzTNCU=">AAACGHicbVA9SwNBEJ3zM8avU0ubxUSwindptJKAjXYRzAfkQtjb20uW7O0eu3tCCPkZNv4VGwtFbNP5b9wkV2jig4HHezPMzAtTzrTxvG9nbX1jc2u7sFPc3ds/OHSPjptaZorQBpFcqnaINeVM0IZhhtN2qihOQk5b4fB25reeqNJMikczSmk3wX3BYkawsVLPvQyEZCKiwqD7GJUDbkcj3PPKSNF+xrFCqfUNktaMywj13JJX8eZAq8TPSQly1HvuNIgkyRK7gXCsdcf3UtMdY2UY4XRSDDJNU0yGuE87lgqcUN0dzx+boHOrRCiWypY9Yq7+nhjjROtREtrOBJuBXvZm4n9eJzPxdXfMRJoZKshiUZxxZCSapYQipigxfGQJJorZWxEZYIWJsVkWbQj+8surpFmt+F7Ff6iWajd5HAU4hTO4AB+uoAZ3UIcGEHiGV3iHD+fFeXM+na9F65qTz5zAHzjTH3wOnhg=</latexit><latexit sha1_base64="BPRrKaTyKzwjdJoUEYc62rzTNCU=">AAACGHicbVA9SwNBEJ3zM8avU0ubxUSwindptJKAjXYRzAfkQtjb20uW7O0eu3tCCPkZNv4VGwtFbNP5b9wkV2jig4HHezPMzAtTzrTxvG9nbX1jc2u7sFPc3ds/OHSPjptaZorQBpFcqnaINeVM0IZhhtN2qihOQk5b4fB25reeqNJMikczSmk3wX3BYkawsVLPvQyEZCKiwqD7GJUDbkcj3PPKSNF+xrFCqfUNktaMywj13JJX8eZAq8TPSQly1HvuNIgkyRK7gXCsdcf3UtMdY2UY4XRSDDJNU0yGuE87lgqcUN0dzx+boHOrRCiWypY9Yq7+nhjjROtREtrOBJuBXvZm4n9eJzPxdXfMRJoZKshiUZxxZCSapYQipigxfGQJJorZWxEZYIWJsVkWbQj+8surpFmt+F7Ff6iWajd5HAU4hTO4AB+uoAZ3UIcGEHiGV3iHD+fFeXM+na9F65qTz5zAHzjTH3wOnhg=</latexit>

R 6= 0<latexit sha1_base64="gMaHUhcgp5P6k573XGOEeE6Ki/A=">AAAB/XicbVC7TsMwFHXKq5RXeGwsFi0SU5V0gQlVYmEsiD6kNqoc56a16jjBdpBKVPErLAwgxMp/sPE3uG0GaDmSpaNz7rWPj59wprTjfFuFldW19Y3iZmlre2d3z94/aKk4lRSaNOax7PhEAWcCmpppDp1EAol8Dm1/dDX12w8gFYvFnR4n4EVkIFjIKNFG6ttHPREzEYDQuHLbE3CPnQru22Wn6syAl4mbkzLK0ejbX70gpmlkrqGcKNV1nUR7GZGaUQ6TUi9VkBA6IgPoGipIBMrLZukn+NQoAQ5jaY6JMVN/b2QkUmoc+WYyInqoFr2p+J/XTXV44WVMJKkGQecPhSnHOsbTKnDAJFDNx4YQKpnJiumQSEK1KaxkSnAXv7xMWrWq61Tdm1q5fpnXUUTH6ASdIRedozq6Rg3URBQ9omf0it6sJ+vFerc+5qMFK985RH9gff4AjemT8g==</latexit><latexit sha1_base64="gMaHUhcgp5P6k573XGOEeE6Ki/A=">AAAB/XicbVC7TsMwFHXKq5RXeGwsFi0SU5V0gQlVYmEsiD6kNqoc56a16jjBdpBKVPErLAwgxMp/sPE3uG0GaDmSpaNz7rWPj59wprTjfFuFldW19Y3iZmlre2d3z94/aKk4lRSaNOax7PhEAWcCmpppDp1EAol8Dm1/dDX12w8gFYvFnR4n4EVkIFjIKNFG6ttHPREzEYDQuHLbE3CPnQru22Wn6syAl4mbkzLK0ejbX70gpmlkrqGcKNV1nUR7GZGaUQ6TUi9VkBA6IgPoGipIBMrLZukn+NQoAQ5jaY6JMVN/b2QkUmoc+WYyInqoFr2p+J/XTXV44WVMJKkGQecPhSnHOsbTKnDAJFDNx4YQKpnJiumQSEK1KaxkSnAXv7xMWrWq61Tdm1q5fpnXUUTH6ASdIRedozq6Rg3URBQ9omf0it6sJ+vFerc+5qMFK985RH9gff4AjemT8g==</latexit><latexit sha1_base64="gMaHUhcgp5P6k573XGOEeE6Ki/A=">AAAB/XicbVC7TsMwFHXKq5RXeGwsFi0SU5V0gQlVYmEsiD6kNqoc56a16jjBdpBKVPErLAwgxMp/sPE3uG0GaDmSpaNz7rWPj59wprTjfFuFldW19Y3iZmlre2d3z94/aKk4lRSaNOax7PhEAWcCmpppDp1EAol8Dm1/dDX12w8gFYvFnR4n4EVkIFjIKNFG6ttHPREzEYDQuHLbE3CPnQru22Wn6syAl4mbkzLK0ejbX70gpmlkrqGcKNV1nUR7GZGaUQ6TUi9VkBA6IgPoGipIBMrLZukn+NQoAQ5jaY6JMVN/b2QkUmoc+WYyInqoFr2p+J/XTXV44WVMJKkGQecPhSnHOsbTKnDAJFDNx4YQKpnJiumQSEK1KaxkSnAXv7xMWrWq61Tdm1q5fpnXUUTH6ASdIRedozq6Rg3URBQ9omf0it6sJ+vFerc+5qMFK985RH9gff4AjemT8g==</latexit><latexit sha1_base64="gMaHUhcgp5P6k573XGOEeE6Ki/A=">AAAB/XicbVC7TsMwFHXKq5RXeGwsFi0SU5V0gQlVYmEsiD6kNqoc56a16jjBdpBKVPErLAwgxMp/sPE3uG0GaDmSpaNz7rWPj59wprTjfFuFldW19Y3iZmlre2d3z94/aKk4lRSaNOax7PhEAWcCmpppDp1EAol8Dm1/dDX12w8gFYvFnR4n4EVkIFjIKNFG6ttHPREzEYDQuHLbE3CPnQru22Wn6syAl4mbkzLK0ejbX70gpmlkrqGcKNV1nUR7GZGaUQ6TUi9VkBA6IgPoGipIBMrLZukn+NQoAQ5jaY6JMVN/b2QkUmoc+WYyInqoFr2p+J/XTXV44WVMJKkGQecPhSnHOsbTKnDAJFDNx4YQKpnJiumQSEK1KaxkSnAXv7xMWrWq61Tdm1q5fpnXUUTH6ASdIRedozq6Rg3URBQ9omf0it6sJ+vFerc+5qMFK985RH9gff4AjemT8g==</latexit>

If �0 singular point, we generally expect R = 0<latexit sha1_base64="Hu/Akl82CRkcbOAgW5NGnVf5cn0=">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</latexit><latexit sha1_base64="Hu/Akl82CRkcbOAgW5NGnVf5cn0=">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</latexit><latexit sha1_base64="Hu/Akl82CRkcbOAgW5NGnVf5cn0=">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</latexit><latexit sha1_base64="Hu/Akl82CRkcbOAgW5NGnVf5cn0=">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</latexit>

7

G(n)(x1, x2, . . . xn; ~) =Z

D��(x1)�(x2) . . .�(xn) e�S(�)/~

For ~ ! 0� Green functions blow up �! ~ = 0 is non-analytic

Observables of interest are the n-point correlation functions of local operators.

The loopwise expansion has zero radius of convergence!

Consider the loopwise expansion in .~<latexit sha1_base64="LFzqSQ7RcFqeiDAyvpHAIH8Bz6g=">AAAB7HicdVBNS8NAEN3Ur1q/qh69LBbBU9jE0NaLFLx4rGDaQhvKZrtpl242YXcjlNDf4MWDIl79Qd78N27aCir6YODx3gwz88KUM6UR+rBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4XfveeSsUScadnKQ1iPBYsYgRrI/mDSYjlsFpD9mWz7np1iGyEGo7rFMRteBcedIxSoAZWaA+r74NRQrKYCk04VqrvoFQHOZaaEU7nlUGmaIrJFI9p31CBY6qCfHHsHJ4ZZQSjRJoSGi7U7xM5jpWaxaHpjLGeqN9eIf7l9TMdNYOciTTTVJDloijjUCew+ByOmKRE85khmEhmboVkgiUm2uRTMSF8fQr/Jx3XdpDt3Hq11tUqjjI4AafgHDigAVrgBrSBDwhg4AE8gWdLWI/Wi/W6bC1Zq5lj8APW2yckPY7j</latexit><latexit sha1_base64="LFzqSQ7RcFqeiDAyvpHAIH8Bz6g=">AAAB7HicdVBNS8NAEN3Ur1q/qh69LBbBU9jE0NaLFLx4rGDaQhvKZrtpl242YXcjlNDf4MWDIl79Qd78N27aCir6YODx3gwz88KUM6UR+rBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4XfveeSsUScadnKQ1iPBYsYgRrI/mDSYjlsFpD9mWz7np1iGyEGo7rFMRteBcedIxSoAZWaA+r74NRQrKYCk04VqrvoFQHOZaaEU7nlUGmaIrJFI9p31CBY6qCfHHsHJ4ZZQSjRJoSGi7U7xM5jpWaxaHpjLGeqN9eIf7l9TMdNYOciTTTVJDloijjUCew+ByOmKRE85khmEhmboVkgiUm2uRTMSF8fQr/Jx3XdpDt3Hq11tUqjjI4AafgHDigAVrgBrSBDwhg4AE8gWdLWI/Wi/W6bC1Zq5lj8APW2yckPY7j</latexit><latexit sha1_base64="LFzqSQ7RcFqeiDAyvpHAIH8Bz6g=">AAAB7HicdVBNS8NAEN3Ur1q/qh69LBbBU9jE0NaLFLx4rGDaQhvKZrtpl242YXcjlNDf4MWDIl79Qd78N27aCir6YODx3gwz88KUM6UR+rBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4XfveeSsUScadnKQ1iPBYsYgRrI/mDSYjlsFpD9mWz7np1iGyEGo7rFMRteBcedIxSoAZWaA+r74NRQrKYCk04VqrvoFQHOZaaEU7nlUGmaIrJFI9p31CBY6qCfHHsHJ4ZZQSjRJoSGi7U7xM5jpWaxaHpjLGeqN9eIf7l9TMdNYOciTTTVJDloijjUCew+ByOmKRE85khmEhmboVkgiUm2uRTMSF8fQr/Jx3XdpDt3Hq11tUqjjI4AafgHDigAVrgBrSBDwhg4AE8gWdLWI/Wi/W6bC1Zq5lj8APW2yckPY7j</latexit><latexit sha1_base64="LFzqSQ7RcFqeiDAyvpHAIH8Bz6g=">AAAB7HicdVBNS8NAEN3Ur1q/qh69LBbBU9jE0NaLFLx4rGDaQhvKZrtpl242YXcjlNDf4MWDIl79Qd78N27aCir6YODx3gwz88KUM6UR+rBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4XfveeSsUScadnKQ1iPBYsYgRrI/mDSYjlsFpD9mWz7np1iGyEGo7rFMRteBcedIxSoAZWaA+r74NRQrKYCk04VqrvoFQHOZaaEU7nlUGmaIrJFI9p31CBY6qCfHHsHJ4ZZQSjRJoSGi7U7xM5jpWaxaHpjLGeqN9eIf7l9TMdNYOciTTTVJDloijjUCew+ByOmKRE85khmEhmboVkgiUm2uRTMSF8fQr/Jx3XdpDt3Hq11tUqjjI4AafgHDigAVrgBrSBDwhg4AE8gWdLWI/Wi/W6bC1Zq5lj8APW2yckPY7j</latexit>

S = S0 + g�S<latexit sha1_base64="sUc3qAjcXLwXdBAL5Il/RM3uA3g=">AAAB+3icdVDLSgMxFM3UV62vsS7dBIsgCCUpYtuFUtCFy0ptLbTDkEkzbWjmQZIRy9BfceNCEbf+iDv/xkxbQUUPXDiccy/33uPFgiuN0IeVW1peWV3Lrxc2Nre2d+zdYkdFiaSsTSMRya5HFBM8ZG3NtWDdWDISeILdeuOLzL+9Y1LxKLzRk5g5ARmG3OeUaCO5drF11nIRPIZD2L9kQhPYcu0SKiOEMMYwI7h6igyp12sVXIM4swxKYIGma7/3BxFNAhZqKohSPYxi7aREak4Fmxb6iWIxoWMyZD1DQxIw5aSz26fw0CgD6EfSVKjhTP0+kZJAqUngmc6A6JH67WXiX14v0X7NSXkYJ5qFdL7ITwTUEcyCgAMuGdViYgihkptbIR0RSag2cRVMCF+fwv9Jp1LGqIyvT0qN80UcebAPDsARwKAKGuAKNEEbUHAPHsATeLam1qP1Yr3OW3PWYmYP/ID19glSB5Ku</latexit><latexit sha1_base64="sUc3qAjcXLwXdBAL5Il/RM3uA3g=">AAAB+3icdVDLSgMxFM3UV62vsS7dBIsgCCUpYtuFUtCFy0ptLbTDkEkzbWjmQZIRy9BfceNCEbf+iDv/xkxbQUUPXDiccy/33uPFgiuN0IeVW1peWV3Lrxc2Nre2d+zdYkdFiaSsTSMRya5HFBM8ZG3NtWDdWDISeILdeuOLzL+9Y1LxKLzRk5g5ARmG3OeUaCO5drF11nIRPIZD2L9kQhPYcu0SKiOEMMYwI7h6igyp12sVXIM4swxKYIGma7/3BxFNAhZqKohSPYxi7aREak4Fmxb6iWIxoWMyZD1DQxIw5aSz26fw0CgD6EfSVKjhTP0+kZJAqUngmc6A6JH67WXiX14v0X7NSXkYJ5qFdL7ITwTUEcyCgAMuGdViYgihkptbIR0RSag2cRVMCF+fwv9Jp1LGqIyvT0qN80UcebAPDsARwKAKGuAKNEEbUHAPHsATeLam1qP1Yr3OW3PWYmYP/ID19glSB5Ku</latexit><latexit sha1_base64="sUc3qAjcXLwXdBAL5Il/RM3uA3g=">AAAB+3icdVDLSgMxFM3UV62vsS7dBIsgCCUpYtuFUtCFy0ptLbTDkEkzbWjmQZIRy9BfceNCEbf+iDv/xkxbQUUPXDiccy/33uPFgiuN0IeVW1peWV3Lrxc2Nre2d+zdYkdFiaSsTSMRya5HFBM8ZG3NtWDdWDISeILdeuOLzL+9Y1LxKLzRk5g5ARmG3OeUaCO5drF11nIRPIZD2L9kQhPYcu0SKiOEMMYwI7h6igyp12sVXIM4swxKYIGma7/3BxFNAhZqKohSPYxi7aREak4Fmxb6iWIxoWMyZD1DQxIw5aSz26fw0CgD6EfSVKjhTP0+kZJAqUngmc6A6JH67WXiX14v0X7NSXkYJ5qFdL7ITwTUEcyCgAMuGdViYgihkptbIR0RSag2cRVMCF+fwv9Jp1LGqIyvT0qN80UcebAPDsARwKAKGuAKNEEbUHAPHsATeLam1qP1Yr3OW3PWYmYP/ID19glSB5Ku</latexit><latexit sha1_base64="sUc3qAjcXLwXdBAL5Il/RM3uA3g=">AAAB+3icdVDLSgMxFM3UV62vsS7dBIsgCCUpYtuFUtCFy0ptLbTDkEkzbWjmQZIRy9BfceNCEbf+iDv/xkxbQUUPXDiccy/33uPFgiuN0IeVW1peWV3Lrxc2Nre2d+zdYkdFiaSsTSMRya5HFBM8ZG3NtWDdWDISeILdeuOLzL+9Y1LxKLzRk5g5ARmG3OeUaCO5drF11nIRPIZD2L9kQhPYcu0SKiOEMMYwI7h6igyp12sVXIM4swxKYIGma7/3BxFNAhZqKohSPYxi7aREak4Fmxb6iWIxoWMyZD1DQxIw5aSz26fw0CgD6EfSVKjhTP0+kZJAqUngmc6A6JH67WXiX14v0X7NSXkYJ5qFdL7ITwTUEcyCgAMuGdViYgihkptbIR0RSag2cRVMCF+fwv9Jp1LGqIyvT0qN80UcebAPDsARwKAKGuAKNEEbUHAPHsATeLam1qP1Yr3OW3PWYmYP/ID19glSB5Ku</latexit>

Upon rescalings the loopwise expansion is equivalent to a proper expansion in the coupling constant:

A modern generalization of Dyson’s argument can be obtained by considering QFT in euclidean signature in a path integral approach.

Perturbative expansions in QFT are generally asymptotic with zero radius of convergence [Dyson, 1952]

8

Mathematically perturbation theory does not make sense!

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<latexit sha1_base64="oItMpXytKDppPmn/CLMor/fdBh4=">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</latexit><latexit sha1_base64="oItMpXytKDppPmn/CLMor/fdBh4=">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</latexit><latexit sha1_base64="oItMpXytKDppPmn/CLMor/fdBh4=">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</latexit><latexit sha1_base64="oItMpXytKDppPmn/CLMor/fdBh4=">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</latexit>

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gpG(n)p

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6=<latexit sha1_base64="MgIZW3FHVsm/pqP9klWVDU40dd4=">AAAB7HicdVBNS8NAEN34WetX1aOXxSJ4CpsY23orePFYwbSFNpTNdtMu3Wzi7kYoob/BiwdFvPqDvPlv3LQVVPTBwOO9GWbmhSlnSiP0Ya2srq1vbJa2yts7u3v7lYPDtkoySahPEp7IbogV5UxQXzPNaTeVFMchp51wclX4nXsqFUvErZ6mNIjxSLCIEayN5MO+oHeDShXZl42a69UgshGqO65TELfunXvQMUqBKliiNai894cJyWIqNOFYqZ6DUh3kWGpGOJ2V+5miKSYTPKI9QwWOqQry+bEzeGqUIYwSaUpoOFe/T+Q4Vmoah6YzxnqsfnuF+JfXy3TUCHIm0kxTQRaLooxDncDiczhkkhLNp4ZgIpm5FZIxlphok0/ZhPD1KfyftF3bQbZz41WbF8s4SuAYnIAz4IA6aIJr0AI+IICBB/AEni1hPVov1uuidcVazhyBH7DePgHKHY6h</latexit><latexit sha1_base64="MgIZW3FHVsm/pqP9klWVDU40dd4=">AAAB7HicdVBNS8NAEN34WetX1aOXxSJ4CpsY23orePFYwbSFNpTNdtMu3Wzi7kYoob/BiwdFvPqDvPlv3LQVVPTBwOO9GWbmhSlnSiP0Ya2srq1vbJa2yts7u3v7lYPDtkoySahPEp7IbogV5UxQXzPNaTeVFMchp51wclX4nXsqFUvErZ6mNIjxSLCIEayN5MO+oHeDShXZl42a69UgshGqO65TELfunXvQMUqBKliiNai894cJyWIqNOFYqZ6DUh3kWGpGOJ2V+5miKSYTPKI9QwWOqQry+bEzeGqUIYwSaUpoOFe/T+Q4Vmoah6YzxnqsfnuF+JfXy3TUCHIm0kxTQRaLooxDncDiczhkkhLNp4ZgIpm5FZIxlphok0/ZhPD1KfyftF3bQbZz41WbF8s4SuAYnIAz4IA6aIJr0AI+IICBB/AEni1hPVov1uuidcVazhyBH7DePgHKHY6h</latexit><latexit sha1_base64="MgIZW3FHVsm/pqP9klWVDU40dd4=">AAAB7HicdVBNS8NAEN34WetX1aOXxSJ4CpsY23orePFYwbSFNpTNdtMu3Wzi7kYoob/BiwdFvPqDvPlv3LQVVPTBwOO9GWbmhSlnSiP0Ya2srq1vbJa2yts7u3v7lYPDtkoySahPEp7IbogV5UxQXzPNaTeVFMchp51wclX4nXsqFUvErZ6mNIjxSLCIEayN5MO+oHeDShXZl42a69UgshGqO65TELfunXvQMUqBKliiNai894cJyWIqNOFYqZ6DUh3kWGpGOJ2V+5miKSYTPKI9QwWOqQry+bEzeGqUIYwSaUpoOFe/T+Q4Vmoah6YzxnqsfnuF+JfXy3TUCHIm0kxTQRaLooxDncDiczhkkhLNp4ZgIpm5FZIxlphok0/ZhPD1KfyftF3bQbZz41WbF8s4SuAYnIAz4IA6aIJr0AI+IICBB/AEni1hPVov1uuidcVazhyBH7DePgHKHY6h</latexit><latexit sha1_base64="MgIZW3FHVsm/pqP9klWVDU40dd4=">AAAB7HicdVBNS8NAEN34WetX1aOXxSJ4CpsY23orePFYwbSFNpTNdtMu3Wzi7kYoob/BiwdFvPqDvPlv3LQVVPTBwOO9GWbmhSlnSiP0Ya2srq1vbJa2yts7u3v7lYPDtkoySahPEp7IbogV5UxQXzPNaTeVFMchp51wclX4nXsqFUvErZ6mNIjxSLCIEayN5MO+oHeDShXZl42a69UgshGqO65TELfunXvQMUqBKliiNai894cJyWIqNOFYqZ6DUh3kWGpGOJ2V+5miKSYTPKI9QwWOqQry+bEzeGqUIYwSaUpoOFe/T+Q4Vmoah6YzxnqsfnuF+JfXy3TUCHIm0kxTQRaLooxDncDiczhkkhLNp4ZgIpm5FZIxlphok0/ZhPD1KfyftF3bQbZz41WbF8s4SuAYnIAz4IA6aIJr0AI+IICBB/AEni1hPVov1uuidcVazhyBH7DePgHKHY6h</latexit>

Exact answer Perturbative answer

The series is never uniformly convergent and not allowed to exchange sum and integration

Asymptotic series

Z(�) ⇠P1

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Z(�)�NX

n=0

Zn�n = O(�N+1) , as � ! 0

such that

Equality sign not allowed: in generalP1

n=0 Zn�n = 1<latexit sha1_base64="dEFDYh6gGIzqt32IqANVby6ESzc=">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</latexit><latexit sha1_base64="dEFDYh6gGIzqt32IqANVby6ESzc=">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</latexit><latexit sha1_base64="dEFDYh6gGIzqt32IqANVby6ESzc=">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</latexit><latexit sha1_base64="dEFDYh6gGIzqt32IqANVby6ESzc=">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</latexit>

Contrast with convergent series: 8� 2 D<latexit sha1_base64="HHiy0ECJX929O9HygbDtSatgJa0=">AAACBnicbVDLSsNAFJ34rPUVdSnCYCu4Kkk3iquCLlxWsA9oQrmZTNqhk0mYmQgldOXGX3HjQhG3foM7/8Zpm4W2Hhg4nHMPc+8JUs6Udpxva2V1bX1js7RV3t7Z3du3Dw7bKskkoS2S8ER2A1CUM0FbmmlOu6mkEAecdoLR9dTvPFCpWCLu9TilfgwDwSJGQBupb59UvSiRwDn2uEmF4DGBc48AxzeTat+uODVnBrxM3IJUUIFm3/7ywoRkMRWacFCq5zqp9nOQmhFOJ2UvUzQFMoIB7RkqIKbKz2dnTPCZUUJs1jFPaDxTfydyiJUax4GZjEEP1aI3Ff/zepmOLv2ciTTTVJD5R1HGsU7wtBMcMkmJ5mNDgEhmdsVkCBKINs2VTQnu4snLpF2vuU7NvatXGldFHSV0jE7ROXLRBWqgW9RELUTQI3pGr+jNerJerHfrYz66YhWZI/QH1ucPyVGX/w==</latexit><latexit sha1_base64="HHiy0ECJX929O9HygbDtSatgJa0=">AAACBnicbVDLSsNAFJ34rPUVdSnCYCu4Kkk3iquCLlxWsA9oQrmZTNqhk0mYmQgldOXGX3HjQhG3foM7/8Zpm4W2Hhg4nHMPc+8JUs6Udpxva2V1bX1js7RV3t7Z3du3Dw7bKskkoS2S8ER2A1CUM0FbmmlOu6mkEAecdoLR9dTvPFCpWCLu9TilfgwDwSJGQBupb59UvSiRwDn2uEmF4DGBc48AxzeTat+uODVnBrxM3IJUUIFm3/7ywoRkMRWacFCq5zqp9nOQmhFOJ2UvUzQFMoIB7RkqIKbKz2dnTPCZUUJs1jFPaDxTfydyiJUax4GZjEEP1aI3Ff/zepmOLv2ciTTTVJD5R1HGsU7wtBMcMkmJ5mNDgEhmdsVkCBKINs2VTQnu4snLpF2vuU7NvatXGldFHSV0jE7ROXLRBWqgW9RELUTQI3pGr+jNerJerHfrYz66YhWZI/QH1ucPyVGX/w==</latexit><latexit sha1_base64="HHiy0ECJX929O9HygbDtSatgJa0=">AAACBnicbVDLSsNAFJ34rPUVdSnCYCu4Kkk3iquCLlxWsA9oQrmZTNqhk0mYmQgldOXGX3HjQhG3foM7/8Zpm4W2Hhg4nHMPc+8JUs6Udpxva2V1bX1js7RV3t7Z3du3Dw7bKskkoS2S8ER2A1CUM0FbmmlOu6mkEAecdoLR9dTvPFCpWCLu9TilfgwDwSJGQBupb59UvSiRwDn2uEmF4DGBc48AxzeTat+uODVnBrxM3IJUUIFm3/7ywoRkMRWacFCq5zqp9nOQmhFOJ2UvUzQFMoIB7RkqIKbKz2dnTPCZUUJs1jFPaDxTfydyiJUax4GZjEEP1aI3Ff/zepmOLv2ciTTTVJD5R1HGsU7wtBMcMkmJ5mNDgEhmdsVkCBKINs2VTQnu4snLpF2vuU7NvatXGldFHSV0jE7ROXLRBWqgW9RELUTQI3pGr+jNerJerHfrYz66YhWZI/QH1ucPyVGX/w==</latexit><latexit sha1_base64="HHiy0ECJX929O9HygbDtSatgJa0=">AAACBnicbVDLSsNAFJ34rPUVdSnCYCu4Kkk3iquCLlxWsA9oQrmZTNqhk0mYmQgldOXGX3HjQhG3foM7/8Zpm4W2Hhg4nHMPc+8JUs6Udpxva2V1bX1js7RV3t7Z3du3Dw7bKskkoS2S8ER2A1CUM0FbmmlOu6mkEAecdoLR9dTvPFCpWCLu9TilfgwDwSJGQBupb59UvSiRwDn2uEmF4DGBc48AxzeTat+uODVnBrxM3IJUUIFm3/7ywoRkMRWacFCq5zqp9nOQmhFOJ2UvUzQFMoIB7RkqIKbKz2dnTPCZUUJs1jFPaDxTfydyiJUax4GZjEEP1aI3Ff/zepmOLv2ciTTTVJD5R1HGsU7wtBMcMkmJ5mNDgEhmdsVkCBKINs2VTQnu4snLpF2vuU7NvatXGldFHSV0jE7ROXLRBWqgW9RELUTQI3pGr+jNerJerHfrYz66YhWZI/QH1ucPyVGX/w==</latexit>

N ! 1<latexit sha1_base64="mdrHBFaUZJ6LU8D8QcZ5ZBYkbYU=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5GSyCq5KIoLgquHElFewDmlAm00k7dDIJMzdKDMVfceNCEbf+hzv/xmmbhbYeuHA4517uvSdIBNfgON9WaWl5ZXWtvF7Z2Nza3rF391o6ThVlTRqLWHUCopngkjWBg2CdRDESBYK1g9HVxG/fM6V5LO8gS5gfkYHkIacEjNSzD248xQdDIErFD9jjMoQM9+yqU3OmwIvELUgVFWj07C+vH9M0YhKoIFp3XScBPycKOBVsXPFSzRJCR2TAuoZKEjHt59Prx/jYKH0cxsqUBDxVf0/kJNI6iwLTGREY6nlvIv7ndVMIL/ycyyQFJulsUZgKDDGeRIH7XDEKIjOEUMXNrZgOiSIUTGAVE4I7//IiaZ3WXKfm3p5V65dFHGV0iI7QCXLROaqja9RATUTRI3pGr+jNerJerHfrY9ZasoqZffQH1ucPZK6VJA==</latexit><latexit sha1_base64="mdrHBFaUZJ6LU8D8QcZ5ZBYkbYU=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5GSyCq5KIoLgquHElFewDmlAm00k7dDIJMzdKDMVfceNCEbf+hzv/xmmbhbYeuHA4517uvSdIBNfgON9WaWl5ZXWtvF7Z2Nza3rF391o6ThVlTRqLWHUCopngkjWBg2CdRDESBYK1g9HVxG/fM6V5LO8gS5gfkYHkIacEjNSzD248xQdDIErFD9jjMoQM9+yqU3OmwIvELUgVFWj07C+vH9M0YhKoIFp3XScBPycKOBVsXPFSzRJCR2TAuoZKEjHt59Prx/jYKH0cxsqUBDxVf0/kJNI6iwLTGREY6nlvIv7ndVMIL/ycyyQFJulsUZgKDDGeRIH7XDEKIjOEUMXNrZgOiSIUTGAVE4I7//IiaZ3WXKfm3p5V65dFHGV0iI7QCXLROaqja9RATUTRI3pGr+jNerJerHfrY9ZasoqZffQH1ucPZK6VJA==</latexit><latexit sha1_base64="mdrHBFaUZJ6LU8D8QcZ5ZBYkbYU=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5GSyCq5KIoLgquHElFewDmlAm00k7dDIJMzdKDMVfceNCEbf+hzv/xmmbhbYeuHA4517uvSdIBNfgON9WaWl5ZXWtvF7Z2Nza3rF391o6ThVlTRqLWHUCopngkjWBg2CdRDESBYK1g9HVxG/fM6V5LO8gS5gfkYHkIacEjNSzD248xQdDIErFD9jjMoQM9+yqU3OmwIvELUgVFWj07C+vH9M0YhKoIFp3XScBPycKOBVsXPFSzRJCR2TAuoZKEjHt59Prx/jYKH0cxsqUBDxVf0/kJNI6iwLTGREY6nlvIv7ndVMIL/ycyyQFJulsUZgKDDGeRIH7XDEKIjOEUMXNrZgOiSIUTGAVE4I7//IiaZ3WXKfm3p5V65dFHGV0iI7QCXLROaqja9RATUTRI3pGr+jNerJerHfrY9ZasoqZffQH1ucPZK6VJA==</latexit><latexit sha1_base64="mdrHBFaUZJ6LU8D8QcZ5ZBYkbYU=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5GSyCq5KIoLgquHElFewDmlAm00k7dDIJMzdKDMVfceNCEbf+hzv/xmmbhbYeuHA4517uvSdIBNfgON9WaWl5ZXWtvF7Z2Nza3rF391o6ThVlTRqLWHUCopngkjWBg2CdRDESBYK1g9HVxG/fM6V5LO8gS5gfkYHkIacEjNSzD248xQdDIErFD9jjMoQM9+yqU3OmwIvELUgVFWj07C+vH9M0YhKoIFp3XScBPycKOBVsXPFSzRJCR2TAuoZKEjHt59Prx/jYKH0cxsqUBDxVf0/kJNI6iwLTGREY6nlvIv7ndVMIL/ycyyQFJulsUZgKDDGeRIH7XDEKIjOEUMXNrZgOiSIUTGAVE4I7//IiaZ3WXKfm3p5V65dFHGV0iI7QCXLROaqja9RATUTRI3pGr+jNerJerHfrY9ZasoqZffQH1ucPZK6VJA==</latexit>

Z(�)�NX

n=0

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9

This explains the spectacular success of high precision computations in quantum field theories, such as the computation of g-2 in QED

If is small enough, asymptotic series capture accurately the exact result�<latexit sha1_base64="/FyonCCmqMS860zRdQiuJOMh+mc=">AAAB7nicbVDLSsNAFL2pr1pfVZduBovgqiQiVFwV3LisYB/QhjKZTNqhk0mYuRFK6Ee4caGIW7/HnX/jtM1CWw8MHM45l7n3BKkUBl332yltbG5t75R3K3v7B4dH1eOTjkkyzXibJTLRvYAaLoXibRQoeS/VnMaB5N1gcjf3u09cG5GoR5ym3I/pSIlIMIpW6g6kjYZ0WK25dXcBsk68gtSgQGtY/RqECctirpBJakzfc1P0c6pRMMlnlUFmeErZhI5431JFY278fLHujFxYJSRRou1TSBbq74mcxsZM48AmY4pjs+rNxf+8fobRjZ8LlWbIFVt+FGWSYELmt5NQaM5QTi2hTAu7K2FjqilD21DFluCtnrxOOld1z617D9e15m1RRxnO4BwuwYMGNOEeWtAGBhN4hld4c1LnxXl3PpbRklPMnMIfOJ8/Ol2PdQ==</latexit><latexit sha1_base64="/FyonCCmqMS860zRdQiuJOMh+mc=">AAAB7nicbVDLSsNAFL2pr1pfVZduBovgqiQiVFwV3LisYB/QhjKZTNqhk0mYuRFK6Ee4caGIW7/HnX/jtM1CWw8MHM45l7n3BKkUBl332yltbG5t75R3K3v7B4dH1eOTjkkyzXibJTLRvYAaLoXibRQoeS/VnMaB5N1gcjf3u09cG5GoR5ym3I/pSIlIMIpW6g6kjYZ0WK25dXcBsk68gtSgQGtY/RqECctirpBJakzfc1P0c6pRMMlnlUFmeErZhI5431JFY278fLHujFxYJSRRou1TSBbq74mcxsZM48AmY4pjs+rNxf+8fobRjZ8LlWbIFVt+FGWSYELmt5NQaM5QTi2hTAu7K2FjqilD21DFluCtnrxOOld1z617D9e15m1RRxnO4BwuwYMGNOEeWtAGBhN4hld4c1LnxXl3PpbRklPMnMIfOJ8/Ol2PdQ==</latexit><latexit sha1_base64="/FyonCCmqMS860zRdQiuJOMh+mc=">AAAB7nicbVDLSsNAFL2pr1pfVZduBovgqiQiVFwV3LisYB/QhjKZTNqhk0mYuRFK6Ee4caGIW7/HnX/jtM1CWw8MHM45l7n3BKkUBl332yltbG5t75R3K3v7B4dH1eOTjkkyzXibJTLRvYAaLoXibRQoeS/VnMaB5N1gcjf3u09cG5GoR5ym3I/pSIlIMIpW6g6kjYZ0WK25dXcBsk68gtSgQGtY/RqECctirpBJakzfc1P0c6pRMMlnlUFmeErZhI5431JFY278fLHujFxYJSRRou1TSBbq74mcxsZM48AmY4pjs+rNxf+8fobRjZ8LlWbIFVt+FGWSYELmt5NQaM5QTi2hTAu7K2FjqilD21DFluCtnrxOOld1z617D9e15m1RRxnO4BwuwYMGNOEeWtAGBhN4hld4c1LnxXl3PpbRklPMnMIfOJ8/Ol2PdQ==</latexit><latexit sha1_base64="/FyonCCmqMS860zRdQiuJOMh+mc=">AAAB7nicbVDLSsNAFL2pr1pfVZduBovgqiQiVFwV3LisYB/QhjKZTNqhk0mYuRFK6Ee4caGIW7/HnX/jtM1CWw8MHM45l7n3BKkUBl332yltbG5t75R3K3v7B4dH1eOTjkkyzXibJTLRvYAaLoXibRQoeS/VnMaB5N1gcjf3u09cG5GoR5ym3I/pSIlIMIpW6g6kjYZ0WK25dXcBsk68gtSgQGtY/RqECctirpBJakzfc1P0c6pRMMlnlUFmeErZhI5431JFY278fLHujFxYJSRRou1TSBbq74mcxsZM48AmY4pjs+rNxf+8fobRjZ8LlWbIFVt+FGWSYELmt5NQaM5QTi2hTAu7K2FjqilD21DFluCtnrxOOld1z617D9e15m1RRxnO4BwuwYMGNOEeWtAGBhN4hld4c1LnxXl3PpbRklPMnMIfOJ8/Ol2PdQ==</latexit>

Care has to be paid with asymptotic series, where summing more and more terms is not a good idea

Zn ⇠ n!anGenerally in QFT for best accuracy for Z(�) is obtained by keeping<latexit sha1_base64="IjYSrgXygjPzQSA591DlU/pLzl8=">AAACHnicbVC7SgNBFJ31bXxFLW0Go6BN2BVEsRJsLBWMikkId2fvxiGzM8vMXWEJfomNv2JjoYhgpX/jJKbwdarDOfd54lxJR2H4EYyNT0xOTc/MVubmFxaXqssr584UVmBDGGXsZQwOldTYIEkKL3OLkMUKL+Le0cC/uEHrpNFnVObYzqCrZSoFkJc61d0YHXEQorAgSp4ayzeutlrKT0hge4NLx01M4KcnPC55DzGXutup1sJ6OAT/S6IRqbERTjrVt1ZiRJGhJqHAuWYU5tTugyUpFN5WWoXDHEQPutj0VEOGrt0fvnfLN72SDE9LjSY+VL939CFzrsxiX5kBXbvf3kD8z2sWlO63+1LnBaEWX4vSQnEyfJAVT6RFQar0BISV/lYursEHRT7Rig8h+v3yX3K+U4/CenS6Uzs8GMUxw9bYOttiEdtjh+yYnbAGE+yOPbAn9hzcB4/BS/D6VToWjHpW2Q8E759qzKFj</latexit><latexit sha1_base64="IjYSrgXygjPzQSA591DlU/pLzl8=">AAACHnicbVC7SgNBFJ31bXxFLW0Go6BN2BVEsRJsLBWMikkId2fvxiGzM8vMXWEJfomNv2JjoYhgpX/jJKbwdarDOfd54lxJR2H4EYyNT0xOTc/MVubmFxaXqssr584UVmBDGGXsZQwOldTYIEkKL3OLkMUKL+Le0cC/uEHrpNFnVObYzqCrZSoFkJc61d0YHXEQorAgSp4ayzeutlrKT0hge4NLx01M4KcnPC55DzGXutup1sJ6OAT/S6IRqbERTjrVt1ZiRJGhJqHAuWYU5tTugyUpFN5WWoXDHEQPutj0VEOGrt0fvnfLN72SDE9LjSY+VL939CFzrsxiX5kBXbvf3kD8z2sWlO63+1LnBaEWX4vSQnEyfJAVT6RFQar0BISV/lYursEHRT7Rig8h+v3yX3K+U4/CenS6Uzs8GMUxw9bYOttiEdtjh+yYnbAGE+yOPbAn9hzcB4/BS/D6VToWjHpW2Q8E759qzKFj</latexit><latexit sha1_base64="IjYSrgXygjPzQSA591DlU/pLzl8=">AAACHnicbVC7SgNBFJ31bXxFLW0Go6BN2BVEsRJsLBWMikkId2fvxiGzM8vMXWEJfomNv2JjoYhgpX/jJKbwdarDOfd54lxJR2H4EYyNT0xOTc/MVubmFxaXqssr584UVmBDGGXsZQwOldTYIEkKL3OLkMUKL+Le0cC/uEHrpNFnVObYzqCrZSoFkJc61d0YHXEQorAgSp4ayzeutlrKT0hge4NLx01M4KcnPC55DzGXutup1sJ6OAT/S6IRqbERTjrVt1ZiRJGhJqHAuWYU5tTugyUpFN5WWoXDHEQPutj0VEOGrt0fvnfLN72SDE9LjSY+VL939CFzrsxiX5kBXbvf3kD8z2sWlO63+1LnBaEWX4vSQnEyfJAVT6RFQar0BISV/lYursEHRT7Rig8h+v3yX3K+U4/CenS6Uzs8GMUxw9bYOttiEdtjh+yYnbAGE+yOPbAn9hzcB4/BS/D6VToWjHpW2Q8E759qzKFj</latexit><latexit sha1_base64="IjYSrgXygjPzQSA591DlU/pLzl8=">AAACHnicbVC7SgNBFJ31bXxFLW0Go6BN2BVEsRJsLBWMikkId2fvxiGzM8vMXWEJfomNv2JjoYhgpX/jJKbwdarDOfd54lxJR2H4EYyNT0xOTc/MVubmFxaXqssr584UVmBDGGXsZQwOldTYIEkKL3OLkMUKL+Le0cC/uEHrpNFnVObYzqCrZSoFkJc61d0YHXEQorAgSp4ayzeutlrKT0hge4NLx01M4KcnPC55DzGXutup1sJ6OAT/S6IRqbERTjrVt1ZiRJGhJqHAuWYU5tTugyUpFN5WWoXDHEQPutj0VEOGrt0fvnfLN72SDE9LjSY+VL939CFzrsxiX5kBXbvf3kD8z2sWlO63+1LnBaEWX4vSQnEyfJAVT6RFQar0BISV/lYursEHRT7Rig8h+v3yX3K+U4/CenS6Uzs8GMUxw9bYOttiEdtjh+yYnbAGE+yOPbAn9hzcB4/BS/D6VToWjHpW2Q8E759qzKFj</latexit>

NBest ⇡1

a�

⇠ e�1a�

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Intrinsic error associated to the asymptotic series

The higher the coupling the less terms you should compute

terms Optimal truncation

Most useful method is so called Borel resummation

Extra input is needed to possibly resum the series using resummation methods

Z(�)<latexit sha1_base64="uy4NRZBA1MqzhkfiYhR4yCwF5Qw=">AAAB9XicbVC7TsMwFL0pr1JeBUYWixapLFXSBcRUiYWxSPQh2lA5jtNadZzIdkBV1P9gYQAhVv6Fjb/BaTNAy5EsHZ1zru718WLOlLbtb6uwtr6xuVXcLu3s7u0flA+POipKJKFtEvFI9jysKGeCtjXTnPZiSXHocdr1JteZ332kUrFI3OlpTN0QjwQLGMHaSA/V+9qAm7SPz6sIDcsVu27PgVaJk5MK5GgNy18DPyJJSIUmHCvVd+xYuymWmhFOZ6VBomiMyQSPaN9QgUOq3HR+9QydGcVHQSTNExrN1d8TKQ6VmoaeSYZYj9Wyl4n/ef1EB5duykScaCrIYlGQcKQjlFWAfCYp0XxqCCaSmVsRGWOJiTZFlUwJzvKXV0mnUXfsunPbqDSv8jqKcAKnUAMHLqAJN9CCNhCQ8Ayv8GY9WS/Wu/WxiBasfOYY/sD6/AEhi5Ds</latexit><latexit sha1_base64="uy4NRZBA1MqzhkfiYhR4yCwF5Qw=">AAAB9XicbVC7TsMwFL0pr1JeBUYWixapLFXSBcRUiYWxSPQh2lA5jtNadZzIdkBV1P9gYQAhVv6Fjb/BaTNAy5EsHZ1zru718WLOlLbtb6uwtr6xuVXcLu3s7u0flA+POipKJKFtEvFI9jysKGeCtjXTnPZiSXHocdr1JteZ332kUrFI3OlpTN0QjwQLGMHaSA/V+9qAm7SPz6sIDcsVu27PgVaJk5MK5GgNy18DPyJJSIUmHCvVd+xYuymWmhFOZ6VBomiMyQSPaN9QgUOq3HR+9QydGcVHQSTNExrN1d8TKQ6VmoaeSYZYj9Wyl4n/ef1EB5duykScaCrIYlGQcKQjlFWAfCYp0XxqCCaSmVsRGWOJiTZFlUwJzvKXV0mnUXfsunPbqDSv8jqKcAKnUAMHLqAJN9CCNhCQ8Ayv8GY9WS/Wu/WxiBasfOYY/sD6/AEhi5Ds</latexit><latexit sha1_base64="uy4NRZBA1MqzhkfiYhR4yCwF5Qw=">AAAB9XicbVC7TsMwFL0pr1JeBUYWixapLFXSBcRUiYWxSPQh2lA5jtNadZzIdkBV1P9gYQAhVv6Fjb/BaTNAy5EsHZ1zru718WLOlLbtb6uwtr6xuVXcLu3s7u0flA+POipKJKFtEvFI9jysKGeCtjXTnPZiSXHocdr1JteZ332kUrFI3OlpTN0QjwQLGMHaSA/V+9qAm7SPz6sIDcsVu27PgVaJk5MK5GgNy18DPyJJSIUmHCvVd+xYuymWmhFOZ6VBomiMyQSPaN9QgUOq3HR+9QydGcVHQSTNExrN1d8TKQ6VmoaeSYZYj9Wyl4n/ef1EB5duykScaCrIYlGQcKQjlFWAfCYp0XxqCCaSmVsRGWOJiTZFlUwJzvKXV0mnUXfsunPbqDSv8jqKcAKnUAMHLqAJN9CCNhCQ8Ayv8GY9WS/Wu/WxiBasfOYY/sD6/AEhi5Ds</latexit><latexit sha1_base64="uy4NRZBA1MqzhkfiYhR4yCwF5Qw=">AAAB9XicbVC7TsMwFL0pr1JeBUYWixapLFXSBcRUiYWxSPQh2lA5jtNadZzIdkBV1P9gYQAhVv6Fjb/BaTNAy5EsHZ1zru718WLOlLbtb6uwtr6xuVXcLu3s7u0flA+POipKJKFtEvFI9jysKGeCtjXTnPZiSXHocdr1JteZ332kUrFI3OlpTN0QjwQLGMHaSA/V+9qAm7SPz6sIDcsVu27PgVaJk5MK5GgNy18DPyJJSIUmHCvVd+xYuymWmhFOZ6VBomiMyQSPaN9QgUOq3HR+9QydGcVHQSTNExrN1d8TKQ6VmoaeSYZYj9Wyl4n/ef1EB5duykScaCrIYlGQcKQjlFWAfCYp0XxqCCaSmVsRGWOJiTZFlUwJzvKXV0mnUXfsunPbqDSv8jqKcAKnUAMHLqAJN9CCNhCQ8Ayv8GY9WS/Wu/WxiBasfOYY/sD6/AEhi5Ds</latexit>

Z(�) + f(�) exp(�c/�n)<latexit sha1_base64="cYDVeYyZYKJYTXqS4eZqcxTwEUw=">AAACFXicbVBLSwMxGMzWV62vVY9egq3QotbdXhRPBS8eK9gHdteSTbNtaDa7JFmxLP0TXvwrXjwo4lXw5r8xbRfU1oHAZGY+km+8iFGpLOvLyCwsLi2vZFdza+sbm1vm9k5DhrHApI5DFoqWhyRhlJO6ooqRViQICjxGmt7gYuw374iQNOTXahgRN0A9Tn2KkdJSxzwq3BQdpvNdVIKH0P+5OOQ+Kh7jk1S45aVCx8xbZWsCOE/slORBilrH/HS6IY4DwhVmSMq2bUXKTZBQFDMyyjmxJBHCA9QjbU05Coh0k8lWI3iglS70Q6EPV3Ci/p5IUCDlMPB0MkCqL2e9sfif146Vf+YmlEexIhxPH/JjBlUIxxXBLhUEKzbUBGFB9V8h7iOBsNJF5nQJ9uzK86RRKdtW2b6q5KvnaR1ZsAf2QRHY4BRUwSWogTrA4AE8gRfwajwaz8ab8T6NZox0Zhf8gfHxDSqEnEo=</latexit><latexit sha1_base64="cYDVeYyZYKJYTXqS4eZqcxTwEUw=">AAACFXicbVBLSwMxGMzWV62vVY9egq3QotbdXhRPBS8eK9gHdteSTbNtaDa7JFmxLP0TXvwrXjwo4lXw5r8xbRfU1oHAZGY+km+8iFGpLOvLyCwsLi2vZFdza+sbm1vm9k5DhrHApI5DFoqWhyRhlJO6ooqRViQICjxGmt7gYuw374iQNOTXahgRN0A9Tn2KkdJSxzwq3BQdpvNdVIKH0P+5OOQ+Kh7jk1S45aVCx8xbZWsCOE/slORBilrH/HS6IY4DwhVmSMq2bUXKTZBQFDMyyjmxJBHCA9QjbU05Coh0k8lWI3iglS70Q6EPV3Ci/p5IUCDlMPB0MkCqL2e9sfif146Vf+YmlEexIhxPH/JjBlUIxxXBLhUEKzbUBGFB9V8h7iOBsNJF5nQJ9uzK86RRKdtW2b6q5KvnaR1ZsAf2QRHY4BRUwSWogTrA4AE8gRfwajwaz8ab8T6NZox0Zhf8gfHxDSqEnEo=</latexit><latexit sha1_base64="cYDVeYyZYKJYTXqS4eZqcxTwEUw=">AAACFXicbVBLSwMxGMzWV62vVY9egq3QotbdXhRPBS8eK9gHdteSTbNtaDa7JFmxLP0TXvwrXjwo4lXw5r8xbRfU1oHAZGY+km+8iFGpLOvLyCwsLi2vZFdza+sbm1vm9k5DhrHApI5DFoqWhyRhlJO6ooqRViQICjxGmt7gYuw374iQNOTXahgRN0A9Tn2KkdJSxzwq3BQdpvNdVIKH0P+5OOQ+Kh7jk1S45aVCx8xbZWsCOE/slORBilrH/HS6IY4DwhVmSMq2bUXKTZBQFDMyyjmxJBHCA9QjbU05Coh0k8lWI3iglS70Q6EPV3Ci/p5IUCDlMPB0MkCqL2e9sfif146Vf+YmlEexIhxPH/JjBlUIxxXBLhUEKzbUBGFB9V8h7iOBsNJF5nQJ9uzK86RRKdtW2b6q5KvnaR1ZsAf2QRHY4BRUwSWogTrA4AE8gRfwajwaz8ab8T6NZox0Zhf8gfHxDSqEnEo=</latexit><latexit sha1_base64="cYDVeYyZYKJYTXqS4eZqcxTwEUw=">AAACFXicbVBLSwMxGMzWV62vVY9egq3QotbdXhRPBS8eK9gHdteSTbNtaDa7JFmxLP0TXvwrXjwo4lXw5r8xbRfU1oHAZGY+km+8iFGpLOvLyCwsLi2vZFdza+sbm1vm9k5DhrHApI5DFoqWhyRhlJO6ooqRViQICjxGmt7gYuw374iQNOTXahgRN0A9Tn2KkdJSxzwq3BQdpvNdVIKH0P+5OOQ+Kh7jk1S45aVCx8xbZWsCOE/slORBilrH/HS6IY4DwhVmSMq2bUXKTZBQFDMyyjmxJBHCA9QjbU05Coh0k8lWI3iglS70Q6EPV3Ci/p5IUCDlMPB0MkCqL2e9sfif146Vf+YmlEexIhxPH/JjBlUIxxXBLhUEKzbUBGFB9V8h7iOBsNJF5nQJ9uzK86RRKdtW2b6q5KvnaR1ZsAf2QRHY4BRUwSWogTrA4AE8gRfwajwaz8ab8T6NZox0Zhf8gfHxDSqEnEo=</latexit>

and have identical asymptotic seriesE.g.

Borel Resummation

BZ(t) =1X

n=0

Zn

n!tn ZB(�) =

Z 1

0dt e�tBZ(t�)

Divide original series by a factorial term to get a convergent series

10

Borel series has generally finite radius of convergence, so operation not allowed (and we would get back to the starting point!)

Warning: do not expand and exchange sum with integralZ 1

0e�tBZ(�t) =

Z 1

0e�t

1X

n=0

Zn

n!�ntn

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n=0

Zn�n

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?

But , if known, can be analytically continued over the whole complex t-plane (Borel plane) and can then be calculable

BZ(t)<latexit sha1_base64="o2jSSq7GBybjgNhikOaD3Xifexo=">AAAB9XicbVDLTgJBEOz1ifhCPXqZCCZ4IbtcNJ6IXjxiIo8IK5kdZmHC7CMzvRqy4T+8eNAYr/6LN//GAfagYCWdVKq6093lxVJotO1va2V1bX1jM7eV397Z3dsvHBw2dZQoxhsskpFqe1RzKULeQIGSt2PFaeBJ3vJG11O/9ciVFlF4h+OYuwEdhMIXjKKRHkppl1FJrib3ZTwr9QpFu2LPQJaJk5EiZKj3Cl/dfsSSgIfIJNW649gxuilVKJjkk3w30TymbEQHvGNoSAOu3XR29YScGqVP/EiZCpHM1N8TKQ20Hgee6QwoDvWiNxX/8zoJ+hduKsI4QR6y+SI/kQQjMo2A9IXiDOXYEMqUMLcSNqSKMjRB5U0IzuLLy6RZrTh2xbmtFmuXWRw5OIYTKIMD51CDG6hDAxgoeIZXeLOerBfr3fqYt65Y2cwR/IH1+QOwrpFJ</latexit><latexit sha1_base64="o2jSSq7GBybjgNhikOaD3Xifexo=">AAAB9XicbVDLTgJBEOz1ifhCPXqZCCZ4IbtcNJ6IXjxiIo8IK5kdZmHC7CMzvRqy4T+8eNAYr/6LN//GAfagYCWdVKq6093lxVJotO1va2V1bX1jM7eV397Z3dsvHBw2dZQoxhsskpFqe1RzKULeQIGSt2PFaeBJ3vJG11O/9ciVFlF4h+OYuwEdhMIXjKKRHkppl1FJrib3ZTwr9QpFu2LPQJaJk5EiZKj3Cl/dfsSSgIfIJNW649gxuilVKJjkk3w30TymbEQHvGNoSAOu3XR29YScGqVP/EiZCpHM1N8TKQ20Hgee6QwoDvWiNxX/8zoJ+hduKsI4QR6y+SI/kQQjMo2A9IXiDOXYEMqUMLcSNqSKMjRB5U0IzuLLy6RZrTh2xbmtFmuXWRw5OIYTKIMD51CDG6hDAxgoeIZXeLOerBfr3fqYt65Y2cwR/IH1+QOwrpFJ</latexit><latexit sha1_base64="o2jSSq7GBybjgNhikOaD3Xifexo=">AAAB9XicbVDLTgJBEOz1ifhCPXqZCCZ4IbtcNJ6IXjxiIo8IK5kdZmHC7CMzvRqy4T+8eNAYr/6LN//GAfagYCWdVKq6093lxVJotO1va2V1bX1jM7eV397Z3dsvHBw2dZQoxhsskpFqe1RzKULeQIGSt2PFaeBJ3vJG11O/9ciVFlF4h+OYuwEdhMIXjKKRHkppl1FJrib3ZTwr9QpFu2LPQJaJk5EiZKj3Cl/dfsSSgIfIJNW649gxuilVKJjkk3w30TymbEQHvGNoSAOu3XR29YScGqVP/EiZCpHM1N8TKQ20Hgee6QwoDvWiNxX/8zoJ+hduKsI4QR6y+SI/kQQjMo2A9IXiDOXYEMqUMLcSNqSKMjRB5U0IzuLLy6RZrTh2xbmtFmuXWRw5OIYTKIMD51CDG6hDAxgoeIZXeLOerBfr3fqYt65Y2cwR/IH1+QOwrpFJ</latexit><latexit sha1_base64="o2jSSq7GBybjgNhikOaD3Xifexo=">AAAB9XicbVDLTgJBEOz1ifhCPXqZCCZ4IbtcNJ6IXjxiIo8IK5kdZmHC7CMzvRqy4T+8eNAYr/6LN//GAfagYCWdVKq6093lxVJotO1va2V1bX1jM7eV397Z3dsvHBw2dZQoxhsskpFqe1RzKULeQIGSt2PFaeBJ3vJG11O/9ciVFlF4h+OYuwEdhMIXjKKRHkppl1FJrib3ZTwr9QpFu2LPQJaJk5EiZKj3Cl/dfsSSgIfIJNW649gxuilVKJjkk3w30TymbEQHvGNoSAOu3XR29YScGqVP/EiZCpHM1N8TKQ20Hgee6QwoDvWiNxX/8zoJ+hduKsI4QR6y+SI/kQQjMo2A9IXiDOXYEMqUMLcSNqSKMjRB5U0IzuLLy6RZrTh2xbmtFmuXWRw5OIYTKIMD51CDG6hDAxgoeIZXeLOerBfr3fqYt65Y2cwR/IH1+QOwrpFJ</latexit>

ZB(�)<latexit sha1_base64="GQ7i1IewFw2abAOb1+HyUcmwFJs=">AAAB9XicbVDLTgIxFL2DL8QX6tJNI5jghsywwbgiunGJiTwijKTT6UBDpzNpOxoy4T/cuNAYt/6LO//GArNQ8CRNTs45N/f2eDFnStv2t5VbW9/Y3MpvF3Z29/YPiodHbRUlktAWiXgkux5WlDNBW5ppTruxpDj0OO144+uZ33mkUrFI3OlJTN0QDwULGMHaSA/l+8FVpc9N3sfn5UGxZFftOdAqcTJSggzNQfGr70ckCanQhGOleo4dazfFUjPC6bTQTxSNMRnjIe0ZKnBIlZvOr56iM6P4KIikeUKjufp7IsWhUpPQM8kQ65Fa9mbif14v0cGFmzIRJ5oKslgUJBzpCM0qQD6TlGg+MQQTycytiIywxESbogqmBGf5y6ukXas6dtW5rZUal1kdeTiBU6iAA3VowA00oQUEJDzDK7xZT9aL9W59LKI5K5s5hj+wPn8AtLSRTQ==</latexit><latexit sha1_base64="GQ7i1IewFw2abAOb1+HyUcmwFJs=">AAAB9XicbVDLTgIxFL2DL8QX6tJNI5jghsywwbgiunGJiTwijKTT6UBDpzNpOxoy4T/cuNAYt/6LO//GArNQ8CRNTs45N/f2eDFnStv2t5VbW9/Y3MpvF3Z29/YPiodHbRUlktAWiXgkux5WlDNBW5ppTruxpDj0OO144+uZ33mkUrFI3OlJTN0QDwULGMHaSA/l+8FVpc9N3sfn5UGxZFftOdAqcTJSggzNQfGr70ckCanQhGOleo4dazfFUjPC6bTQTxSNMRnjIe0ZKnBIlZvOr56iM6P4KIikeUKjufp7IsWhUpPQM8kQ65Fa9mbif14v0cGFmzIRJ5oKslgUJBzpCM0qQD6TlGg+MQQTycytiIywxESbogqmBGf5y6ukXas6dtW5rZUal1kdeTiBU6iAA3VowA00oQUEJDzDK7xZT9aL9W59LKI5K5s5hj+wPn8AtLSRTQ==</latexit><latexit sha1_base64="GQ7i1IewFw2abAOb1+HyUcmwFJs=">AAAB9XicbVDLTgIxFL2DL8QX6tJNI5jghsywwbgiunGJiTwijKTT6UBDpzNpOxoy4T/cuNAYt/6LO//GArNQ8CRNTs45N/f2eDFnStv2t5VbW9/Y3MpvF3Z29/YPiodHbRUlktAWiXgkux5WlDNBW5ppTruxpDj0OO144+uZ33mkUrFI3OlJTN0QDwULGMHaSA/l+8FVpc9N3sfn5UGxZFftOdAqcTJSggzNQfGr70ckCanQhGOleo4dazfFUjPC6bTQTxSNMRnjIe0ZKnBIlZvOr56iM6P4KIikeUKjufp7IsWhUpPQM8kQ65Fa9mbif14v0cGFmzIRJ5oKslgUJBzpCM0qQD6TlGg+MQQTycytiIywxESbogqmBGf5y6ukXas6dtW5rZUal1kdeTiBU6iAA3VowA00oQUEJDzDK7xZT9aL9W59LKI5K5s5hj+wPn8AtLSRTQ==</latexit><latexit sha1_base64="GQ7i1IewFw2abAOb1+HyUcmwFJs=">AAAB9XicbVDLTgIxFL2DL8QX6tJNI5jghsywwbgiunGJiTwijKTT6UBDpzNpOxoy4T/cuNAYt/6LO//GArNQ8CRNTs45N/f2eDFnStv2t5VbW9/Y3MpvF3Z29/YPiodHbRUlktAWiXgkux5WlDNBW5ppTruxpDj0OO144+uZ33mkUrFI3OlJTN0QDwULGMHaSA/l+8FVpc9N3sfn5UGxZFftOdAqcTJSggzNQfGr70ckCanQhGOleo4dazfFUjPC6bTQTxSNMRnjIe0ZKnBIlZvOr56iM6P4KIikeUKjufp7IsWhUpPQM8kQ65Fa9mbif14v0cGFmzIRJ5oKslgUJBzpCM0qQD6TlGg+MQQTycytiIywxESbogqmBGf5y6ukXas6dtW5rZUal1kdeTiBU6iAA3VowA00oQUEJDzDK7xZT9aL9W59LKI5K5s5hj+wPn8AtLSRTQ==</latexit>

Singularity dangerous or not depending on the sign of a:

a<0 (alternating series) singularity for t < 0

a>0 (same sign series) singularity for t>0 ☹

11

The chances of success of this resummation procedure can be guessed from the form of the asymptotic series

for Zn ⇠ n!an, BZ(t) =X

n

(at)n ⇠ 1

1� atExample:

😐

Sometimes an alternating series is said to be Borel resummable

This is misleading, because as we have seen there is no way to recover the original function with no further input, not being uniquely defined

Further input requires to know the analyticity properties of the function close to the origin

12

�<latexit sha1_base64="/FyonCCmqMS860zRdQiuJOMh+mc=">AAAB7nicbVDLSsNAFL2pr1pfVZduBovgqiQiVFwV3LisYB/QhjKZTNqhk0mYuRFK6Ee4caGIW7/HnX/jtM1CWw8MHM45l7n3BKkUBl332yltbG5t75R3K3v7B4dH1eOTjkkyzXibJTLRvYAaLoXibRQoeS/VnMaB5N1gcjf3u09cG5GoR5ym3I/pSIlIMIpW6g6kjYZ0WK25dXcBsk68gtSgQGtY/RqECctirpBJakzfc1P0c6pRMMlnlUFmeErZhI5431JFY278fLHujFxYJSRRou1TSBbq74mcxsZM48AmY4pjs+rNxf+8fobRjZ8LlWbIFVt+FGWSYELmt5NQaM5QTi2hTAu7K2FjqilD21DFluCtnrxOOld1z617D9e15m1RRxnO4BwuwYMGNOEeWtAGBhN4hld4c1LnxXl3PpbRklPMnMIfOJ8/Ol2PdQ==</latexit><latexit sha1_base64="/FyonCCmqMS860zRdQiuJOMh+mc=">AAAB7nicbVDLSsNAFL2pr1pfVZduBovgqiQiVFwV3LisYB/QhjKZTNqhk0mYuRFK6Ee4caGIW7/HnX/jtM1CWw8MHM45l7n3BKkUBl332yltbG5t75R3K3v7B4dH1eOTjkkyzXibJTLRvYAaLoXibRQoeS/VnMaB5N1gcjf3u09cG5GoR5ym3I/pSIlIMIpW6g6kjYZ0WK25dXcBsk68gtSgQGtY/RqECctirpBJakzfc1P0c6pRMMlnlUFmeErZhI5431JFY278fLHujFxYJSRRou1TSBbq74mcxsZM48AmY4pjs+rNxf+8fobRjZ8LlWbIFVt+FGWSYELmt5NQaM5QTi2hTAu7K2FjqilD21DFluCtnrxOOld1z617D9e15m1RRxnO4BwuwYMGNOEeWtAGBhN4hld4c1LnxXl3PpbRklPMnMIfOJ8/Ol2PdQ==</latexit><latexit sha1_base64="/FyonCCmqMS860zRdQiuJOMh+mc=">AAAB7nicbVDLSsNAFL2pr1pfVZduBovgqiQiVFwV3LisYB/QhjKZTNqhk0mYuRFK6Ee4caGIW7/HnX/jtM1CWw8MHM45l7n3BKkUBl332yltbG5t75R3K3v7B4dH1eOTjkkyzXibJTLRvYAaLoXibRQoeS/VnMaB5N1gcjf3u09cG5GoR5ym3I/pSIlIMIpW6g6kjYZ0WK25dXcBsk68gtSgQGtY/RqECctirpBJakzfc1P0c6pRMMlnlUFmeErZhI5431JFY278fLHujFxYJSRRou1TSBbq74mcxsZM48AmY4pjs+rNxf+8fobRjZ8LlWbIFVt+FGWSYELmt5NQaM5QTi2hTAu7K2FjqilD21DFluCtnrxOOld1z617D9e15m1RRxnO4BwuwYMGNOEeWtAGBhN4hld4c1LnxXl3PpbRklPMnMIfOJ8/Ol2PdQ==</latexit><latexit sha1_base64="/FyonCCmqMS860zRdQiuJOMh+mc=">AAAB7nicbVDLSsNAFL2pr1pfVZduBovgqiQiVFwV3LisYB/QhjKZTNqhk0mYuRFK6Ee4caGIW7/HnX/jtM1CWw8MHM45l7n3BKkUBl332yltbG5t75R3K3v7B4dH1eOTjkkyzXibJTLRvYAaLoXibRQoeS/VnMaB5N1gcjf3u09cG5GoR5ym3I/pSIlIMIpW6g6kjYZ0WK25dXcBsk68gtSgQGtY/RqECctirpBJakzfc1P0c6pRMMlnlUFmeErZhI5431JFY278fLHujFxYJSRRou1TSBbq74mcxsZM48AmY4pjs+rNxf+8fobRjZ8LlWbIFVt+FGWSYELmt5NQaM5QTi2hTAu7K2FjqilD21DFluCtnrxOOld1z617D9e15m1RRxnO4BwuwYMGNOEeWtAGBhN4hld4c1LnxXl3PpbRklPMnMIfOJ8/Ol2PdQ==</latexit>

It is in general hard to argue the analyticity property of Z, which is unknown!

In general, if Z(�) is analytic in the light blue circle<latexit sha1_base64="0lnYLYi11mLX53siooiu59v37TI=">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</latexit><latexit sha1_base64="0lnYLYi11mLX53siooiu59v37TI=">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</latexit><latexit sha1_base64="0lnYLYi11mLX53siooiu59v37TI=">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</latexit><latexit sha1_base64="0lnYLYi11mLX53siooiu59v37TI=">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</latexit>

[Early ‘900 theorem, rediscovered in 1980!]

ZB(�) = Z(�) 🙂

In special cases, like in certain 2d or 3d scalar field theories, perturbation theory has been proved to be Borel resummable

[Eckmann, Magnen and Seneor, 1975; Magnen and Seneor, 1977; MS, Spada, Villadoro, 2018]

Unfortunately interesting theories such as 4d gauge theories are not expected to be Borel resummable, at least in a simple way.

13

When a perturbative series is not Borel resummable because of a singularity in the real positive axis, one might deform the contour. The result however presents a non-perturbative ambiguity of order exp(�a/�n)

<latexit sha1_base64="eOyWRlVNfEpT1OF/XZC7GrJ7A3s=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBahLqyJCLosunFZwT6giWUymbRDJ5MwM5GW0l9x40IRt/6IO//GaZuFVg8MHM65h3vnBClnSjvOl1VYWV1b3yhulra2d3b37P1ySyWZJLRJEp7IToAV5UzQpmaa004qKY4DTtvB8Gbmtx+pVCwR93qcUj/GfcEiRrA2Us8ue3SUVk/xmcdNKMQP4qRnV5yaMwf6S9ycVCBHo2d/emFCspgKTThWqus6qfYnWGpGOJ2WvEzRFJMh7tOuoQLHVPmT+e1TdGyUEEWJNE9oNFd/JiY4VmocB2Yyxnqglr2Z+J/XzXR05U+YSDNNBVksijKOdIJmRaCQSUo0HxuCiWTmVkQGWGKiTV0lU4K7/OW/pHVec52ae3dRqV/ndRThEI6gCi5cQh1uoQFNIDCCJ3iBV2tqPVtv1vtitGDlmQP4BevjG8ukk58=</latexit><latexit sha1_base64="eOyWRlVNfEpT1OF/XZC7GrJ7A3s=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBahLqyJCLosunFZwT6giWUymbRDJ5MwM5GW0l9x40IRt/6IO//GaZuFVg8MHM65h3vnBClnSjvOl1VYWV1b3yhulra2d3b37P1ySyWZJLRJEp7IToAV5UzQpmaa004qKY4DTtvB8Gbmtx+pVCwR93qcUj/GfcEiRrA2Us8ue3SUVk/xmcdNKMQP4qRnV5yaMwf6S9ycVCBHo2d/emFCspgKTThWqus6qfYnWGpGOJ2WvEzRFJMh7tOuoQLHVPmT+e1TdGyUEEWJNE9oNFd/JiY4VmocB2Yyxnqglr2Z+J/XzXR05U+YSDNNBVksijKOdIJmRaCQSUo0HxuCiWTmVkQGWGKiTV0lU4K7/OW/pHVec52ae3dRqV/ndRThEI6gCi5cQh1uoQFNIDCCJ3iBV2tqPVtv1vtitGDlmQP4BevjG8ukk58=</latexit><latexit sha1_base64="eOyWRlVNfEpT1OF/XZC7GrJ7A3s=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBahLqyJCLosunFZwT6giWUymbRDJ5MwM5GW0l9x40IRt/6IO//GaZuFVg8MHM65h3vnBClnSjvOl1VYWV1b3yhulra2d3b37P1ySyWZJLRJEp7IToAV5UzQpmaa004qKY4DTtvB8Gbmtx+pVCwR93qcUj/GfcEiRrA2Us8ue3SUVk/xmcdNKMQP4qRnV5yaMwf6S9ycVCBHo2d/emFCspgKTThWqus6qfYnWGpGOJ2WvEzRFJMh7tOuoQLHVPmT+e1TdGyUEEWJNE9oNFd/JiY4VmocB2Yyxnqglr2Z+J/XzXR05U+YSDNNBVksijKOdIJmRaCQSUo0HxuCiWTmVkQGWGKiTV0lU4K7/OW/pHVec52ae3dRqV/ndRThEI6gCi5cQh1uoQFNIDCCJ3iBV2tqPVtv1vtitGDlmQP4BevjG8ukk58=</latexit><latexit sha1_base64="eOyWRlVNfEpT1OF/XZC7GrJ7A3s=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBahLqyJCLosunFZwT6giWUymbRDJ5MwM5GW0l9x40IRt/6IO//GaZuFVg8MHM65h3vnBClnSjvOl1VYWV1b3yhulra2d3b37P1ySyWZJLRJEp7IToAV5UzQpmaa004qKY4DTtvB8Gbmtx+pVCwR93qcUj/GfcEiRrA2Us8ue3SUVk/xmcdNKMQP4qRnV5yaMwf6S9ycVCBHo2d/emFCspgKTThWqus6qfYnWGpGOJ2WvEzRFJMh7tOuoQLHVPmT+e1TdGyUEEWJNE9oNFd/JiY4VmocB2Yyxnqglr2Z+J/XzXR05U+YSDNNBVksijKOdIJmRaCQSUo0HxuCiWTmVkQGWGKiTV0lU4K7/OW/pHVec52ae3dRqV/ndRThEI6gCi5cQh1uoQFNIDCCJ3iBV2tqPVtv1vtitGDlmQP4BevjG8ukk58=</latexit>

If one is able to find a semi-classical instanton like configuration (and its whole series) leading to the same factors, one might hope to remove the ambiguity

A systematic way to proceed along these lines uses the theory of resurgence [Ecalle, 1981]

Some progress has been achieved in the last years using the theory of resurgence to relate classical and instanton series among each other but it

is unfortunately hard to pursue this direction in generic QFTs [Review by Aniceto, Basar, Schiappa, 1802.10441]

On the other hand Borel resummation (no need of resurgence) can be useful even when we do not know if the series is resummable or

not, and we do not know its large order behaviour Indeed, while optimal truncation is given by the singularity closest to the origin in the Borel plane (wherever it is), the non-Borel summability of a

series is determined by its singularities on the positive real axis

Moreover, non-ambiguous non-perturbative contributions might be present, which would be ``invisible” in perturbation theory

14

t

t1t1 > |t0|

Optimal truncation error ⇠ e�|t0|�

Indeterminacy in the Borel function ⇠ e�t1� < e�

|t0|�

It is useful to Borel resum a series even when it is not resummable!

We will use this result when going back to the conformal window in QCD

t0 =1

a

15

2) O(n!) contributions of O(1) Feynman diagrams

These are governed by complex instantons whose action determine the factor a. If real instantons are also present, perturbative series is not Borel resummable because

of singularities appearing on the positive real axis of the Borel plane

Generally expected in QFT with marginal couplings. Not known (if any) the real or complex semi-classical configurations associated. Singularities in the Borel function

are called renormalons and are related to the RG flow of the marginal couplings

[Gross&Neveu 1974, Lautrup 1977,’t Hooft 1977]

Instantons and Renormalons

In QFT series are asymptotics because of factorial growth of their coefficients

[Vainhstein 1964, Lam 1968, Bender&Wu 1969, Lipatov 1976, …]

There are two known sources for the factorial growth:

1) O(1) contributions of O(n!) Feynman diagrams

16

Nature of the series of RG functions in 4d non-abelian gauge theories

Perturbative gauge coupling expansion of physical observables in 4d gauge theories is generally asymptotic and non-Borel resummable

Singularities in the positive real axis due both to instantons (or instanton-anti instanton pairs) and renormalons (more precisely IR renormalons)

Not much is known about RG functions such as beta-function or anomalous dimensions (i.e. of fermion bilinear)

Established results only in the limit of large number of flavours

These functions are renormalization scheme dependent so we expect that their large order behaviour depends on the scheme.

nf ! 1, ↵ ! 0 fixed

[Palanques-Mestre, Pascual, 1984; Gracey, 1996]

� = nf↵

In this limit the O(1/nf ) terms in �(�) and �(�) can be computed exactly.

17

�(�) =2

3�2 +

1

nf�(1)(�) +O

⇣n�2f

⌘

� =1

nf�(1)(�) +O

⇣n�2f

⌘

In particular in MS

�(1), �(1), known functions of �

Interestingly enough, these are analytic at , which implies a convergent series!� = 0

In other more physical schemes, such as on-shell or momentum subtraction, the series are divergent asymptotic

nc ! 1 � = nc↵↵ ! 0

nf ! 1 x = nf

nc

’t Hooft:

↵ ! 0

fixed

nc ! 1Veneziano: � = nc↵ fixed

For finite number of flavours and colors, and in other large n limits (’t Hooft and Veneziano) the nature of the series for and is unknown � �MS

18

QCD Conformal Window

After this long preparatory journey, we can come back to our original problem of studying the conformal window in QCD

Of course we can’t have a conformal window in the large flavour or large color limit, but we can and do have a conformal window in the

Veneziano limit. So we will also consider this limit of QCD.

n⇤f Method References

12 Schwinger-Dyson [Appelquist et al, hep-ph/9602385+…]

10 Exact RG [Gies, Jaeckel, hep-ph/0507171+ …]

9 Perturbation Theory [Ryttov,Shrock, 1607.06866]

10 Conformal Expansion [Kim, Hong, Lee, 2001.02690]

9 Padè/Resummations [Ryttov,Shrock,2018; Antipin et al, 2019]

QCDPrevious (non-lattice) results

19

x⇤

3.7÷ 4.2

Method References

4 Schwinger-Dyson [Appelquist et al,hep-ph/9602385+…]

4 Exact RG [Gies, Jaeckel, hep-ph/0507171+ …]

3 Padè Pert. Theory [Ryttov,Shrock,1710.06944]

Bottom-up holographic [Jarvinen, Kiritsis, 1112.1261+ …]

Veneziano

Previous results made predictions without an estimate of the associated error, so it is hard to assess their reliability.

We will argue that it is too optimistic to hope to reach the lower end of the conformal window using perturbation theory techniques only

20

This is a hard problem in the lattice and no consensus yet on the value of n⇤f

In particular it is still debated whether QCD with flavours is conformal or notnf = 12

No lattice results are available in the Veneziano limit

Our approachWe start from the available 5-loop coefficients of the -function MS �

We assume the worst case scenario in which the series diverges and is not-Borel resummable. Then we perform a Borel resummation using Padè

approximants to reconstruct the whole Borel function (Padè-Borel method)

[Baikov, Chetyrkin, Kuhn,1606.08659; Herzog et al, 1701.01404][Luthe et al, 1709.07718; Chetyrkin et al, 1709.08541]

We then look for zeros of the Borel resummed beta-function

Waiting for the conformal bootstrap, lattice is the only first principle method to address the QCD conformal window.

[See e.g. review by DeGrand, 1510.05018]

[Fodor et al, 1811.05024]

21

Remarks:

- It has been conjectured that conformality is lost because two nearby fixed points merge and disappear into the complex plane. In the Veneziano limit this implies that one (or more) double trace operator (presumably four fermion operators) approaches marginality

[Kaplan et al, 0905.4752]

- This is in agreement with earlier results using gap equations according to which in the IR |� | ⇡ 1 for nf = n⇤

f

[Yamawaki, Bando, Matumoto,1986; Appelquist, Lane, Mahanta,1988; Cohen, Georgi,1989 ]

- For this reason and also because this is an observable measured on the lattice, we will also compute the mass anomalous dimensions at the IR fixed point ( )�⇤

Recall that � 4 ⇡ 2� 2 at large n

22

A fixed point can also be found using the Banks-Zaks conformal expansion

Assume that the coefficient of the first available order is accidentally small so that a “violation” of perturbation theory is allowed

�(↵) = �0↵2 + �1↵

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[Caswell,1974; Banks,Zaks,1982]

↵⇤ ⇡ ��0

�1/ ✏

✏ / (n+f � nf ) ✏ / (x+ � x)(QCD) (Veneziano)

↵⇤ =1X

n=1

bn✏n �⇤ =

1X

n=1

gn✏n

are scheme-independent coefficientsgnConformal expansion series seems to be better behaved than ordinary coupling

expansions, but its nature (convergent or not) follows that of the ordinary expansion

As double check, we also Borel resum the conformal expansion and compare the results with those obtained using the ordinary coupling series

23

Even if both the conformal and the ordinary series expansions would be non-Borel resummable, we expect an improvement with respect to optimal truncation

t

t0 t1

t1 > |t0|

Optimal truncation error ⇠ e�|t0|�

Indeterminacy in the Borel function ⇠ e�t1� < e�

|t0|�

Perturbation theory of the 5-loop beta-function breaks down for nf 13

Loop Order nf = 12 nf = 13 nf = 14 nf = 15 nf = 16

2 6 · 10�2 3.7 · 10�2 2.2 · 10�2 1.1 · 10�2 3.3 · 10�3

3 3.5 · 10�2 2.5 · 10�2 1.7 · 10�2 9.8 · 10�3 3.2 · 10�3

4 3.5 · 10�2 2.7 · 10�2 1.8 · 10�2 1.0 · 10�2 3.2 · 10�3

5 �5.5 · 10�6 3.2 · 10�2 1.9 · 10�2 1.0 · 10�2 3.2 · 10�3

Table 1: Approximate values of the QCD Caswell-Banks-Zaks fixed point coupling a⇤ as a function of

nf obtained using di↵erent loop orders. The red color indicates values that should be taken with care,because of a possible breakdown of perturbation theory.

contour prescription to avoid the singularities present along the t axis. Let us denote by t1 the

distance from the origin of the closest singularity on the positive real axis. We end up having an

equivalence class of functions {FB,n(a)}, one for each di↵erent prescription, and an ambiguous

partial result. For simplicity let us consider the typical situation where the order of magnitude of

the ambiguity is the same as the leading non-perturbative correction which is anyhow missed.11

The di↵erence �B between the exact value of the function F and any FB in the class {FB,n(a)}

is roughly given by

�B(a) ⇠ e� t1

a . (3.6)

The error (3.6) should be compared with the best one we can obtain in perturbation theory

using optimal truncation, given in eq. (3.3). Crucially, the singularity at |t0| and the one at

t1 are generally di↵erent. Since by definition t1 � |t0|, Borel resumming a formally non-Borel

resummable function might lead to a better accuracy in the ending result.

After this general discussion, we can come back to the situation at hand, and see what

ordinary gauge coupling perturbation theory predicts for the zeroes of the MS �-function. We

report in table 1 the values a⇤ where �(a⇤) vanishes for QCD for di↵erent values of nf and using

di↵erent loop orders. As expected, the closer we are to the upper edge n+

f = 33/2 of the QCD

conformal window, the more reliable perturbation theory is. The five-loop term for nf = 12 and

nf = 13 is larger than the lower-order (four-loop) term for the values of a where the four-(or

lower-)loop � function has zeroes. For this reason they are reported in red and should be taken

with care. Barring numerical accidents and assuming the series entered its asymptotic form, a

higher-order term larger in magnitude than a lower-order one in a series signals either that i) we

are outside the convergence radius of a series or that ii) the series is asymptotic. As discussed

in the previous section, we will assume the case ii) in the following.

The values of t0 and t1 are not known for �(a). If non-perturbative corrections show up as

ambiguities in the Borel resummation, as is often the case, based on the arguments in section

2 we expect that t1 � 2/�0. In particular, the singularity at t1 = 2/�0 would be an IR renor-

11Whenever the singularities are due to semi-classical configurations such as instantons, one might hope tocancel the ambiguities by combining in a single trans-series the asymptotic series arising from each semi-classicalconfiguration. In the best case scenario, one could be able in this way to recover all missing non-perturbativee↵ects and reproduce the exact result. This is the subject of resurgence, see [43] for a recent review.

11

[from 2003.01742]

24

We do not expect that the perturbative -function (independently of whether it is convergent or divergent) can capture well the IR physics for any

MS �n⇤f nf n+

f

MS is a mass-independent scheme and no mass scale can enter in RG functions

to all orders in perturbation theory

Non-perturbatively, however, contributions proportional to

⇤QCD

µ⇡ e

1�0↵ could appear

By dimensional analysis an irrelevant operator with UV dimension 4+k anddimensionless MS coupling h can appear in � function as

�� ⇠ h

✓⇤QCD

µ

◆k

= h ek

�0↵

We expect leading contributions arise from k=2, due to four fermion operators. When these are sizable our analysis break down.

We add this systematic uncertainty to our error estimate, that includes the numerical error due to the reconstruction of the Borel function out of few perturbative coefficients

25

�������

0.000 0.005 0.010 0.015 0.020 0.025

-0.0001

0.0000

0.0001

0.0002

0.00032 Loops

3 Loops

4 Loops

5 Loops

[3,1]

[1,3]

[2,2]

Large exact results are better reproduced by [2/2] approximantnf

Results

26

�������

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

-0.0015

-0.0010

-0.0005

0.0000

0.0005

0.0010

0.00152 Loops

3 Loops

4 Loops

5 Loops

[3/1]

[1/3]

[2/2]

We see conformality at nf = 12

27

Banks-Zaks conformal expansion

10 11 12 13 14 15

0.00

0.05

0.10

0.15

10 11 12 13 14 15

0.0

0.2

0.4

0.6

0.8

1.0

2 Loops

3 Loops

4 Loops

[2/1]

[0/3]

[1/2]

Evidence of conformality also for nf = 11

28

�������

12 13 14 15 16

0.01

0.02

0.03

0.04

0.05

Conformal

Coupling

�������

12 13 14 15 16

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Conformal

Coupling

29

Ref. |�⇤| Method

This work

0.1 0.2 0.3 0.4 0.5 0.6

0.320(85) PB coupling0.345(47) PB conformal

[44] 0.23(6) Gradient flow

[45]0.47(10)

Top. susceptibility0.33(6)

[46] 0.32(3) Dirac eigenmodes[47] 0.235(46)

Finite-size scaling[48] 0.235(15)[49] 0.45(5)[50] 0.403(13)

Table 2: Comparison between the results of our Pade-Borel (PB) resummation for the mass-anomalousdimension for QCD with nf = 12 –both using the coupling expansion and the Banks-Zaks conformalexpansion, and averaging over all available Pade approximants in each case– and lattice results.

some care is needed before jumping too quickly to a conclusion. When nf < 12 the contribution

(C.9) to the full error, which is sub-leading for higher values of nf , becomes sizable. The results

shown have cnp = 10. In both the ordinary coupling and conformal expansions, for nf � 13

the choice of cnp is essentially irrelevant, unless one considers unreasonably large values of this

parameter. For nf = 12 we have to take cnp ⇠ 50 to enlarge the error so that this is compatible

with no fixed point in the coupling expansion. In the conformal expansion this value reaches

cnp ⇠ 5⇥ 104. We think that a non-perturbative correction of this size is unreasonable and that

the evidence for a fixed point at nf = 12 is overwhelming. On the other hand, the fixed point

for nf = 11 in the conformal expansion is compatible with no fixed point for cnp ⇠ 50, while

in the ordinary coupling expansion the error is dominated by the contribution associated to the

numerical reconstruction of the Borel function, namely it is compatible with no fixed point even

if one takes cnp = 0. We take these results as evidence that nf = 11 is conformal, but we do not

consider it enough to make a strong claim. If we trust this evidence, we can use the resummation

of the conformal expansion to obtain |�⇤| = 0.485(143) and �

⇤g = 0.36(19) (averaging all the

available Pade’s weighted by their errors and combining the errors in quadrature). Needless to

say, we do not commit ourselves with an estimate for n⇤f .

We would finally like to conclude with a general observation about the use of Pade-Borel

versus simple Pade approximants. In the former case, we would not expect a gain in considering

Borel-resummation, because �(a) would be analytic at a = 0 and an ordinary Pade approximant

on �(a) should su�ce. On the contrary, for a convergent series the Borel function is analytic

everywhere and expected to have an exponential behaviour at infinity, and functions of this

kind are not well approximated by low-order Pade approximants. We have verified that by

taking ordinary Pade approximants our results remain qualitatively unchanged, though Pade-

Borel approximants give slightly more accurate results. This can be seen as a sort of indirect

numerical evidence of the non-convergence of the MS �-function. See appendix D for a more

19

[from 2003.01742]

[44] - Carosso, Hasenfratz, Neil 1806.01385[45] - Aoki et al. 1601.04687[46] - Cheng, Hasenfratz, Petropoulos, Schaich, 1301.1355[47] - Lombardo, Miura, Nunes da Silva, Pallante, 1410.0298[48] - Cheng et al, 1401.0195[49] - Aoki et al, 1207.3060[50] - Appelquist et al, 1106.2148

30

Similar results have been derived in the Veneziano limit of QCD

�������

4.2 4.4 4.6 4.8 5.0

0.04

0.06

0.08

0.10

Conformal

Coupling

�������

4.2 4.4 4.6 4.8 5.0

0.10

0.15

0.20

0.25

0.30

Conformal

Coupling

If the series is Borel resummable we get an approximation of the exact result valid beyond perturbation theory.

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Perturbation theory is the most notable analytical tool to study QFTs.

Conclusions and future perspectives

Its natural regime of validity, by definition, is weak coupling, where it allows us to do accurate predictions, despite the

generic non-convergence of the associated series.

However, computing a sufficient number of higher order terms allows us to estimate the Borel transform of the function and resum the series.

This is not the case if the series is not Borel resummable, yet we might do better than perturbation theory.

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We have shown this in the study of the conformal window of QCD-like theories. In particular, we have ample evidence that QCD with flavours (a debated case in the lattice community) flows in the IR to a CFT

nf = 12

Note that we are not claiming that . Indeed we have some evidence that is also conformal

n⇤f = 12

nf = 11

Below that, perturbative methods break down and a genuine non-perturbative approach, like lattice or conformal bootstrap, is required

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Future perspectives

3.Generalizations to theories with other gauge theories or fermion representations (straightforward).

2.Keep going in computing higher and higher loops, necessary for a better and better resummation and to get insights for point 1.

1.Although our results point toward a divergent asymptotic nature of the RG functions, it would be nice to firmly establish their nature. MS

Thank You

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