New method for the conformal bootstrap with OPE truncations

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1 New method for the conformal bootstrap with OPE truncations Wenliang LI Kyung Hee University, Seoul, South Korea Symposium: "Bootstrap Approach to Conformal Field Theories and Applications, OIST, March19-23, 2018 based on

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3 Outline OPE truncation OPE hierarchy Effective Field Theory analogue Variational method: error minimization

4 Outline OPE truncation OPE hierarchy Effective Field Theory analogue Variational method: error minimization

5 Conformal bootstrap Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson s Operator Product Expansion Proposed around 1973 by Ferrara-Grillo-Gatto & Polyakov 2d success in 1984 by Belavin-Polyakov-Zamolodchikov Revived in 2008 by Rattazzi-Rychkov-Tonni-Vichi Carving out the parameter space of unitary CFTs 3d Ising CFT at the kink, later in a tiny island, and a lot of progress in other CFTs Extract the spectrum from extremal functionals

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sha1_base64="m0iqlcwe4gciwaieneiwmsqr6v0=">aaacb3icbvdltgjbejzff+il9ohlijgbg2txmkghe6icpgiiqsismjvbcxnmh5mzxccgh/afvorvk/hqz3jzuxxgdwpw0p1kvxe6u07emvsm+wvkvlbx1jeym7mt7z3dvxxh/0ggsadqoceprcshejglokgy4tckbbdf4db0bjdtvzkeivky3ktrbb2f9almmuqulrr5wvcqzj2my7rzdhf4xo7mi2bfnaeveyslrzsi3s1/225iyx8crtmrsm2zkeokrchgouxydiwhinraetdwnca+ye4ye32cj7xiyi8uugkfz+rvjyt4uo58r0/6rpxlojcv//pasfiuogklolhbqoehvjhjfejpdthlaqjii00ifuz/immfcekvtitnu+dztcsuavdkoioxfgnyjo3tymxfujsrvq/tblloeb2herlqoaqiw1rhdutri3pgl+jvedlejhfjyz6amdkda/qhxucpmaixna==</latexit> <latexit sha1_base64="m0iqlcwe4gciwaieneiwmsqr6v0=">aaacb3icbvdltgjbejzff+il9ohlijgbg2txmkghe6icpgiiqsismjvbcxnmh5mzxccgh/afvorvk/hqz3jzuxxgdwpw0p1kvxe6u07emvsm+wvkvlbx1jeym7mt7z3dvxxh/0ggsadqoceprcshejglokgy4tckbbdf4db0bjdtvzkeivky3ktrbb2f9almmuqulrr5wvcqzj2my7rzdhf4xo7mi2bfnaeveyslrzsi3s1/225iyx8crtmrsm2zkeokrchgouxydiwhinraetdwnca+ye4ye32cj7xiyi8uugkfz+rvjyt4uo58r0/6rpxlojcv//pasfiuogklolhbqoehvjhjfejpdthlaqjii00ifuz/immfcekvtitnu+dztcsuavdkoioxfgnyjo3tymxfujsrvq/tblloeb2herlqoaqiw1rhdutri3pgl+jvedlejhfjyz6amdkda/qhxucpmaixna==</latexit> OPE truncation Let us go back to the original question: solve the bootstrap equation (approximately, in certain regime) Why truncate OPEs? Difficult problem (infinitely many unknowns) becomes simpler (finitely) Experiments/ Monte Carlo simulations measure low-lying CFT data Does it work? Gliozzi s determinant method Universality for different CFTs OPE hierarchy, error minimization (in this talk) <latexit sha1_base64="ayrd+zian2/+sgqlnv33qhsnmis=">aaab/3icbvbns8naen3ur1q/qh69lbbbg5rebpugfo3bywvjc0kom82kxbr5yhcjtkex/4nxvxosr/4ub/4ut20o2vpg4pheddpz/jqzquzzyygtla+srpxxkxubw9s71d29b5lkgojne56ijk8kcbadrzji0ekfkmjn0pyhnxo//qhcsis+v8muvij0yhyyspswnoxq5j64phf41k3wzlo5bv4kvkfqqecrw/12g4rmecskcikly5mp8niifkmcxhu3k5asoia9cdsnsqtsy6cnj/grvgicjkjxrpbu/t2rk0jkyetrzoiovpz3juj/npop8mllwzxmcmi6wxrmhkset/7harnafr9qqqhg+lzm+0qqqnrkftea0g3mbho4immditufwckxt+uxdevurna4lripown0ii6rhc5ra92ifrirrql6ri/o1xgy3ox342pwwjkkmx30b8bnd957llg=</latexit> Focus on 4-point scalar correlators & Euclidean regime u = z z v =(1 z)(1 z) z = z

7 Determinant method Gliozzi ) Truncate the OPE to a few terms 2) Set the truncated bootstrap equation to zero 3) Linear dependence of truncated equations (from derivatives) indicates vanishing determinants of minors Less numerical demand & not using unitarity/sign-definiteness (Gliozzi-Rago 14, Gliozzi-Liendo-Meineri-Rago 15, Nakayama 16, Gliozzi 16, Esterlis-Fitzpatrick-Ramirez 16, Hikami 17 & 18, LeClair-Squires 18) Error estimation is subtle

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9 <latexit sha1_base64="wp5db8c0w4g1ux7kuzsxymonvku=">aaacexicbzdlssnafiynxmu9rv12eyycijskcopcknqfywrgfppsjtotdujkwsyjueixvopv4fa3rsstt+dor3hazqgtpwx8/occzpnftwrxantfxtlyyuraemgjulm1vbnr7u3fqzivdfwwi1i2fkpa8ahc5cigluigos+g6q+vj/xma0jf4+gorwl0qtqpemazrw11zzkxdhjx8zchogz86zxmodo1y3bfnspabcehmsnv6jrfxi9maqgrmkgvajt2gp2msurmwljopqosyoa0d22nedu7o9n0e2prsds9k4ilfhfau/f3rezdpuahrztdigm1x5uy/9xakqbnnyxhsyoqsdmiibuwxtykeavhjtauiw2usa5vtdiasspq51b0ehb49cyrg0a61pe48wesglutxfsc29ny7srppkbk5jace4eckrq5iq3iekyeytn5ia/gk/fmvbsfs9yli585ih9kfp4aheqc6q==</latexit> <latexit 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sha1_base64="wp5db8c0w4g1ux7kuzsxymonvku=">aaacexicbzdlssnafiynxmu9rv12eyycijskcopcknqfywrgfppsjtotdujkwsyjueixvopv4fa3rsstt+dor3hazqgtpwx8/occzpnftwrxantfxtlyyuraemgjulm1vbnr7u3fqzivdfwwi1i2fkpa8ahc5cigluigos+g6q+vj/xma0jf4+gorwl0qtqpemazrw11zzkxdhjx8zchogz86zxmodo1y3bfnspabcehmsnv6jrfxi9maqgrmkgvajt2gp2msurmwljopqosyoa0d22nedu7o9n0e2prsds9k4ilfhfau/f3rezdpuahrztdigm1x5uy/9xakqbnnyxhsyoqsdmiibuwxtykeavhjtauiw2usa5vtdiasspq51b0ehb49cyrg0a61pe48wesglutxfsc29ny7srppkbk5jace4eckrq5iq3iekyeytn5ia/gk/fmvbsfs9yli585ih9kfp4aheqc6q==</latexit> <latexit sha1_base64="gtul+drdugk6v53qezetuzk/z6g=">aaab/3icbvbns8naen3ur1q/qh69bivgqss1ob6eoj14rgbsiqlls5m0szfzslsrsujf/+bvr57eqz/fmz/fbzudtj6y4fheddv7gprrqszryyitrk6tb5q3k1vbo7t71f2db8kzqcahnhhrc7aerhnwffumeqkahacmushozup3h0fiypn7nu7bj/egorelwgnjpfpa/czvu3e7x61zdwsgc5nybamhap1+9dsloclisbrhwerxtlll51goshhmkl4micvkhafgaprggksfz06emcdacc2ic12jmmfq740cx1ko40bpxlgn5ai3ff/z3exff35okzrtkjd5q1hgtmxn6f/nkaogio01wurqfatjhlhgonrkfs+eygvnxhuywhmdib0ywdjxgvxlun3xrlwui2zk6agdo1nko3puqreogxxeeefp6aw9gk/gm/fufmxhs0axc4j+wpj8ayr/lx4=</latexit> <latexit sha1_base64="gtul+drdugk6v53qezetuzk/z6g=">aaab/3icbvbns8naen3ur1q/qh69bivgqss1ob6eoj14rgbsiqlls5m0szfzslsrsujf/+bvr57eqz/fmz/fbzudtj6y4fheddv7gprrqszryyitrk6tb5q3k1vbo7t71f2db8kzqcahnhhrc7aerhnwffumeqkahacmushozup3h0fiypn7nu7bj/egorelwgnjpfpa/czvu3e7x61zdwsgc5nybamhap1+9dsloclisbrhwerxtlll51goshhmkl4micvkhafgaprggksfz06emcdacc2ic12jmmfq740cx1ko40bpxlgn5ai3ff/z3exff35okzrtkjd5q1hgtmxn6f/nkaogio01wurqfatjhlhgonrkfs+eygvnxhuywhmdib0ywdjxgvxlun3xrlwui2zk6agdo1nko3puqreogxxeeefp6aw9gk/gm/fufmxhs0axc4j+wpj8ayr/lx4=</latexit> <latexit sha1_base64="gtul+drdugk6v53qezetuzk/z6g=">aaab/3icbvbns8naen3ur1q/qh69bivgqss1ob6eoj14rgbsiqlls5m0szfzslsrsujf/+bvr57eqz/fmz/fbzudtj6y4fheddv7gprrqszryyitrk6tb5q3k1vbo7t71f2db8kzqcahnhhrc7aerhnwffumeqkahacmushozup3h0fiypn7nu7bj/egorelwgnjpfpa/czvu3e7x61zdwsgc5nybamhap1+9dsloclisbrhwerxtlll51goshhmkl4micvkhafgaprggksfz06emcdacc2ic12jmmfq740cx1ko40bpxlgn5ai3ff/z3exff35okzrtkjd5q1hgtmxn6f/nkaogio01wurqfatjhlhgonrkfs+eygvnxhuywhmdib0ywdjxgvxlun3xrlwui2zk6agdo1nko3puqreogxxeeefp6aw9gk/gm/fufmxhs0axc4j+wpj8ayr/lx4=</latexit> <latexit sha1_base64="gtul+drdugk6v53qezetuzk/z6g=">aaab/3icbvbns8naen3ur1q/qh69bivgqss1ob6eoj14rgbsiqlls5m0szfzslsrsujf/+bvr57eqz/fmz/fbzudtj6y4fheddv7gprrqszryyitrk6tb5q3k1vbo7t71f2db8kzqcahnhhrc7aerhnwffumeqkahacmushozup3h0fiypn7nu7bj/egorelwgnjpfpa/czvu3e7x61zdwsgc5nybamhap1+9dsloclisbrhwerxtlll51goshhmkl4micvkhafgaprggksfz06emcdacc2ic12jmmfq740cx1ko40bpxlgn5ai3ff/z3exff35okzrtkjd5q1hgtmxn6f/nkaogio01wurqfatjhlhgonrkfs+eygvnxhuywhmdib0ywdjxgvxlun3xrlwui2zk6agdo1nko3puqreogxxeeefp6aw9gk/gm/fufmxhs0axc4j+wpj8ayr/lx4=</latexit> <latexit sha1_base64="8pxd3drarazo58thqkbav60tu6c=">aaace3icbzdlssnafiynxmu9rv2kecycg0tsc+pcknqfywrgfpoqjtotdujkwsxekker38f3cktbv+lwb3dnozhps9dwh4b55z/nmdofnzaqpgl+aquls8srq6w18vrg5ta2vrn7l+kue7bjzgle8beariowjzumogkhhpom2v7woq+3h4algkd3cpsag+j+ranksfsrpx+0vnqle3bmsrrt9kxxdqq22sn04okvs2pozmwbqzavvkjl6d9olyzpcjekdavrtcxeuhnmkhig47ktckgwgei+djwncajczsbfgbthkukzqczviqqxsx9pzdguyht6qjpecibma3n4x62byudczwiupbiimr0osjkhyynnyvqobylzsblmofvvncgakyzskss7pqiczuy0gumci7fmacwbu1a9qfq39urjqmbtqvvoeb0jc52hbrpblwqjgh7rm3pbr9qt9qa9ax/t1gwtmnldf6r9/gagdjy9</latexit> <latexit sha1_base64="8pxd3drarazo58thqkbav60tu6c=">aaace3icbzdlssnafiynxmu9rv2kecycg0tsc+pcknqfywrgfpoqjtotdujkwsxekker38f3cktbv+lwb3dnozhps9dwh4b55z/nmdofnzaqpgl+aquls8srq6w18vrg5ta2vrn7l+kue7bjzgle8beariowjzumogkhhpom2v7woq+3h4algkd3cpsag+j+ranksfsrpx+0vnqle3bmsrrt9kxxdqq22sn04okvs2pozmwbqzavvkjl6d9olyzpcjekdavrtcxeuhnmkhig47ktckgwgei+djwncajczsbfgbthkukzqczviqqxsx9pzdguyht6qjpecibma3n4x62byudczwiupbiimr0osjkhyynnyvqobylzsblmofvvncgakyzskss7pqiczuy0gumci7fmacwbu1a9qfq39urjqmbtqvvoeb0jc52hbrpblwqjgh7rm3pbr9qt9qa9ax/t1gwtmnldf6r9/gagdjy9</latexit> <latexit sha1_base64="8pxd3drarazo58thqkbav60tu6c=">aaace3icbzdlssnafiynxmu9rv2kecycg0tsc+pcknqfywrgfpoqjtotdujkwsxekker38f3cktbv+lwb3dnozhps9dwh4b55z/nmdofnzaqpgl+aquls8srq6w18vrg5ta2vrn7l+kue7bjzgle8beariowjzumogkhhpom2v7woq+3h4algkd3cpsag+j+ranksfsrpx+0vnqle3bmsrrt9kxxdqq22sn04okvs2pozmwbqzavvkjl6d9olyzpcjekdavrtcxeuhnmkhig47ktckgwgei+djwncajczsbfgbthkukzqczviqqxsx9pzdguyht6qjpecibma3n4x62byudczwiupbiimr0osjkhyynnyvqobylzsblmofvvncgakyzskss7pqiczuy0gumci7fmacwbu1a9qfq39urjqmbtqvvoeb0jc52hbrpblwqjgh7rm3pbr9qt9qa9ax/t1gwtmnldf6r9/gagdjy9</latexit> <latexit sha1_base64="8pxd3drarazo58thqkbav60tu6c=">aaace3icbzdlssnafiynxmu9rv2kecycg0tsc+pcknqfywrgfpoqjtotdujkwsxekker38f3cktbv+lwb3dnozhps9dwh4b55z/nmdofnzaqpgl+aquls8srq6w18vrg5ta2vrn7l+kue7bjzgle8beariowjzumogkhhpom2v7woq+3h4algkd3cpsag+j+ranksfsrpx+0vnqle3bmsrrt9kxxdqq22sn04okvs2pozmwbqzavvkjl6d9olyzpcjekdavrtcxeuhnmkhig47ktckgwgei+djwncajczsbfgbthkukzqczviqqxsx9pzdguyht6qjpecibma3n4x62byudczwiupbiimr0osjkhyynnyvqobylzsblmofvvncgakyzskss7pqiczuy0gumci7fmacwbu1a9qfq39urjqmbtqvvoeb0jc52hbrpblwqjgh7rm3pbr9qt9qa9ax/t1gwtmnldf6r9/gagdjy9</latexit> Universality for CFTs Truncated OPE a approximate relations for low-lying CFT data Universal: consistent with different CFTs in various dimensions (W. Li, JHEP01(2018)077) Inverse bootstrap: require general bootstrap solutions reproduce truncated OPEs Ansatz (Alday-Zhiboedov 15): Toy example 1 1 =1+ 2 <latexit sha1_base64="wp5db8c0w4g1ux7kuzsxymonvku=">aaacexicbzdlssnafiynxmu9rv12eyycijskcopcknqfywrgfppsjtotdujkwsyjueixvopv4fa3rsstt+dor3hazqgtpwx8/occzpnftwrxantfxtlyyuraemgjulm1vbnr7u3fqzivdfwwi1i2fkpa8ahc5cigluigos+g6q+vj/xma0jf4+gorwl0qtqpemazrw11zzkxdhjx8zchogz86zxmodo1y3bfnspabcehmsnv6jrfxi9maqgrmkgvajt2gp2msurmwljopqosyoa0d22nedu7o9n0e2prsds9k4ilfhfau/f3rezdpuahrztdigm1x5uy/9xakqbnnyxhsyoqsdmiibuwxtykeavhjtauiw2usa5vtdiasspq51b0ehb49cyrg0a61pe48wesglutxfsc29ny7srppkbk5jace4eckrq5iq3iekyeytn5ia/gk/fmvbsfs9yli585ih9kfp4aheqc6q==</latexit> P 2 = In 2d, the degenerate case 3 2 =4 1 (Esterlis-Fitzpatrick-Ramirez, 16) was found by the determinant method

10 <latexit sha1_base64="/xhnihfmg0nwxgn82ff3wgaoomm=">aaace3icbzdlssnafiyn9vbrrepshgarhejjiqauhkjdukzqwqejzti9aydolsycccv05tv4dm5160rc+gdufbsnbrba+spax3/o4zz5vvhwhzb1zeswlldw1/lrhy3nre2d4u7enyosyadfihhje48qedyefniucb9loienoo0nryf19gnixaowiamy3id2q+5zrlfb3ekhew9413aqb6bmfgmxp1atn7vfklwxpjixwc6grdi1usvvpxexjiaqmabkdwwrrjelejktmc44iykysihtq0djspvwn51+y2wea6dn+phul0rz6v6esgmg1cjwdgdacadmaxpzv1onqf/ctxkyjwghmy3ye2fize4ymxtcakmx0kcz5ppwkw2opax1cgwnb75tt506ckrjhyk9h8aitkqvi4p9e1qqxwxz5mkbosinxcznpezusio0ccop5jm8kffjyxgz3o2pwwvoygb2yr8znz9bo518</latexit> <latexit sha1_base64="/xhnihfmg0nwxgn82ff3wgaoomm=">aaace3icbzdlssnafiyn9vbrrepshgarhejjiqauhkjdukzqwqejzti9aydolsycccv05tv4dm5160rc+gdufbsnbrba+spax3/o4zz5vvhwhzb1zeswlldw1/lrhy3nre2d4u7enyosyadfihhje48qedyefniucb9loienoo0nryf19gnixaowiamy3id2q+5zrlfb3ekhew9413aqb6bmfgmxp1atn7vfklwxpjixwc6grdi1usvvpxexjiaqmabkdwwrrjelejktmc44iykysihtq0djspvwn51+y2wea6dn+phul0rz6v6esgmg1cjwdgdacadmaxpzv1onqf/ctxkyjwghmy3ye2fize4ymxtcakmx0kcz5ppwkw2opax1cgwnb75tt506ckrjhyk9h8aitkqvi4p9e1qqxwxz5mkbosinxcznpezusio0ccop5jm8kffjyxgz3o2pwwvoygb2yr8znz9bo518</latexit> <latexit sha1_base64="/xhnihfmg0nwxgn82ff3wgaoomm=">aaace3icbzdlssnafiyn9vbrrepshgarhejjiqauhkjdukzqwqejzti9aydolsycccv05tv4dm5160rc+gdufbsnbrba+spax3/o4zz5vvhwhzb1zeswlldw1/lrhy3nre2d4u7enyosyadfihhje48qedyefniucb9loienoo0nryf19gnixaowiamy3id2q+5zrlfb3ekhew9413aqb6bmfgmxp1atn7vfklwxpjixwc6grdi1usvvpxexjiaqmabkdwwrrjelejktmc44iykysihtq0djspvwn51+y2wea6dn+phul0rz6v6esgmg1cjwdgdacadmaxpzv1onqf/ctxkyjwghmy3ye2fize4ymxtcakmx0kcz5ppwkw2opax1cgwnb75tt506ckrjhyk9h8aitkqvi4p9e1qqxwxz5mkbosinxcznpezusio0ccop5jm8kffjyxgz3o2pwwvoygb2yr8znz9bo518</latexit> <latexit sha1_base64="/xhnihfmg0nwxgn82ff3wgaoomm=">aaace3icbzdlssnafiyn9vbrrepshgarhejjiqauhkjdukzqwqejzti9aydolsycccv05tv4dm5160rc+gdufbsnbrba+spax3/o4zz5vvhwhzb1zeswlldw1/lrhy3nre2d4u7enyosyadfihhje48qedyefniucb9loienoo0nryf19gnixaowiamy3id2q+5zrlfb3ekhew9413aqb6bmfgmxp1atn7vfklwxpjixwc6grdi1usvvpxexjiaqmabkdwwrrjelejktmc44iykysihtq0djspvwn51+y2wea6dn+phul0rz6v6esgmg1cjwdgdacadmaxpzv1onqf/ctxkyjwghmy3ye2fize4ymxtcakmx0kcz5ppwkw2opax1cgwnb75tt506ckrjhyk9h8aitkqvi4p9e1qqxwxz5mkbosinxcznpezusio0ccop5jm8kffjyxgz3o2pwwvoygb2yr8znz9bo518</latexit> Add stress tensor 1 1 = T Slightly larger errors (2-3%) for 2d minimal models with m = 4, 5, 6, 7. Universal because effective OPEs coincide

11 Outline OPE truncation OPE hierarchy Effective Field Theory analogue Variational method: error minimization

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sha1_base64="fun3b8t2/6adkzkqhvxih40njlm=">aaacexicbzdlsgmxgiuz9vbrbdrln8eiviqyuwrqqijahcsk1hy6twtsf9rqziuki9shc9/bd3crw1fi1idw56oyabvq1gobj3p+nythjtityrk+jmzs8srqwny9t7g5tb1j7u7dyjawfbo05kfouuqczwe0ffmcwpea4rscmu7wms2b9yakc4mbnyqg45n+wdxgidjw18w/ndueibqnfucmwnfsti9tolorj7tmwspze+ffsgdqqdpvu+a30wtp7eogkcdstm0rup2ecmuoh3hoisveha5jh9oaa+kd7cstt4zxoxz62aufpohce/f3rkj8kue+qyd9ogzypkvn/7j2rlzttskckfyq0olfxsyxcnhaco4xavtxkqzcbdnvxxradcdk95zzeua5tcspavckrcsel2arguxswcm+rhsqf7nusiipdlar2egevdevqqmgougrpamx9go8gw/gu/exhc0ys5199efg5w/dzpze</latexit> <latexit sha1_base64="fun3b8t2/6adkzkqhvxih40njlm=">aaacexicbzdlsgmxgiuz9vbrbdrln8eiviqyuwrqqijahcsk1hy6twtsf9rqziuki9shc9/bd3crw1fi1idw56oyabvq1gobj3p+nythjtityrk+jmzs8srqwny9t7g5tb1j7u7dyjawfbo05kfouuqczwe0ffmcwpea4rscmu7wms2b9yakc4mbnyqg45n+wdxgidjw18w/ndueibqnfucmwnfsti9tolorj7tmwspze+ffsgdqqdpvu+a30wtp7eogkcdstm0rup2ecmuoh3hoisveha5jh9oaa+kd7cstt4zxoxz62aufpohce/f3rkj8kue+qyd9ogzypkvn/7j2rlzttskckfyq0olfxsyxcnhaco4xavtxkqzcbdnvxxradcdk95zzeua5tcspavckrcsel2arguxswcm+rhsqf7nusiipdlar2egevdevqqmgougrpamx9go8gw/gu/exhc0ys5199efg5w/dzpze</latexit> OPE convergence In unitary CFTs, the OPE convergence is exponentially rapid (Pappadopulo-Rychkov-Espin-Rattazzi, 12) X i> P i F i,l i (u, v) < (2 )4 (4 + 1) (z =1/2) = 3 2 p <latexit 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sha1_base64="/jq6hprmng8ehekzr6pin4dvuv0=">aaacfxicbva9swnben3zm8avqkwfi0hqwngxbbuibe1hqwbmibfc3myuwdz7chdoiedk/4p/wvzbk7g1tvonuikp1phg2cd7m8zm82ipnnr2hzuxotu9m5uzy84vlc4t51zwr3suka5vhsli1t2mqyoqqihqqj1wwajpqs27ph34tvtqwkthjfziaaasewpfcizgauu2xnwntu9kzl5xp7s/w3t1jujqphfqu+actnj5u2apqcejmyj5msj5k/fptioebbail0zrhmph2eyzqsel9lnuoifm/jp1ogfoyalqzxr4sj9ugavn/uizfyidqj87uhzo3qs8uxkw7oq/3kd8z2sk6b81uxhgculivwf5iaqy0ueqtc0ucjq9qxhxwuxkezcpxtfkl3xb4luv1k2arny3kth/axgn1wlhuobchotlj6nsmmsdbjjt4pbduizn5jxucsf35je8kwfrwxqxxq2379ija9szrn7bev8c3xecgw==</latexit> <latexit sha1_base64="/jq6hprmng8ehekzr6pin4dvuv0=">aaacfxicbva9swnben3zm8avqkwfi0hqwngxbbuibe1hqwbmibfc3myuwdz7chdoiedk/4p/wvzbk7g1tvonuikp1phg2cd7m8zm82ipnnr2hzuxotu9m5uzy84vlc4t51zwr3suka5vhsli1t2mqyoqqihqqj1wwajpqs27ph34tvtqwkthjfziaaasewpfcizgauu2xnwntu9kzl5xp7s/w3t1jujqphfqu+actnj5u2apqcejmyj5msj5k/fptioebbail0zrhmph2eyzqsel9lnuoifm/jp1ogfoyalqzxr4sj9ugavn/uizfyidqj87uhzo3qs8uxkw7oq/3kd8z2sk6b81uxhgculivwf5iaqy0ueqtc0ucjq9qxhxwuxkezcpxtfkl3xb4luv1k2arny3kth/axgn1wlhuobchotlj6nsmmsdbjjt4pbduizn5jxucsf35je8kwfrwxqxxq2379ija9szrn7bev8c3xecgw==</latexit> The prefactor is not optimal, concrete CFTs work even better For the bootstrap equation, we want to know X v F P i,li(u, v) (u $ v) i <? v u i> use max v (z),u (1 z) will overestimate the truncation error (smaller due to the minus sign) z = 4 (1 + ) 2

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sha1_base64="gacbfkewefmzfomuxur6eyov0qu=">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</latexit> <latexit sha1_base64="gacbfkewefmzfomuxur6eyov0qu=">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</latexit> Radial coordinate Using the radial coordinates, conformal blocks are rapidly convergent power series (Hogervorst-Rychkov, 13) G,l (u, v) = 1X n=0 z +n X j An estimate at z=1/2 v r (u $ v) v u z!1/2 A n,j C j (cos ) = exponential decay for large exponents Sign flipping due to Gegenbauer polynomials = = re i <latexit 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sha1_base64="b9jqtusleaszfv7urexy4yehsjw=">aaaccxicbzdlssnafiyn9vbrldqlm2arxehjrfaxqteuxfywttdempmcnemnf2zohbl6br6dw926erc+htsfxfsy0nyfbj7+/xzm8hup4apn80srls2vrk6v1ysbm1vbo/ru3r1kmsnazolizmejcgspwuaoajqpbbp5atre4hqctx9bkp7edzhmwy1op+ybzxqlq6dxhrkml9i5hoecoxgc0lfpr5l1cyjjeawz1mhmrz7+7fgjyykikqmqvncyu3rzkpezaaokkylikrvqpnqljgkeys0nx4+mw8lxjscrxyvrmli/n3iaktwmvgiyohiq+wxs/pd1mwzo3zzhayyqs+lhqsymtixxe4bpjtauwwiok7y41wahlzrh0vff8sfwmrntbdgtxjovybhsk/pf3bo9rtwuzt2uyt45iefeimekqw5ii9ieksf5ji/kvxvs3rr37wm6wtjmo1xyr9rnd0emmmw=</latexit> 1X n=0 r +n X j B n,j C j (cos ) p 1 (3 2 p 2) 2

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sha1_base64="b+yr4vxzm0v98s0r+qwue/jw/vi=">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</latexit> <latexit sha1_base64="b+yr4vxzm0v98s0r+qwue/jw/vi=">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</latexit> <latexit sha1_base64="b+yr4vxzm0v98s0r+qwue/jw/vi=">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</latexit> <latexit sha1_base64="b+yr4vxzm0v98s0r+qwue/jw/vi=">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</latexit> Example: 2d Ising Spin-spin OPE Bootstrap equation at z = 1/2 =1+ + T +[4, 4] + [4, 0] + [5, 4] + [6, 6] + [6, 2] + [7, 6] =1 e 0.17 e 1.86 e 6.14 e 6.71 e 8.94 e 9.95 e 10.6 e apple [ <latexit sha1_base64="s7wm1o0xojqvtlukzmdoa8n3p9y=">aaacbxicbzbls8nafiunpmt9norszwar3fisulb3rbtwwchaqhplzhltdp1mwsxekkf7/4nb3bost/4od/4up4+fth4y+djnxuzygpqzpr3ny1pzxvvf2cxsfbd3dvdk9v7bvuoysaffe57itkauccagpznm0eklkdjg0a6g15o8/qhssutc6vekfkz6gkwmem2snl2ch/ysovyui7ftcws9u+xunknwmrhzkko5mj372wstmsugnoveqa7rpnrpidsmchgxvuxbsuiq9kfrujaylj9pdx/je+oeoeqkeuljqft7iyexuqm4mjmx0qo1me3m/7jupqmlp2ciztqiovsoyjjwcz60gemmgwo+mkcozozwtadeeqpnv0uvhmhr5f4duczju4m7wmaytkqvy4p7wyvxr+bdfnarokanyexnqi5uubo1eeuzekyv6nv6st6sd+tjnrpizxco0r9znz8uhpd0</latexit> i,l i ]! log e Only when close to z=1/2? P i v F i,l i (u, v) (u $ v) v u z!1/2

15 Real axis By exact 2d blocks, examine each contribution, not their sums/truncated sum rules z ϵ T [4,4] [4,0] [5,4] [6,6] [6,2] [7,6] large gap below T due to decoupling of a spin-2 primary (sparse spectrum)

16 <latexit sha1_base64="a6atnyqnwat7jvtyinprcydk5cs=">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</latexit> <latexit sha1_base64="a6atnyqnwat7jvtyinprcydk5cs=">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</latexit> <latexit sha1_base64="a6atnyqnwat7jvtyinprcydk5cs=">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</latexit> <latexit sha1_base64="a6atnyqnwat7jvtyinprcydk5cs=">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</latexit> Imaginary axis θ ϵ T [4,4] [4,0] [5,4] -15 [ i,l i ]! log P i v F i,l i (u, v) (u $ v) v u Re[z]=1/2

17 Some comments OPE hierarchy most clear at z=1/2 but persists in a large region Delicate balance of different contributions Remarkable consequence: OPE truncation error is dominated by 1 conformal multiplet (come back to this later) 2d Lee-Yang CFT is similar (non-unitary) z [-0.4,0] -2 T -4 [3.6,4] -6 [4,0] -8 [5.6,6] [7,6] [-0.4,0] T [3.6,4] [4,0] -15

18 Outline OPE truncation OPE hierarchy Effective Field Theory analogue Variational method: error minimization

19 Effective Field Theory Another great legacy of Ken Wilson EFT is useful when a large hierarchy Large dimension operators suppressed by small coupling constants A few terms in effective Lagrangians capture main physics Effective large N CFTs are related to AdS locality (Heemskerk-Penedones-Polchinski-Sully 09, Fitzpatrick-Katz-Poland- Simmons-Duffin 10)

20 Effective OPE Truncated/effective OPE 34 effective Lagrangian OPE coefficient 34 coupling constant OPE hierarchy: high-lying operators suppressed by small OPE coefficients Effective OPEs simplify the strong coupling problem Shifman Vainshtein Zakharov sum rules in QCD Gliozzi s determinant method in conformal bootstrap

21 Energy scale What are the analogue of energy scale? Low energy limit: crossing symmetric point z=1/2 OPE hierarchies are largest Effective OPEs remains good approximations in a large region around z=1/2 EFT needs to know when effective description breaks down

22 UV cutoff High energy limit: branch cuts (z>1, z<0, infinity) Approaching branch cuts, high-lying operators become more and more important Gradually, an effective OPE does not make sense: one of the omitted operators becomes comparable -3.0 Example: 2d Ising -3.5 Cutoff also depends on precision z -4.0 ϵ T [4,4] [4,0] [5,4] [6,6] [6,2] [7,6]

23 <latexit sha1_base64="jtzl5cxjnxvlmrmhclrb8ysgdgy=">aaacfxicbza9swnbeib34lemx1flcw+diatcnqhqiqrnyrkhmyfcchubuwtj3t6xoyeei6x/wf9gq62v2frb+vpcfbsa+mlcwzszzozrx4jrdjwvk7o0vlk6ll3pbwxube/kd/fudzqobnuwiug1fapbcal15ciggsugos+g4q9uxvxgayjni1ndyqztkpykdzijakxo/tdtvbdsd3kiespxbtgdwhmrywktky84jwciexhcgrtitnvo/tvrriwjqsitvouw68tytqlczgsmcl6iiazsqhvqmiip2dtojx8z2cfg6dpbpmytae/c3xmpdbuehr7pdcn29xxtbp5xayuyxlrtlumeqblpoiarnkb2obw7yxuwfemdlclubrvznyrk0gsx87oqejxuq4baojkrupmblel9thrzcu/ocuxrwtzzckcoyalxytkpk1tsjxxcycn5ji/k1xqy3qx362pamrfmm/vkj6zph0rxnzg=</latexit> <latexit sha1_base64="jtzl5cxjnxvlmrmhclrb8ysgdgy=">aaacfxicbza9swnbeib34lemx1flcw+diatcnqhqiqrnyrkhmyfcchubuwtj3t6xoyeei6x/wf9gq62v2frb+vpcfbsa+mlcwzszzozrx4jrdjwvk7o0vlk6ll3pbwxube/kd/fudzqobnuwiug1fapbcal15ciggsugos+g4q9uxvxgayjni1ndyqztkpykdzijakxo/tdtvbdsd3kiespxbtgdwhmrywktky84jwciexhcgrtitnvo/tvrriwjqsitvouw68tytqlczgsmcl6iiazsqhvqmiip2dtojx8z2cfg6dpbpmytae/c3xmpdbuehr7pdcn29xxtbp5xayuyxlrtlumeqblpoiarnkb2obw7yxuwfemdlclubrvznyrk0gsx87oqejxuq4baojkrupmblel9thrzcu/ocuxrwtzzckcoyalxytkpk1tsjxxcycn5ji/k1xqy3qx362pamrfmm/vkj6zph0rxnzg=</latexit> <latexit sha1_base64="jtzl5cxjnxvlmrmhclrb8ysgdgy=">aaacfxicbza9swnbeib34lemx1flcw+diatcnqhqiqrnyrkhmyfcchubuwtj3t6xoyeei6x/wf9gq62v2frb+vpcfbsa+mlcwzszzozrx4jrdjwvk7o0vlk6ll3pbwxube/kd/fudzqobnuwiug1fapbcal15ciggsugos+g4q9uxvxgayjni1ndyqztkpykdzijakxo/tdtvbdsd3kiespxbtgdwhmrywktky84jwciexhcgrtitnvo/tvrriwjqsitvouw68tytqlczgsmcl6iiazsqhvqmiip2dtojx8z2cfg6dpbpmytae/c3xmpdbuehr7pdcn29xxtbp5xayuyxlrtlumeqblpoiarnkb2obw7yxuwfemdlclubrvznyrk0gsx87oqejxuq4baojkrupmblel9thrzcu/ocuxrwtzzckcoyalxytkpk1tsjxxcycn5ji/k1xqy3qx362pamrfmm/vkj6zph0rxnzg=</latexit> <latexit sha1_base64="jtzl5cxjnxvlmrmhclrb8ysgdgy=">aaacfxicbza9swnbeib34lemx1flcw+diatcnqhqiqrnyrkhmyfcchubuwtj3t6xoyeei6x/wf9gq62v2frb+vpcfbsa+mlcwzszzozrx4jrdjwvk7o0vlk6ll3pbwxube/kd/fudzqobnuwiug1fapbcal15ciggsugos+g4q9uxvxgayjni1ndyqztkpykdzijakxo/tdtvbdsd3kiespxbtgdwhmrywktky84jwciexhcgrtitnvo/tvrriwjqsitvouw68tytqlczgsmcl6iiazsqhvqmiip2dtojx8z2cfg6dpbpmytae/c3xmpdbuehr7pdcn29xxtbp5xayuyxlrtlumeqblpoiarnkb2obw7yxuwfemdlclubrvznyrk0gsx87oqejxuq4baojkrupmblel9thrzcu/ocuxrwtzzckcoyalxytkpk1tsjxxcycn5ji/k1xqy3qx362pamrfmm/vkj6zph0rxnzg=</latexit> Example: 2d Ising z ϵ T [4,4] [4,0] [5,4] [6,6] [6,2] [7,6] : UV cutoff for the effective OPE =1+ + T

24 Outline OPE truncation OPE hierarchy Effective Field Theory analogue Variational method: error minimization

25 How to tame the truncation error? From a review talk by Slava Rychkov in 2015