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)
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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 .
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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. The simula ion
poin s a e i ed o a [1, 2] Padé (χ2¼17.05 o 18 deg ees o
eedom) and a [4, 0] Padé (χ2¼15.42 o 17 deg ees o
eedom). The di e en i unc ions a e ha d o dis inguish.
SLOW RUNNING OF THE GRADIENT FLOW COUPLING …PHYSICAL REVIEW D 95, 014507 (2017)
014507-17
[1] M. Dalla B ida, P. F i zsch, T. Ko zec, A. Ramos, S. Sin ,
and R. Somme , The Accu acy o QCD Pe u ba ion Theo y
a High Ene gies, Phys. Re . Le . 117, 182001 (2016).
[2] R. Somme , Non-pe u ba i e eno maliza ion o QCD,
P oc. Sci., LATTICE2014 (2015) 017.
[3] M. Lüsche , P. Weisz, and U. Wol , A nume ical me hod o
compu e he unning coupling in asymp o ically ee he-
o ies, Nucl. Phys. B359, 221 (1991).
[4] M. Lüsche , R. Na ayanan, P. Weisz, and U. Wol , The
Sch ödinge Func ional: a eno malizable p obe o non-
abelian gauge heo ies, Nucl. Phys. B384, 168 (1992).
[5] S. Sin , On he Sch ödinge unc ional in QCD, Nucl. Phys.
B421, 135 (1994).
[6] M. Lüsche , R. Somme , P. Weisz, and U. Wol , A p ecise
de e mina ion o he unning coupling in he SU(3) Yang-
Mills heo y, Nucl. Phys. B413, 481 (1994).
[7] M. Della Mo e, R. F ezzo i, J. Hei ge , J. Rol , R. Somme ,
and U. Wol (ALPHA Collabo a ion), Compu a ion o
he s ong coupling in QCD wi h wo dynamical la o s,
Nucl. Phys. B713, 378 (2005).
[8] F. Tekin, R. Somme , and U. Wol (ALPHA Collabo a ion),
The unning coupling o QCD wi h ou la o s, Nucl. Phys.
B840, 114 (2010).
[9] M. Lüsche , P ope ies and uses o he Wilson low in la ice
QCD, J. High Ene gy Phys. 08 (2010) 071.
[10] A. Ramos, The Yang-Mills g adien low and eno maliza-
ion, P oc. Sci., LATTICE2014 (2015) 017.
[11] Z. Fodo , K. Holland, J. Ku i, D. Nog adi, and C. H. Wong,
The Yang-Mills g adien low in ini e olume, J. High
Ene gy Phys. 11 (2012) 007.
[12] P. F i zsch and A. Ramos, The g adien low coupling in he
Sch ödinge Func ional, J. High Ene gy Phys. 10 (2013)
008.
[13] A. Ramos, The g adien low unning coupling wi h wis ed
bounda y condi ions, J. High Ene gy Phys. 11 (2014) 101.
[14] M. Lüsche , S ep scaling and he Yang-Mills g adien low,
J. High Ene gy Phys. 06 (2014) 105.
[15] M. B uno e al., Simula ion o QCD wi h N ¼2þ1
la o s o non-pe u ba i ely imp o ed Wilson e mions,
J. High Ene gy Phys. 02 (2015) 043.
[16] M. Dalla B ida, P. F i zsch, T. Ko zec, A. Ramos, S. Sin ,
and R. Somme , A s a us upda e on he de e mina ion o
ΛN ¼3
¯
MS by he ALPHA collabo a ion, P oc. Sci.,LAT-
TICE2015 (2015) 248.
[17] I. Campos, P. F i zsch, C. Pena, D. P e i, A. Ramos, and A.
Vladikas, P ospec s and s a us o qua k mass eno maliza-
ion in h ee- la o QCD, P oc. Sci., LATTICE2015 (2015)
249.
[18] S. Sin and P. Weisz (ALPHA Collabo a ion), The unning
qua k mass in he SF scheme and i s wo loop anomalous
dimension, Nucl. Phys. B545, 529 (1999).
[19] R. Na ayanan and H. Neube ge , In ini e N phase ansi ions
in con inuum Wilson loop ope a o s, J. High Ene gy Phys.
03 (2006) 064.
[20] M. Lüsche and P. Weisz, Pe u ba i e analysis o he
g adien low in non-abelian gauge heo ies, J. High Ene gy
Phys. 02 (2011) 051.
[21] L. Del Debbio, G. M. Manca, and E. Vica i, C i ical slowing
down o opological modes, Phys. Le . B B594, 315
(2004).
[22] S. Schae e , R. Somme , and F. Vi o a (ALPHA Collabo-
a ion), C i ical slowing down and e o analysis in la ice
QCD simula ions, Nucl. Phys. B845, 93 (2011).
[23] P. F i zsch, A. Ramos, and F. S ollenwe k, C i ical slowing
down and he g adien low coupling in he Sch ödinge
unc ional, P oc. Sci., LATTICE2013 (2013) 461.
[24] M. Lüsche and P. Weisz, Compu a ion o he ac ion o on-
shell imp o ed la ice gauge heo ies a weak coupling,
Phys. Le . 158B, 250 (1985).
[25] S. Aoki, R. F ezzo i, and P. Weisz, Compu a ion o he
imp o emen coe icien cSW o one loop wi h imp o ed
gluon ac ions, Nucl. Phys. B540, 501 (1999).
[26] M. Lüsche and S. Schae e , La ice QCD wi h open
bounda y condi ions and wis ed-mass eweigh ing,
Compu . Phys. Commun. 184, 519 (2013).
[27] B. Sheikholeslami and R. Wohle , Imp o ed con inuum
limi la ice ac ion o QCD wi h Wilson e mions, Nucl.
Phys. B259, 572 (1985).
[28] M. Lüsche , S. Sin , R. Somme , and P. Weisz, Chi al
symme y and O(a) imp o emen in la ice QCD, Nucl.
Phys. B478, 365 (1996).
[29] J. Bula a and S. Schae e , Imp o emen o N ¼3la ice
QCD wi h Wilson e mions and ee-le el imp o ed gauge
ac ion, Nucl. Phys. B874, 188 (2013).
[30] A. Ramos and S. Sin , Symanzik imp o emen o he
g adien low in la ice gauge heo ies, Eu . Phys. J. C
76, 15 (2016).
[31] S. Takeda, S. Aoki, and K. Ide, A Pe u ba i e de e mi-
na ion o OðaÞbounda y imp o emen coe icien s o
he Sch ödinge unc ional coupling a one loop wi h
imp o ed gauge ac ions, Phys. Re . D 68, 014505 (2003).
[32] P. Vilaseca (p i a e communica ion).
[33] P. F i zsch and T. Ko zec, Simula ing he QCD Sch ödinge
Func ional wi h h ee massless qua k la o s (unpublished).
[34] M. Dalla B ida, P. F i zsch, T. Ko zec, A. Ramos, S. Sin ,
and R. Somme (unpublished).
[35] M. B uno, P. Ko cyl, T. Ko zec, S. Lo ini, and S.
Schae e , On he ex ac ion o spec al quan i ies wi h
open bounda y condi ions, P oc. Sci., LATTICE2014
(2014) 089.
[36] M. B uno, T. Ko zec, and S. Schae e , Se ing he scale o
he CLS 2þ1 la o ensembles, a Xi :1608.08900.
[37] M. Dalla B ida e al. (ALPHA Collabo a ion) (unpub-
lished).
[38] R. Somme and U. Wol , Non-pe u ba i e compu a ion o
he s ong coupling cons an on he la ice, Nucl. Pa . Phys.
P oc. 261–262, 155 (2015).
[39] M. Hasenbusch, Speeding up he hyb id Mon e Ca lo
algo i hm o dynamical e mions, Phys. Le . B 519, 177
(2001).
[40] M. Hasenbusch and K. Jansen, Speeding up la ice
QCD simula ions wi h clo e imp o ed Wilson e mions,
Nucl. Phys. B659, 299 (2003).
[41] A. D. Kennedy, I. Ho a h, and S. Sin , A new exac me hod
o dynamical e mion compu a ions wi h nonlocal ac ions,
Nucl. Phys. B, P oc. Suppl. 73, 934 (1999).
[42] M. A. Cla k and A. D. Kennedy, Accele a ing Dynamical
Fe mion Compu a ions using he Ra ional Hyb id
Mon e Ca lo (RHMC) Algo i hm wi h Mul iple Pseudo e -
mion Fields, Phys. Re . Le . 98, 051601 (2007).
MATTIA DALLA BRIDA e al. PHYSICAL REVIEW D 95, 014507 (2017)
014507-18
[43] M. Cè, C. Consonni, G. P. Engel, and L. Gius i, Non-
Gaussiani ies in he opological cha ge dis ibu ion o he
SU(3) Yang–Mills heo y, Phys. Re . D 92, 074502 (2015).
[44] N. I. Achieze , Theo y o App oxima ion (Do e , New Yo k,
1992).
[45] U. Wol (ALPHA Collabo a ion), Mon e Ca lo e o s
wi h less e o s, Compu . Phys. Commun. 156, 143
(2004).
[46] M. Lüsche and S. Schae e , La ice QCD wi hou opology
ba ie s, J. High Ene gy Phys. 07 (2011) 036.
SLOW RUNNING OF THE GRADIENT FLOW COUPLING …PHYSICAL REVIEW D 95, 014507 (2017)
014507-19