Slow running of the Gradient Flow coupling from 200 MeV to 4 GeV in $N_{\rm f}=3$ QCD

Slow unning o he g adien low coupling om 200 MeV o 4 GeV in N = 3 QCD Ma ia Dalla B ida,1Pa ick F i zsch,2Tomasz Ko zec,3Albe o Ramos,4S e an Sin ,5and Raine Somme 1,6 1John on Neumann Ins i u e o Compu ing (NIC), DESY, Pla anenallee 6, 15738 Zeu hen, Ge many 2Ins i u o de Física Teó ica UAM/CSIC, Uni e sidad Au ónoma de Mad id, C/Nicolás Cab e a 13-15, Can oblanco, Mad id 28049, Spain 3Depa men o Physics, Be gische Uni e si ä Wuppe al, Gaußs . 20, 42119 Wuppe al, Ge many 4Theo e ical Physics Depa men , CERN, CH 1211 Gene a 23, Swi ze land 5School o Ma hema ics, T ini y College Dublin, Dublin 2, I eland 6Ins i u ü Physik, Humbold -Uni e si ä zu Be lin, New ons . 15, 12489 Be lin, Ge many (Recei ed 17 No embe 2016; published 25 Janua y 2017) Using a ini e olume g adien low eno maliza ion scheme wi h Sch ödinge Func ional bounda y condi ions, we compu e he nonpe u ba i e unning coupling in he ange 2.2≲¯ g2 GFðLÞ≲13. Ca e ul con inuum ex apola ions u n ou o be c ucial o each ou high accu acy. The unning o he coupling is always be ween one loop and wo loop and e y close o one loop in he egion o 200 MeV ≲μ¼ 1=L ≲4GeV. While he e is no con incing con ac o wo-loop unning, we ma ch nonpe u ba i ely o he Sch ödinge unc ional coupling wi h backg ound ield. In his case, we know he μ-dependence up o ∼100 GeV and can hus connec o he Λ-pa ame e . DOI: 10.1103/PhysRe D.95.014507 I. INTRODUCTION The ene gy dependence o he s ong coupling cons an αsðμÞin a physical scheme p o ides in o ma ion on how o connec he low and high ene gy egimes o QCD. Rela ing hese e y di e en domains o he s ong in e ac ions is key o p o iding a solid de e mina ion o he undamen al pa ame e s o he S anda d Model [1]. La ice QCD is in p inciple an ideal ool o such s udies. Obse ables de ined a sho Euclidean dis ances can be used o a nonpe u ba i e physical coupling de ini ion (see o example Re . [2] and e e ences ci ed he ein), and i s alue can be ex ac ed accu a ely ia Mon e Ca lo simula ions. A di ec implemen a ion o his p og am has o ace he so-called window p oblem: he sho Euclidean dis ance used o de ine he eno maliza ion scale has o be bo h la ge compa ed o he la ice spacing aand small compa ed o he o al size o he box (deno ed by L)used in he simula ion. Since he box has o be la ge enough o desc ibe had onic physics, compu a ional cons ain s se e ely limi he ange o eno maliza ion scales ha one can s udy. Fini e size scaling p o ides an elegan solu ion o his p oblem [3]. Rela ing he eno maliza ion scale μwi h he ini e size o he box ia μ¼1=L, he coupling ¯ g2ðLÞ depends on only one scale.1La ices o di e en olumes can be ma ched, allowing us o compu e he s ep scaling unc ion σðuÞ[3]. I measu es how much he coupling changes when he eno maliza ion scale changes by a ixed ac o , which we se o 2, σðuÞ¼¯ g2ð2LÞj¯ g2ðLÞ¼u:ð1:1Þ I can be conside ed a disc e e e sion o he eno maliza- ion g oup β- unc ion. The exac ela ion is logð2Þ¼−Zffiffiffiffiffiffiffi σðuÞ p ffiffiu p dx βðxÞ;ð1:2Þ wi h he con en ion βð¯ gÞ¼−L∂¯ gðLÞ ∂L∼ ¯ g→0 −b0¯ g3−b1¯ g5þ…;ð1:3Þ whe e he uni e sal coe icien s in he asymp o ic expan- sion ake he alues b0¼9=ð16π2Þand b1¼1=ð4π4Þin N ¼3QCD. Once σðuÞis known, one can choose a e e ence scale L e and se u0¼¯ g2ðL e Þ. The ecu si e ela ion uk¼σ−1ðuk−1Þ;k¼1;…;n sð1:4Þ allows one o ela e nonpe u ba i ely he scale 1=L e wi h he scales 2k=L e o k¼0;…;n s. A ew i e a ions su ice Published by he Ame ican Physical Socie y unde he e ms o he C ea i e Commons A ibu ion 4.0 In e na ional license. Fu he dis ibu ion o his wo k mus main ain a ibu ion o he au ho (s) and he published a icle’s i le, jou nal ci a ion, and DOI. 1We use a massless eno maliza ion scheme. Reno malized couplings and eno maliza ion ac o s o qua k masses and composi e ope a o s a e de ined a ze o qua k mass, and he eno maliza ion g oup unc ions do no depend on he qua k masses. PHYSICAL REVIEW D 95, 014507 (2017) 2470-0010=2017=95(1)=014507(19) 014507-1 Published by he Ame ican Physical Socie y o connec a had onic low ene gy scale wi h he elec o- weak scale. This is he s a egy o he ALPHA Collabo a ion. Using he so-called Sch ödinge Func ional (SF) scheme [4,5], QCD wi h N ¼0, 2 and N ¼4qua k la o s has been s udied [6–8]. O immedia e ele ance o he p esen wo k is he ecen applica ion o his echnique o he high ene gy domain o N ¼3QCD [1]. The e, he ene gy dependence o he s ong coupling was s udied be ween he elec oweak scale and a e e ence scale 1=L e ¼1=L0∼4GeV, de ined by ¯ g2 SFðL0Þ¼2.012, wi h e y high accu acy. This s a egy is heo e ically e y appealing bu mee s some di icul ies i he e e ence scale 1=L e is lowe ed u he in o he had onic egime. The compu a ional cos o measu ing he SF coupling g ows as a low ene gies and in pa icula owa d he con inuum limi . Thus, i is challenging o each he low ene gy domain cha ac e is ic o had onic physics, especially i one aims a main aining he high p ecision achie ed in Re . [1]. The ecen ly p oposed coupling de ini ions based on he g adien low (GF) [9] a e much be e sui ed o his ask. The ela i e p ecision o he GF coupling in a Mon e Ca lo simula ion is ypically high and shows a weak dependence on bo h he ene gy scale and he cu o (see Re . [10] o a ecen e iew and mo e quan i a i e s a emen s). Mo eo e , GF couplings can easily be used in combina ion wi h ini e size scaling and a pa icula choice o bounda y con- di ions [11–14]. In his wo k, we use he GF coupling de ined wi h SF bounda y condi ions [12] [deno ed by ¯ g2 GFðLÞ] o connec nonpe u ba i ely he in e media e ene gy scale 1=L0 wi h a had onic e e ence scale 1=Lhad de ined by he condi ion ¯ g2 GFðLhadÞ¼11.31:ð1:5Þ The main esul o his pape is he ela ion Lhad ¼21.86ð42ÞL0:ð1:6Þ As he eade will see, ou choices o la ice disc e iza ion and scale 1=Lhad a e such ha Lhad can be ela ed wi h he pion and kaon decay cons an s by using he CLS ensembles [15]. This wo k he e o e ep esen s an essen ial s ep in he ALPHA Collabo a ion e o o a i s p inciples de e mi- na ion o he s ong coupling cons an and qua k masses a he elec oweak scale in e ms o low ene gy had onic obse ables [16,17]. The pape is o ganized as ollows. In Sec. II, we ix ou no a ion and in oduce he de ails o ou coupling de i- ni ion. Sec. III discusses gene al aspec s o aking he con inuum limi , while Sec. IV con ains he ex ac ion o he con inuum σðuÞ. A e a i ing a ou main esul in Sec. V, we discuss ou indings in Sec. VI. II. RUNNING COUPLING A. Con inuum We wo k in ou -dimensional Euclidean space and conside s anda d SF bounda y condi ions wi h ze o back- g ound ield [4,5]. In summa y, gauge ields a e pe iodic in he h ee spa ial di ec ions wi h pe iod L, and he spa ial componen s k¼1, 2, 3 o he gauge ield sa is y homo- geneous Di ichle bounda y condi ions in ime, Akð0;xÞ¼AkðT;xÞ¼0:ð2:1Þ Fe mion ields a e equi ed o obey pe iodic bounda y condi ions in space up o a phase, ψðxþL ˆ kÞ¼e{θψðxÞ; ¯ ψðxþL ˆ kÞ¼e−{θ¯ ψðxÞ:ð2:2Þ We choose he alue θ¼1=2[18]. De ining he p ojec o s P¼1 2ð1γ0Þ, he ime bounda y condi ions ead Pþψð0;xÞ¼0¼¯ ψð0;xÞP−; P−ψðT;xÞ¼0¼¯ ψðT;xÞPþ:ð2:3Þ The GF [9,19] de ines a amily o gauge ields Bμð ; xÞ pa ame ized by he low ime ≥0 ia he equa ion2 ∂ Bμð ; xÞ¼DνGνμð ; xÞ;B μð0;xÞ¼AμðxÞ;ð2:4Þ whe e Dμ¼∂μþ½Bμ;·is he co a ian de i a i e and Gμνð ; xÞis he ield s eng h enso o he low ield, Gμν ¼∂μBν−∂νBμþ½Bμ;B ν:ð2:5Þ Gauge in a ian composi e ope a o s de ined om he low ield Bμð ; xÞa e eno malized obse ables; see Re . [20]. In pa icula , ou de ini ion o a unning coupling ollows he p oposal o using he ac ion densi y a posi i e low ime [9]. In a ini e olume and wi h ou choice o bounda y condi ions, he unning coupling was de ined in Re . [12], ¯ g2 GFðLÞ¼N−1ðcÞ 2 4hGa ijð ; xÞGa ijð ; xÞδQ;0i hδQ;0iffiffiffi 8 p¼cL;x0¼T=2 ; ð2:6Þ whe e NðcÞis a known unc ion (see Table I). No e ha we use only he spa ial componen s o he ield s eng h enso o de ine he coupling. As a gued in Re . [12], bounda y e ec s a e smalle o his pa icula coupling de ini ion, 2Unless s a ed o he wise, epea ed g eek indices a e summed om 0 o 3. Repea ed la in indices a e ei he summed om 1 o 8 (a; b; …) o om 1 o 3 (i; j; …). MATTIA DALLA BRIDA e al. PHYSICAL REVIEW D 95, 014507 (2017) 014507-2 while we ha e obse ed ha one does no lose nume ical p ecision. The coupling is de ined by p ojec ing o he sec o o anishing opological cha ge, Q¼ 1 32π2RxϵμνρσGa μνð ; xÞGa ρσð ; xÞ, ia he inse ion o δQ;0in o he pa h in eg al expec a ion alues. This choice is con- enien because la ice simula ions wi h SF bounda y condi ions su e om he opology eezing p oblem a small la ice spacing [21–23]. P ojec ing o he ze o cha ge sec o a oids his p oblem [23]. The eno maliza ion scheme is comple ely de ined by speci ying T¼Land c¼0.3:ð2:7Þ This choice is ixed in his wo k, apa om Sec. III whe e we also conside o he alues o c. B. La ice Fo ou la ice compu a ions, we wo k on a ðL=aÞ3× ðT=aÞla ice wi h la ice spacing a. We use he ee-le el imp o ed Symanzik gauge ac ion [24]. Wi h S0and S1 deno ing he se o 1×1and 2×1o ien ed loops, espec i ely, we ha e SG½U¼1 g2 0X 1 k¼0 ckX C∈Sk wkðCÞ ½1−UðCÞ;ð2:8Þ whe e UðCÞdeno es he p oduc o he link a iables UμðxÞ a ound he loop C. T ee-le el Oða2Þbulk imp o emen is gua an eed by choosing c0¼5=3and c1¼−1=12. Modi ica ions o he gauge ac ion nea he ime bounda ies x0¼0,Tlead o Sch ödinge Func ional bounda y con- di ions in he con inuum. We s ick o op ion B o Re . [25] and choose he weigh s wkðCÞas ollows3: w0ðCÞ¼8 > > < > > : 1=2;all links in Ca e on he ime bounda y c ðg0Þ;Chas one link on he ime bounda y 1;o he wise ; ð2:9aÞ w1ðCÞ¼8 > > < > > : 1=2;all links in Ca e on he ime bounda y 3=2;Chas wo links on he ime bounda y 1;o he wise : ð2:9bÞ The imp o emen coe icien c is inse ed wi h he a ail- able one-loop p ecision; see Sec. III A. We simula e h ee massless la o s o nonpe u ba i ely OðaÞ-imp o ed Wilson e mions wi h ac ion SF½U; ¯ ψ;ψ¼a4X xX N i¼1 ¯ ψiðxÞðDþm0ÞψiðxÞ;ð2:10Þ whe e m0is he ba e qua k mass ha we se o he c i ical alue mc . The Di ac ope a o can be decomposed as D¼DwþδDsw þδDbnd;ð2:11Þ whe e Dwis he usual la ice Wilson-Di ac ope a o , δDswψðxÞ¼acsw { 4σμνFcl μνðxÞψðxÞð2:12Þ is he Sheikholeslami-Wohle e m [27] wi h Fcl μν being he la ice clo e disc e ized e sion o he ield s eng h enso , and inally δDbndψðxÞ¼ð~ c −1Þ1 aðδx0=a;1þδx0=a;T=a−1ÞψðxÞ ð2:13Þ is he con ibu ion o he e mionic bounda y coun e e m [28]. We use he nonpe u ba i ely de e mined cswðg0Þ TABLE I. T ee-le el alues o 2hEmagð ; xÞi o c¼ffiffiffiffi 8 p=L ¼0.3,x0¼T=2and bo h he Oða2Þ ee-le el imp o ed Lüsche -Weisz gauge ac ion and he plaque e gauge ac ion (c . Sec. V). We quo e he wo choices o low disc e iza ions ha we conside in his wo k: Zeu hen low/Lüsche -Weisz (LW) obse able (labelled ZFL) and Wilson low/clo e obse able (labelled WFL). See Sec. III o mo e de ails. The con inuum alue NðcÞis also epo ed. Lüsche -Weisz ac ion Plaque e ac ion L=a ZFL WFL ZFL WFL 89.031615 ×10−37.321606 ×10−39.769331 ×10−37.886150 ×10−3 12 8.650918 ×10−37.985873 ×10−38.964291 ×10−38.273666 ×10−3 16 8.591758 ×10−38.232878 ×10−38.768579 ×10−38.402436 ×10−3 24 8.569388 ×10−38.414731 ×10−38.648576 ×10−38.492610 ×10−3 32 8.565542 ×10−38.479524 ×10−38.610253 ×10−38.523837 ×10−3 Con inuum limi 8.563741 ×10−3 3All simula ions we e pe o med wi h a modi ied e sion o he openQCD 1.0 package [26]. The documen a ion o he package p o ides use ul in o ma ion o he in e es ed eade . SLOW RUNNING OF THE GRADIENT FLOW COUPLING …PHYSICAL REVIEW D 95, 014507 (2017) 014507-3 [29]. Excep a he ime bounda ies, ou ac ion is he same as he one used by he CLS Collabo a ion [15]. Wi h ou choice o bounda y condi ions in ime, he comple e emo al o OðaÞe ec s equi es he knowl- edge o he bounda y imp o emen coe icien s c ,~ c .We use hei alues de e mined in pe u ba ion heo y. Fo an es ima e o he unce ain y o pe u ba ion heo y, we use he las known e m in he pe u ba i e se ies, he one-loop e m (c . Sec. III A). De ails will be discussed la e . SF bounda y condi ions on he la ice a e imposed in comple e analogy o he con inuum coun e pa s. The gauge links obey UkðxÞjx0¼0;T ¼1;k¼1;2;3;ð2:14Þ while he e mion bounda y condi ions emain he same as in he con inuum, Eq. (2.3). The e is much eedom when ansla ing he GF equa- ion (2.4) and he ene gy densi y used o de ine he coupling [see Eq. (2.6)] o he la ice. Di e en choices di e only by cu o e ec s, bu hese can be subs an ial. A popula choice is he Wilson low (no summa ion o e μ) a2ð∂ Vμð ; xÞÞVμð ; xÞ†¼−g2 0∂x;μSW½V; Vμð0;xÞ¼UμðxÞ;ð2:15Þ whe e Vμð ; xÞa e he links a posi i e low ime and ∂x;μSW½Vis he o ce de i ing om he Wilson plaque e gauge ac ion [i.e. Eq. (2.8) wi h he choices c0¼1, c1¼0]. I has been shown [30] ha his choice in oduces Oða2Þcu o e ec s when in eg a ing he low equa ion. They can be a oided by using he Symanzik Oða2Þ imp o ed “Zeu hen low”equa ion (no summa ion o e μ) a2ð∂ Vμð ; xÞÞVμð ; xÞ†¼−g2 01þa2 12 Δμ∂x;μSLW½V; Vμð0;xÞ¼UμðxÞ;ð2:16Þ whe e ∂x;μSLW½Vis he o ce de i ing om he Symanzik ee-le el imp o ed (Lüsche -Weisz) gauge ac ion Eq. (2.8) (see Re . [30] o mo e de ails). We inse he co ec ion e m Δμ¼∇ μ∇μin o he low equa ion o all links ðx; μÞ excep o hose links ðx; 0Þwhe e an end poin ouches one o he SF bounda ies x0¼0,T. Fo hose links, we simply se Δ0¼0. The disc e ized obse able is de ined as he spa ial pa o he Lüsche -Weisz ac ion densi y, c . Eq. (2.8).Ou choices gua an ee ha , neglec ing small e ms coming om he ime bounda ies a x0¼0,T, nei he he g adien low no he de ini ion o he obse able in oduce any Oða2Þ cu o e ec s. The emaining cu o e ec s in ou low quan i ies a e hence p oduced by ou la ice ac ion Eqs. (2.8) and (2.10) and by he ini ial condi ion o he low equa ion a ¼0[30]. Al hough his is ou p e e ed se up, in se e al pa s o he wo k, we will compa e he esul s wi h he mo e s anda d Wilson low/clo e -obse - able disc e iza ion. A nonze o a=L, ou coupling de ini ion eads ¯ g2 GFðLÞ¼¯ g2 0.3ðLÞ;ð2:17Þ wi h ¯ g2 cðLÞ¼ 2 ˆ N−1ðc; a=LÞhEmagð ; xÞˆ δðQÞi hˆ δðQÞi ffiffiffi 8 p¼cL;x0¼T=2 ; ð2:18Þ and Emagð ; xÞ¼1 4½Ga ijð ; xÞGa ijð ; xÞLW:ð2:19Þ Se e al commen s a e in o de . We ha e chosen o de ine he coupling h ough jus he magne ic pa Emag o Esince his choice has a lowe sensi i i y o he bounda y imp o e- men coe icien c and because i s ( ee-le el) Oða2Þ imp o emen does no need any u he e ms.4As in Re . [12], he no maliza ion ac o ˆ Nðc; a=LÞis compu ed on he la ice wi h ou choices o disc e iza ion (ac ion, low, and obse able), such ha in he ela ion ¯ g2 GF ¼ g2 0þOðg4 0Þ he leading e m has all la ice a i ac s emo ed (see Table I). Due o he ac ha we use a ee-le el imp o ed ac ion, and nei he he Zeu hen low equa ion no he Lüsche -Weisz obse able disc e iza ion in oduces any Oða2Þa i ac s, we u he mo e ha e δðc; a=LÞ≡ ˆ Nðc; a=LÞ NðcÞ−1¼Oðða=LÞ4Þ;ð2:20Þ i.e. he la ice no maliza ion in ac only co ec s subleading Oðða=LÞ4Þ e ms. Finally, on he la ice, one has o cla i y wha is mean by p ojec ing o ze o opology. We de ine he opological cha ge by [9] Q¼1 32π2X x ϵμνρσ½Ga μνð ; xÞGa ρσð ; xÞcl;ð2:21Þ using he clo e disc e iza ion o he ield s eng h enso o V. We hen se ffiffiffiffi 8 p¼cL wi h c¼0.3and use he Zeu hen low. Wi h his de ini ion, he opological cha ge is no in ege alued bu app oaches in ege s close o he con inuum limi . The e o e, he K onecke δQ;0o he con inuum de ini ion is eplaced by 4In con as , he elec ic componen s would equi e addi ional e ms o cancel o al de i a i e con ibu ions ha do no anish because o ou Sch ödinge Func ional bounda y condi ions [30]. MATTIA DALLA BRIDA e al. PHYSICAL REVIEW D 95, 014507 (2017) 014507-4 ˆ δðQÞ¼1;i jQj<0.5 0;o he wise:ð2:22Þ III. GENERAL CONSIDERATIONS ON THE CONTINUUM LIMIT OF FLOW QUANTITIES All s udies o ini e size scaling wi h he GF scheme show signi ican cu o e ec s in he ex apola ions o he s ep scaling unc ion (see Re . [10] and e e ences he ein). In ac , one may be conce ned no only by he leading5 Oða2Þe ec s bu also by he subleading highe o de co ec ions [in he p esen case, s a ing a Oða3Þ] ha migh lead o he w ong con inuum limi . Local composi e ields cons uc ed om he low ield ha e a na u al leng h scale gi en by he smoo hing adius ffiffiffiffi 8 p, which is smalle han Lby a ac o c. Hence, he na u al expansion pa ame e o he cu o e ec s is ϵ¼a= ffiffiffiffi 8 p¼a=ðcLÞ. A i s example is p o ided by checking he e ec s a ee le el. This jus amoun s o s udying δðc; a=LÞ, Eq. (2.20). In o de o ge a mo e gene al pic u e, we conside besides ou disc e iza ion o he g adien low and he obse able (“Zeu hen low” o sho ) also he one used in many s udies: Wilson low and clo e disc e iza ion o he ene gy densi y [9] ( o sho “Wilson low”). We ind ha δðc; a=LÞ∼−0.9118ϵ2þ0.4867ϵ4Wilson low 1.7165ϵ4Zeu hen low ; ϵ¼a cL ¼a ffiffiffiffi 8 p;ð3:1Þ whe e ∼holds wi h co ec ions o less han 10−4 o ϵ<0.33 and o cin a ange 0.1–0.4. One may also conside he GF coupling o wis ed pe iodic bounda y condi ions [13]; he abo e numbe s ha dly change. This example no only shows ha in ac he cu o e ec s a e p edominan ly a unc ion o ϵ¼a= ffiffiffiffi 8 pbu also ha he con ibu ion o o de s highe han ϵ2a e only a he le el o a ew pe cen o ϵ<0.3. The e is a clea hie a chy o he di e en o de s a small ϵ, say ϵ<0.3. O cou se, one has o s udy he si ua ion beyond ee- le el pe u ba ion heo y, and in pa icula he scaling p ope ies o he la ice app oxima ion o he s ep scaling unc ion σðuÞo Eq. (1.1). In o de o do so, i is use ul o conside he gene al a io Rc;c0ðu; a=L; sÞ¼ ¯ g2 cðLÞ ¯ g2 c0ðsLÞ¯g2 cðLÞ¼u ;ð3:2Þ ha has a na u al expansion Rc;c0ðu;a=L;sÞ¼Rc;c0ðu;0;sÞ 1þAc;c0ðuÞ½ϵ2−ϵ02þ…g; ð3:3Þ whe e ϵ¼a=ðcLÞand ϵ0¼a=ðc0sLÞ. The connec ion o he s anda d s ep scaling unc ion is σðuÞ¼u=Rc;cðu; 0;2Þ. I is again wo hwhile o i s conside ee le el. To his end, we empo a ily eplace he no maliza ion ˆ Nðc; a=LÞ by he con inuum one, N, in Eq. (2.18); o he wise, all cu o e ec s a e emo ed. Wi h his eplacemen , he ee- le el a io Rc;c0ðu; 0;sÞis o a e y good app oxima ion jus a unc ion o uand he p oduc sc0, while he unc ion Ac;c0ðuÞdepends li le on c,c0. An inspec ion o ou nume ical da a shows ha his is ue also a non anishing coupling. These p ope ies allow us o gain insigh in o he scaling p ope ies o he s ep scaling unc ion by consid- e ing he case s¼1whe e we can use ou ull da a se . As we shall see in Sec. IV, we ha e i e la ice esolu ions L=a ¼8, 12, 16, 24, 32 a ou disposal. Con inuum ex apola ions can in ol e a change o he la ice spacing o up o a ac o 4. Mo eo e , hese a ios can be compu ed e en mo e p ecisely han he s ep scaling unc ion, since hey a e e alua ed on he same ensembles and one bene i s om he s a is ical co ela ion o he da a. Figu es 1and 2show Rc;c0ðu; a=L; 1Þ o all oge he six di e en combina ions, c,c0, and wo alues o u. The da a o igina e om he simula ions desc ibed in Appendix, o ming i s he a ios a he a ailable ¯ g2 cand hen pe o ming a ( e y smoo h) in e pola ion o he wo chosen alues o u. As shown in he igu es, we sepa a ely ex apola e he a ios o he wo di e en disc e iza ions o he low obse ables o he con inuum limi . We use a pu e a2ansa z o he cu o e ec s in he anges FIG. 1. Ra io R0.3;c0ð4.26; a=LÞ o a ious c0and wo di e en disc e iza ions o he obse able. In he de ini ion Eq. (3.2), all quan i ies e e o he same disc e iza ion. Full lines a e linea i s in a2 o da a sa is ying Eq. (3.4). 5The OðaÞe ec s om he SF ime bounda ies will be igno ed in he ollowing discussion bu conside ed la e . SLOW RUNNING OF THE GRADIENT FLOW COUPLING …PHYSICAL REVIEW D 95, 014507 (2017) 014507-5 jϵ2−ϵ02j¼j1−ðc0=cÞ2ja c0L2 <0.10 Wilson low; 0.25 Zeu hen low: ð3:4Þ The da a a e compa ible wi h he linea beha io in a2, and he so-es ima ed con inuum limi s ag ee. The es is a he s ingen because he e he p ecision is highe han in he s ep scaling unc ions, which o m he co e obse ables o he es o his pape . Fo he s ep scaling unc ions, he e is no analogy o he co ela ions o he nume a o and denomina o in Eq. (3.2), which enhance he p ecision o Rc;c0ðu; a=L; 1Þ. Figu e 1and Fig. 2a e a good con i ma ion ha highe o de cu o e ec s a e small, when Eq. (3.4) is sa is ied. T ansla ing he bounds (3.4) o he case o he s ep scaling unc ion, we ha e ½1−ð1=2Þ2a cL2 <0.10 Wilson low; 0.25 Zeu hen low:ð3:5Þ We hen expec he s ep scaling unc ion compu ed using he Zeu hen low o ha e only small co ec ions o an a2 scaling o ϵ2¼a2=ðcLÞ2<0.33. Ou coa ses da a se has L=a ¼8and c¼0.3, which implies ϵ2¼0.17. The di e ence in he bounds (3.5) means ha he mo e p ecise con inuum limi is ob ained o he Zeu hen low. Despi e he ac ha cu o e ec s o he Wilson low a e smalle , hei complica ed unc ional o m makes ex ap- ola ions mo e di icul and less p ecise. In pa icula , he coa se la ices used o de e mine he con inuum s ep scaling unc ion in he nex sec ion would ha e signi ican iola ions o he leading a2scaling i we we e using he Wilson low da a. Howe e , one has o s a e ha he a2co ec ions a e sizable. Since nei he he Zeu hen low equa ion no he e alua ion o a classically imp o ed obse able in oduces any a2cu o e ec s, hese emaining la ice a i ac s a e a consequence o he quan um co ec ions due o he ini ial condi ion o he low equa ion a ¼0and due o he ac ion o he luc ua ing ields in he pa h in eg al [30]. Whe he he e a e p ac ical ways o educe hese emaining a2e ec s subs an ially is an in e es ing p oblem ha dese es u he a en ion in he u u e. A. Bounda y Oða=LÞe ec s Wi h ou choice o SF bounda y condi ions (2.14),(2.3), he comple e emo al o OðaÞcu o e ec s equi es no only he nonpe u ba i e alue o he coe icien csw [29] bu also he de e mina ion o he bounda y coe icien s c , ~ c [4,28]. These a e known only o one loop o ou choice o la ice ac ion [31–33] c ¼1þcð1Þ g2 0þOðg4 0Þ;c ð1Þ ¼0.0326718; ~ c ¼1þ~ cð1Þ g2 0þOðg4 0Þ;~ cð1Þ ¼−0.01505;ð3:6Þ and he e o e we ha e o es ima e he possible e ec s o highe o de e ms in he coupling. Fo his pu pose, i is con enien o ecall ha ou GF coupling is de ined a ime slice x0¼T=2, and wi h ou choice c¼0.3and T¼L, he smea ing adius is ffiffiffiffi 8 p¼cL ¼0.3T. The e o e, we expec bounda y e ec s o be supp essed, since ou obse able is localized a he cen e o he la ice, away om he bounda ies. The issue was in es iga ed in Re . [14] wi h he conclusion ha indeed hese bounda y con ibu ions a e small. He e, we es ima e he e ec quan i a i ely and speci ically o ou obse able. We i s quo e he linea a-e ec s a leading o de in pe u ba ion heo y. They a e ob ained by expanding he ee-le el no m Nin c −1, ea ing c ¼1þOðg2 0Þ. The esul is Σðu; a=LÞ¼Σðu; a=LÞc ¼1þc −1 g2 0cð1Þ Δc Σðu; a=LÞ;ð3:7Þ Δc Σðu; a=LÞ¼ ð1Þ 1Σ2a 2LþOðΣ3Þ;ð3:8Þ wi h ð1Þ 1¼−0.013 in he ele an ange o L=a ≥8.We ha e no malized by he one-loop con ibu ion o c , using he known cð1Þ . In his way, Δc Σgi es he e ec in Σi one akes as an unce ain y he one-loop e m in he pe u ba i e se ies o c . As he e he one-loop e m is he las known one, his is exac ly wha we wan o do in his wo k. As a check on he use o pe u ba ion heo y, we pe o med simula ions on ou smalles la ice L=a ¼8a FIG. 2. Ra io R0.36;c0ð8.24; a=LÞ o a ious c0and wo di e en disc e iza ions o he obse able. In he de ini ion Eq. (3.2), all quan i ies e e o he same disc e iza ion. Full lines a e linea i s in a2 o da a sa is ying Eq. (3.4). MATTIA DALLA BRIDA e al. PHYSICAL REVIEW D 95, 014507 (2017) 014507-6 ¯ g2∼4.5wi h h ee di e en alues o c a ound he one- loop one. We ound ha he e ec i e coe icien e 1≡2L aΣ−2∂Σ ∂c ð3:9Þ e alua es o e 1¼−0.0121ð5Þg2 0cð1Þ Σ2ð3:10Þ when we es ima e i om a nume ical de i a i e a ou cen al simula ion poin c ¼1þcð1Þ g2 0. The ag eemen wi h lowes o de pe u ba ion heo y is good enough o jus ake Eq. (3.7) as ou es ima e o he unce ain y. We p opaga e (by quad a u e) he ull one-loop e ec o his bounda y coun e e m (3.7) o ou e o on Σðu; a=LÞ. No e ha his e ec is subdominan in compa ison wi h ou s a is ical accu acy. The co espond- ing unce ain y due o ~ c will be neglec ed since i is supp essed by a u he powe o g2. IV. CONTINUUM EXTRAPOLATIONS AND THE β-FUNCTION As al eady men ioned, he way o connec nonpe u ba- i ely he had onic scale Lhad and he in e media e scale L0 passes h ough he compu a ion o he s ep scaling unc ion. I is de ined as he con inuum limi σðuÞ¼ lim a=L→0 Σðu; a=LÞð4:1Þ o i s la ice app oxima ion, Σðu; a=LÞ¼¯ g2 GFð2LÞj¯ g2 GFðLÞ¼u;m¼0:ð4:2Þ The condi ion m¼0 ixes he ba e qua k mass o each esolu ion a=L and each alue o he ba e coupling g2 0. The esul ing unc ion is deno ed mc ðg0; a=LÞand desc ibed in Appendix. The second condi ion, ¯ g2 GFðLÞ¼u, ixes g0 o each alue o uand esolu ion a=L conside ed. The doubled la ices, whe e ¯ g2 GFð2LÞis de e mined, sha e he ba e pa ame e s wi h he L=a la ices. A. S a egy and da a se In p ac ice, hese condi ions ha e o be implemen ed by a uning o he ba e pa ame e s such ha he eno malized ones a e ixed as desc ibed. We b ie ly explain ou s a egy o a i e a a p ecise uning o a ew app op ia e alues o u and he es ima es o Σ: (i) The uning o he ba e mass m0was al eady ca ied ou in Re . [33] o he ull ange o ba e couplings and a=L conside ed. In he con inuum limi , he chi al poin o anishing qua k mass is unique; he a=L- dependence is a cu o e ec . Howe e , in o de o ha e a smoo h ex apola ion o he con inuum limi , one i s de ines exac ly which mass is se o ze o a a ixed a=L and hen de e mines he unc ion mc ðg0; a=LÞ. In he ci ed e e ence, his ask was ca ied ou wi h high p ecision. As a esul , we can neglec any de ia ions om he exac c i ical line. The used unc ions mc ðg0; a=LÞa e lis ed in Appendix. (ii) Fo he nex s ep, we pe o med nine p ecise simula ions wi h L=a ¼16. These de e mine nine alues o u¼ i;i¼1;…;9, which we ake as ou p ime a ge s o compu e σð iÞ. We u he need alues o β o L=a ¼8, 12 such ha ¯ g2 GF equals ou a ge alues i. This is achie ed by an in e - pola ion o se e al simula ions desc ibed in de ail in Appendix B. A his poin , we ound o each L=a ¼8, 12, 16 nine alues o βwhe e couplings ¯ g2 GFðLÞma ch a he well. These β- alues a e lis ed in Table II. (iii) We hen ca ied ou simula ions on he doubled la ices a he same alues o β,m0; see columns 4–6 in Table II. The da a o ¯ g2 GFð2LÞin he able a e es ima es o he s ep scaling unc ion Σðu; a=LÞa u¼¯ g2 GFðLÞβ;L=a. Fo ou es ima es o ¯ g2 GFðLÞ, we could ake he numbe s om he in e pola ion in s ep 2. These a e simply he same as hose a L=a ¼16. Howe e , in o de o enhance he p e- cision, we pe o m sepa a ely a each L=a ¼8,12 an in e pola ing i o all a ailable da a o Table IX. These i s de e mine ¯ g2 GFðLÞin Table II. De ails on he e y well-de e mined in e pola ion a e gi en in Appendix B. (i ) Fo he las s ep, we p opaga e he e o s o ¯ g2 GFðLÞ in o hose o Σ. As we will see in Sec. IV B 1, ou nonpe u ba i e da a a e well desc ibed by he unc ional o m 1 Σ−1 u¼cons an ;ð4:3Þ which sugges s using he de i a i e, ∂Σ=∂u¼ Σ2=u2, o he e o p opaga ion. This yields he las column o Table II, whe e uis he cen al alue o ¯ g2 GF wi hou e o . The di e ence o he e o s in columns 4 and 7 is mos ly due o he unce ain y o OðaÞimp o emen , Eq. (3.7); a small pa o he unce ain y is also con ibu ed by he p opaga ed e o s o ¯ g2 GFðLÞ. The las wo ows in Table II a e om addi ional simula ions pe o med wi h he aim o ha ing ¯ g2 GFð2LÞ≈11.3. They will also be use ul below. B. Con inuum ex apola ion o he s ep scaling unc ion The esul s a ini e esolu ion need o be ex apola ed o he con inuum. I is appa en om Table II ha his is an essen ial s ep, since Σchanges by up o 20% in he accessible ange o a=L— a ou side he s a is ical e o s. SLOW RUNNING OF THE GRADIENT FLOW COUPLING …PHYSICAL REVIEW D 95, 014507 (2017) 014507-7 Howe e , ou in es iga ion in Sec. III showed ha he cu o e ec s a e s ongly domina ed by he ða=LÞ2 e ms, which mo i a es ex apola ions linea in his a iable. Gi en he high p ecision which we achie e, his is a c ucial pa o his wo k, and a de ailed analysis will ollow. In pa icula , we i s s udy he sys ema ic e ec s in he con inuum de e mina ion o σðuÞby pe o ming inde- penden ex apola ions a nine ixed alues o u. These can anspa en ly be illus a ed by simple g aphs. 1. σðuÞand sys ema ic e ec s in he con inuum ex apola ions Apa om he las wo ows o Table II, he de ia ions o ¯ g2 GFðLÞ om he nine a ge alues i( he ones a L=a ¼16) a e e y small. We can he e o e simply shi he da a o Σusing Eq. (4.3). The esul ing da a a e shown in Fig. 3. Wi hin he unce ain ies, linea i y in a2is pe ec , and we ex apola e by TABLE II. S ep scaling unc ions. A he speci ied β, we lis ¯ g2ðLÞon he L=a-la ice ob ained om he desc ibed i as well as ¯ g2ð2LÞ on he 2L=a-la ice. Thei e o s do no con ain he unce ain y o c .Nms and NQ e e o he measu emen s on he 2L=a-la ice Simula ions wi h NQ¼∅we e ca ied ou wi h he algo i hm es ic ed o Q¼0.A β¼3.556470 and L=a ¼16, we ha e wo ensembles, wi h and wi hou ixing he opology (bo h ensembles gi e compa ible esul s, and in columns 4 and 7, we quo e as esul s he weigh ed a e age). The las column con ains Σðu; a=LÞwi h uequal o he cen al alue o column 3 and he ull e o ob ained om ¯ g2ðLÞ,¯ g2ð2LÞas well as he unce ain y o c . No e ha e o s in columns 3 and 7 a e co ela ed, as discussed in he ex . L=a β¯ g2ðLÞ¯ g2ð2LÞNms NQΣðu; a=LÞ 8 3.556470 6.5485(60) 11.452(79) 2000, 2000 725;∅11.452(134) 8 3.653850 5.8670(34) 9.250(66) 2000 220 9.250(97) 8 3.754890 5.3009(32) 7.953(44) 2001 30 7.953(68) 8 3.947900 4.4848(25) 6.207(23) 2001 1 6.207(39) 8 4.151900 3.8636(21) 5.070(16) 2001 0 5.070(26) 8 4.457600 3.2040(18) 3.968(11) 2001 0 3.968(17) 8 4.764900 2.7363(14) 3.265(8) 2001 0 3.265(12) 8 5.071000 2.3898(15) 2.772(6) 2001 0 2.772(9) 8 5.371500 2.1275(15) 2.423(5) 2001 0 2.423(7) 12 3.735394 6.5442(82) 12.874(165) 3000 ∅12.874(191) 12 3.833254 5.8728(46) 10.497(78) 2400 ∅10.497(99) 12 3.936816 5.2990(36) 8.686(49) 2400 ∅8.686(64) 12 4.128217 4.4908(32) 6.785(36) 2400 1 6.785(44) 12 4.331660 3.8666(25) 5.380(25) 2400 0 5.380(29) 12 4.634654 3.2058(17) 4.180(14) 2403 0 4.180(17) 12 4.938726 2.7380(15) 3.403(11) 2400 0 3.403(13) 12 5.242465 2.3902(11) 2.896(9) 2400 0 2.896(10) 12 5.543070 2.1235(12) 2.504(8) 2400 0 2.504(9) 16 3.900000 6.5489(155) 13.357(136) 1205 ∅13.357(167) 16 4.000000 5.8673(140) 10.913(118) 1404 ∅10.913(136) 16 4.100000 5.3013(134) 9.077(75) 1403 1 9.077(91) 16 4.300000 4.4901(77) 6.868(40) 2507 0 6.868(48) 16 4.500000 3.8643(63) 5.485(22) 2000 0 5.485(28) 16 4.800000 3.2029(52) 4.263(16) 2000 0 4.263(20) 16 5.100000 2.7359(35) 3.485(11) 2500 0 3.485(14) 16 5.400000 2.3900(30) 2.935(7) 2500 0 2.935(9) 16 5.700000 2.1257(25) 2.536(7) 2500 0 2.536(8) 12 3.793389 6.1291(56) 11.788(132) 2556 ∅11.788(154) 16 3.976400 6.037(14) 11.346(100) 1203 ∅11.346(124) FIG. 3. Con inuum ex apola ion o Σo da a shi ed o nine di e en alues o u. MATTIA DALLA BRIDA e al. PHYSICAL REVIEW D 95, 014507 (2017) 014507-8 Σð i; a=LÞ¼σiþ~ i×ða=LÞ2;ð4:4Þ a each alue i. The quali y o he i s is e y good wi h a o al χ2o 6.3 wi h 9 deg ees o eedom. The i pa ame e s σi, second column o Table III, a e i s es ima es o he con inuum s ep scaling unc ion. I u ns ou ha he nonpe u ba i e esul s a e well desc ibed by 1=σi−1= i≈ −0.083 (see he las wo columns o Table III), which is he unc ional o m o one-loop pe u ba ion heo y, bu wi h a coe icien sligh ly di e en om he pe u ba i e −0.0790. This su p ising beha io holds ou o σðuÞ¼Oð10Þ.We will come o a compa ison wi h pe u ba ion heo y la e . Fo now, his sugges s i ing also 1=Σð i; a=LÞ¼1=σiþ i×ða=LÞ2:ð4:5Þ The quali y o hese i s is as good as he p e ious ones (χ2¼6.3 o 9 deg ees o eedom). Disc imina ing s a- is ically be ween he wo i o ms would equi e a highe p ecision han we ha e. An implici assump ion behind Eq. (4.4) and (4.5) is ha highe o de s in a2a e negligible. When his is he case, he i pa ame e s σiha e o ag ee be ween he wo i s (see Table III). The e is ag eemen a he le el o one s anda d de ia ion. Howe e , he di e ence be ween he wo ex apola ions is o cou se sys ema ic: σia e always la ge when hey a e ex apola ed ollowing Eq. (4.5). This is also appa en in Fig. 3. Fu he mo e, when nonlinea i ies in a2 a e negligible, he e is he mo e s ingen condi ion i¼−~ i=σ2 i. As expec ed, we ind mo e signi ican di e ences be ween hese slope pa ame e s6(see Fig. 4). No e ha he di e ence be ween he unc ional o ms o Eq. (4.4) and Eq. (4.5) is o o de a4. Due o he ela i ely la ge Oða2Þe ec s, hese a e no negligible a la ge alues o he coupling (a small alues o u, we ha e good ag eemen be ween bo h ypes o i s). I is his Oða4Þe ec ha p oduces a sys ema ic shi in he pa ame e s σi, i. A i o 1=σðuÞ−1=u o a cons an p o ides a good desc ip ion o ou con inuum da a (χ2=do <1) in he whole ange u∈½2.1;6.5. Al hough he sys ema ic di e - ence be ween he con inuum i s (4.4) and (4.5) was poin by poin in σibelow ou s a is ical accu acy, he unce ain y in a cons an i o 1=σðuÞ−1=u is educed by a ac o 3 due o he ac ha we use nine independen alues o de e mine i . The sys ema ic e ec hen becomes clea ly no iceable. 2. Fi ing s a egy The p e ious conside a ions illus a e ha he Oða4Þ e ec s a e no la ge bu s ill canno simply be igno ed. The size o he Oða2Þ e m, ha amoun s o 20% a he la ges alue o he coupling umax ¼6.5a L=a ¼8, sugges s ha he e he Oða4Þe ec s a e a ound 5%. Taking in o accoun ha a u-independen e m is emo ed by he no maliza ion o he coupling, his ansla es in o he ough scaling ΔsysΣi¼0.05Σi8a L4u umax :ð4:6Þ This sys ema ic e ec is negligible compa ed wi h ou s a is ical accu acy o he la ices wi h L=a ≥12 a all alues o u(in ac , he di e ences seen in Table III become insigni ican when we pe o m he ex apola ions wi h jus L=a ≥12), bu i becomes dominan a L=a ¼8and la ge alues o u. When i ing o some pa icula unc ional o m, one pe o ms a minimiza ion o a χ2 unc ion, de ined as TABLE III. Examples o he con inuum limi s o he s ep scaling unc ion σi¼lima=L→0Σð i; a=LÞob ained by a ious ex apola ions a ixed alues o u¼ i. The las ow shows i s o columns 4 and 5 o a cons an . These i s o a cons an p o ide an excellen desc ip ion o ou da a. σið1=σi−1= iÞ×102 iEq. (4.4) Eq. (4.5) Eq. (4.4) Eq. (4.5) 6.5489 14.005(175) 14.184(197) −8.13ð10Þ−8.22ð12Þ 5.8673 11.464(123) 11.654(146) −8.32ð10Þ−8.46ð13Þ 5.3013 9.371(79) 9.468(89) −8.19ð11Þ−8.30ð12Þ 4.4901 7.139(47) 7.181(51) −8.26ð11Þ−8.34ð12Þ 3.8643 5.622(28) 5.641(30) −8.09ð10Þ−8.15ð14Þ 3.2029 4.354(19) 4.367(21) −8.25ð12Þ−8.32ð13Þ 2.7359 3.541(14) 3.550(15) −8.31ð12Þ−8.38ð13Þ 2.3900 2.991(10) 2.996(10) −8.40ð12Þ−8.46ð13Þ 2.1257 2.575(9) 2.578(9) −8.21ð14Þ−8.26ð14Þ Cons an i : −8.233ð37Þ−8.316ð42ÞFIG. 4. Slopes io Eq. (4.5) and −~ i=σ2 iwi h ~ io Eq. (4.4). 6No e ha he de e mina ion o asymp o ic alues o io ~ iis no ou goal. We only discuss hem because hey show ha di e ences be ween he con inuum limi s es ima ed om Eq. (4.4) and (4.5) ha e o be aken in o accoun . SLOW RUNNING OF THE GRADIENT FLOW COUPLING …PHYSICAL REVIEW D 95, 014507 (2017) 014507-9 24, 32. Table VII shows he au oco ela ion imes, de e - mined as in Re . [45] in molecula dynamic uni s, while Fig. 10 indica es ha hey oughly ollow he expec ed scaling wi h a−2[46] a cons an ¯ g2 GFðLÞ, i.e. in ixed physical olume. E en he de ia ions om scaling seen a he la ge coupling ha e a plausible explana ion in e ms o a co ec ion o scaling. When he la ice spacing is bigge han a ound 0.05 m, he s anda d HMC s ill shows opological ac i i y [22]. Algo i hm B will he e o e ha e a numbe o a emp s o change opology, which inc eases wi h he la ice spacing. These a emp s a e e oed by he accep ance s ep, educing he accep ance a e and inc eas- ing he au oco ela ions. This easily explains he h ee highes lying poin s in he igu e, bu o cou se he quali y o he da a is no good enough o a quan i a i e s a emen . The leng h o ou Mon e Ca lo chains is always be ween 200τin and 2000τin . Despi e he expec a ion ha au oco - ela ions will e en ually scale a he di e en ly in la ge olume compa ed o ou si ua ion wi h Sch ödinge unc- ional bounda y condi ions, he longes au oco ela ion imes o ou ini e olume simula ions a e compa able o he longes ones obse ed in la ge olume in Re . [15]. 4. C i ical lines Since we wo k in a massless eno maliza ion scheme, we need o de ine and know he c i ical line in he space o ba e la ice pa ame e s ðβ;κ; L=aÞ, o equi alen ly ðg2 0;am 0; L=aÞwi h g2 0¼6=β;am 0¼ð2κÞ−1−4:ðA2Þ The c i ical line, am0¼amc ðg0; a=LÞ, is de ined by m1¼0, whe e m1is a cu en qua k mass in an ðL=aÞ4 la ice. Making mc dependen on L=a in his way and using he same L=a in his de ini ion as in Σ, he cu o e ec s a e gua an eed o disappea as Oða2Þin he imp o ed heo y. De ails on m1as well as on he many p ecise simula ions done o ind he c i ical lines by in e pola ion can be ound in Re . [33]. Fo comple eness, we he e lis he esul s needed o compu e mc . In Table VIII, we p o ide he coe icien s o he in e pola ing unc ions o he c i ical lines, amc ðg0; a=LÞ¼X 6 k¼0 μkg2k 0×X 6 i¼0 ζig2i 0−1 ;ðA3Þ a a gi en alue o L=a, alid o all alues o g2 0used in his pape . These pa ame iza ions gua an ee m1L<0.005. Wi h hese coe icien s, he eade can econs uc he inpu mass pa ame e κc co esponding o ou simula ions. APPENDIX B: TUNING TO SELECTED COUPLINGS In Table IX, we collec ou aw da a o ¯ g2 GFðLÞon he small la ices. FIG. 10. Scaling o τin as a unc ion o ¯ g2ðLÞ o di e en L=a. TABLE VIII. Coe icien s o he pa ame iza ion Eq. (A3). The h ee leading coe icien s ζ0,μ0in he uppe pa o he able a e combina ions o known pe u ba i e coe icien s, while he o he s we e de e mined by a i [33]. Coe icien L=a ¼8L=a ¼12 L=a ¼16 ζ0þ1.005834130000000 þ1.002599440000000 þ1.001463290000000 μ0−0.000022208694999 −0.000004812471537 −0.000001281872601 μ1−0.202388398516844 −0.201746020772477 −0.201520105247962 ζ1−0.560665657872021 −0.802266237327923 −0.892637061391273 ζ2þ3.262872842957498 þ4.027758778155415 þ5.095631719496583 ζ3−5.788275397637978 −6.928207214808553 −8.939546687871335 ζ4þ4.587959856400246 þ5.510985771180077 þ7.046607832794273 ζ5−1.653344785588201 −2.076308895962694 −2.625638312722623 ζ6þ0.227536321065082 þ0.320430672213824 þ0.405387660384441 μ2þ0.090366980657738 þ0.128161834555849 þ0.139461345465939 μ3−0.600952105402754 −0.681097059845447 −0.847457204378732 μ4þ0.934252532135398 þ0.991316994385556 þ1.261676178806362 μ5−0.608706158693056 −0.606597739050552 −0.754644691612547 μ6þ0.140501978953879 þ0.129031928169091 þ0.153135714480269 MATTIA DALLA BRIDA e al. PHYSICAL REVIEW D 95, 014507 (2017) 014507-16 As explained in he main ex , we make maximum use o hese da a by pe o ming smoo h in e pola ions o L=a ¼8, 12. This enables a e y p ecise de e mina ion o ¯ g2 GFðLÞ o hose ba e pa ame e s whe e we ha e compu ed ¯ g2 GFð2LÞ. A ixed L=a, we i ðβÞ¼1=¯ g2 GFðLÞ;ðA4Þ o a Padé ansa z o deg ees ½n1;n 2, ðβÞ¼ Pn1 n¼0anβn 1þPn2 n¼1bnβn;ðA5Þ and ob ain p edic ions ¯ g2 GFðLÞa he desi ed β om he i and hei e o s om he co a iance ma ix o he i pa ame e s. In Fig. 11, we show a couple o ypical i s o all he L=a ¼8da a o a [4, 0] Padé and a [1, 2] one. These i s ha e a good quali y. O he i unc ions we e es ed wi h he esul ha , once he i s ha e a easonable numbe o deg ees o eedom and a good χ2, he in e pola ed alues o ¯ g2 GFðLÞa e en i ely s able wi hin hei e o s. This holds also o he L=a ¼12 da a. Fo inal desc ip ion o ou L=a ¼8, 12 da a, we use he [4, 0] and [3, 0] Padé, i.e. a simple polynomial o deg ees 4 and 3, espec i ely. These choices yield he alues o ¯ g2 GFðLÞlis ed in Table II. TABLE IX. Coupling esul s on he small la ices o a ious β¼6=g2 0and L=a. The sepa a ion o measu emen s is 5–10 MDUs. Nms deno es he numbe o measu emen s ou o which NQha e nonze o cha ge Q. The e ec i e numbe o measu emen s is Nms −NQ. Simula ions wi h NQ¼∅we e ca ied ou wi h Algo i hm B. L=a β¯ g2Nms NQL=a β¯ g2Nms NQ 8 3.50000 7.0271(180) 5001 55 8 4.00000 4.3057(60) 5001 0 8 3.55647 6.5501(149) 5001 34 8 4.15190 3.8619(45) 5001 0 8 3.55800 6.5385(106) 5001 20 8 4.20000 3.7501(49) 5001 0 8 3.60000 6.2343(131) 5001 35 8 4.45760 3.2046(36) 5001 0 8 3.65385 5.8612(126) 5001 7 8 4.50000 3.1250(37) 5001 0 8 3.65452 5.8574(84) 5001 7 8 4.76490 2.7353(31) 5001 0 8 3.70000 5.5990(93) 5001 1 8 4.80000 2.6921(30) 5001 0 8 3.75489 5.3040(88) 5001 0 8 5.07100 2.3910(26) 5001 0 8 3.75709 5.2728(74) 5001 0 8 5.10000 2.3615(26) 5001 0 8 3.80000 5.0959(87) 5001 0 8 5.37150 2.1293(24) 5001 0 8 3.94790 4.4870(56) 5001 0 8 5.40000 2.1037(23) 5001 0 12 3.40000 11.3081(994) 5000 ∅12 4.33166 3.8725(60) 5001 0 12 3.50000 9.1035(284) 5000 ∅12 4.50000 3.4738(54) 5001 0 12 3.70000 6.8400(167) 5001 69 12 4.63465 3.2051(47) 5001 0 12 3.73539 6.5428(176) 5001 18 12 4.80000 2.9255(32) 8000 0 12 3.80000 6.0832(123) 5001 8 12 4.93873 2.7371(38) 5001 0 12 3.83325 5.8685(134) 5001 3 12 5.10000 2.5470(26) 8000 0 12 3.90000 5.4794(106) 5001 2 12 5.24247 2.3919(25) 8000 0 12 3.93682 5.2996(107) 5001 1 12 5.40000 2.2394(22) 8000 0 12 4.00000 4.9991(100) 5001 0 12 5.54307 2.1213(21) 8000 0 12 4.12822 4.4945(75) 5001 0 12 5.60000 2.0823(21) 8001 0 12 4.20000 4.2480(66) 5001 0 16 3.90000 6.5489(155) 4600 15 16 4.80000 3.2029(52) 5000 0 16 4.00000 5.8673(140) 4602 35 16 5.10000 2.7359(35) 6001 0 16 4.10000 5.3013(134) 3200 0 16 5.40000 2.3900(30) 6001 0 16 4.30000 4.4901(77) 5000 0 16 5.70000 2.1257(25) 7001 0 16 4.50000 3.8643(63) 5000 0 16 3.97640 6.0369(142) 4567 0 FIG. 11. 1 ¯g2 GF −β 6as a unc ion o β o L=a ¼8. 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