Opening the Rome-Southampton window for operator mixing matrices

2.4 MB·

by  R. Arthur

Opening the Rome-Southampton window for operator mixing matrices
R. Arthur,1 P.A. Boyle,1 N. Garron,1 C. Kelly,2 and A.T. Lytle3
(RBC and UKQCD Collaborations)
1SUPA, School of Physics, The University of Edinburgh, Edinburgh EH9 3JZ, UK
2Physics Department, Columbia University, New York, NY 10027, USA.
3School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK
(Dated: Dated: December 25, 2011)
We show that the running of operators which mix under renormalization can be computed fully
non-perturbatively as a product of continuum step scaling matrices. These step scaling matrices
are obtained by taking the “ratio” of Z matrices computed at different energies in an RI-MOM
type scheme for which twisted boundary conditions are an essential ingredient. Our method allows
us to relax the bounds of the Rome-Southampton window. We also explain why such a method
is important in view of the light quark physics program of the RBC-UKQCD collaborations. To
illustrate our method, using n = 2+1 domain-wall fermions, we compute the non-perturbative
2 f
running matrix of four-quark operators needed in K → ππ decay and neutral kaon mixing. Our
1
results are then compared to perturbation theory.
0
2

n Introduction of external momenta), and F is the tree level value of
a Lattice QCD has now reached a stage of high precision Λ , one imposes
J O
in flavour physics and plays a crucial rˆole in the quest
4
fornewphysics. Precisionphenomenologyrequiresusing
2 a non-perturbative scheme to renormalize quantities ZOS(µ,a)Zψ−n/2(µ,a)mli→m0ΛO(µ,a,m)=F . (1)
obtained from lattice simulations, avoiding ill conver-
]
t gent and low order lattice perturbation theory. One In this way one obtains, non-perturbatively, the scheme
a
-l possibility is to use the Schr¨odinger functional (SF) and scale dependent renormalization factors ZOS(µ,a).
p scheme, a theoretically appealing method which allows The precise definition of the lattice scheme S depends
e for a smooth connection between low energy - where the onthedetailsoftheimplementation. Inthesecondstep,
h hadronic matrix elements are computed on the lattice one converts the result to a scheme more appropriate for
[ - and very high energy, where the perturbative series phenomenological applications. In the case of an effec-

2 is accurate [1, 2]. This connection is done through tive low-energy theory one can express the Hamiltonian
v the use of step-scaling functions [3–5]. Unfortunately, by a sum of local operators multiplied by some Wilson
3 the range of operators actually computed in the SF coefficients. One must find a scheme where both the
2 scheme is rather limited; perhaps this is due to the fact renormalized matrix elements of these operators and the
2
that a peculiar perturbation theory (different from the correspondingWilsoncoefficientscanbecomputed. Typ-
1
infinite volume one) is required. Another possibility ically, one matches the Z factors obtained in the lattice
.
9 - very popular in the lattice community - is to use a schemeatµ∼2−3GeVtoMS. Equivalently,(ifpertur-
0 RI-MOM type scheme [6]: such a scheme is theoretically bationtheoryisaccurate)onecandividethevertexfunc-
1
sound, relatively easy to implement and the connection tioncomputedinthelatticeschemebythecorresponding
1
to MS or any other perturbative scheme is done using running to obtain the renormalization group invariant Z
:
v continuum infinite volume perturbation theory. Thus factor and then convert to the desired scheme and scale.
Xi one benefits from the recent multi-loop computations It is important to note that the running and the match-
achieved by several groups (see for example [7–9]). In ingareusuallycomputedonlyperturbatively(wediscuss
r
a an RI-MOM type scheme one numerically computes some recent implementations of a non-perturbative run-
the off-shell amputated vertex function G of the ning in the next section).
O
operator of interest O between external states with In order to keep the discretisation effects under con-
given momenta and fixed gauge gluonic configurations. trol, and at the same time access a region where per-
We project this quantity onto its Dirac-colour structure turbation theory can be applied, RI-MOM schemes re-
and obtain the quantity Λ (the choice of projector quire the existence of the so-called Rome-Southampton
O
is in general not unique and the precise definition of window: ideally one would require ΛQCD (cid:28) µ (cid:28) a−1.
the scheme depends on this choice of projector). We In practice, this window can be quite narrow or even
take all the quark masses to be degenerate and equal closedandmostofthelatticecomputationsaredonewith
to m. Then one requires that in the chiral limit the aµ ∼ 1. In fact the size of the discretisation effects de-
renormalized projected-amputated operator matches its pends on the lattice action, and it is usually admitted
tree-level value [6], i.e. if a is the lattice spacing, n is that for an off-shell O(a)-improved action it is sufficient
the number of quark fields, ψ, in the operator and µ is to require (aµ/π)2 (cid:28) 1. Concerning the other side of
the renormalization scale (which depends on the choice the window the situation was greatly improved when it
was realised that a different choice of kinematics, called
2
non-exceptional, strongly suppresses some infrared con- this is not the case for the step scaling matrix which has
tributions to the vertex function [10, 11]. Nevertheless, well-defined continuum limit and depends only on the
thiswindowisstillanimportantlimitation,inparticular choice of renormalization scheme S (and on the number
when considering light pion masses where the physical offlavours). Inthisworkweuseaschemewhichiscalled
volume - and thus the lattice spacing - has to be large. (γ ,γ )-scheme in [13]: it involves non-exceptional kine-
µ µ
This could become a real issue when one tries to tackle matics with a symmetric point. Such a choice greatly
the very challenging computation of K →ππ decays be- suppresses unwanted infrared effects.
cause one needs even larger physical volumes. A new The strategy that we are proposing can be summarised
step scaling method has recently been introduced [12] in the following way:
thatallowstoaddressthewindowproblemforRI-MOM. • Consider a set of simulations with a rather large
This method has already been applied for multiplica- volume of spatial extent L where physical pion
0
tively renormalized operators [13, 14]. masses can be simulated and with moderately
large lattice spacings a , but small enough for the
0
Strategy Symanzik expansion to converge. Compute the
In this work we apply the method presented in [12], gen- bare matrix elements of interest (cid:104)Obare(a0)(cid:105) and
eralised to the case where several operators mix under renormalize them in a lattice scheme S at the
renormalization. The main idea is to separate the scale low energy scale µ0, i.e. compute (cid:104)OS(µ0)(cid:105) =
and the lattice spacing(s) at which the bare matrix el- lim (cid:2)ZOS(µ0,a0)(cid:104)Obare(a0)(cid:105)(cid:3). Of course the scale
a0→0
ements are computed from the ones at which the run- µ should be such that the associated discretisa-
0
ning and the conversion to the perturbative scheme are tion errors are small but, compared to the Rome-
performed. Let us consider the case of four-quark oper- Southampton window, we do not require the non-
ators mixing: following again [6], we define a matrix of perturbative effects to be small. Instead one just
amputated, projected vertex functions Λij = Pj{GOi}, hastoensurethatthefinitevolumeeffectsareneg-
wheretheOi formabasisunderrenormalizationandthe ligible, so the renormalization window becomes
P projects and traces onto the Dirac-colour structure of
i
the corresponding operator Oi. If we denote by F the L−2 (cid:28)µ2 (cid:28)(π/a )2 .
0 0 0
tree-level value of Λ, equation (1) becomes
ZS(µ,a) Λ (µ,a,m) • Iterate the following step, with i=1,2,...,n: con-
ij × lim jk =F , (2)
ZA(a)2 m→0ΛA(µ,a,m)2 ik siderasetofsimulations,withaphysicalvolumeof
spaceextentL <L andasetoflatticespacings
i i−1
where, for convenience, we use the vertex function of the ai. With the requirement that, on each lattice,
axial current Λ to fix Z . Equation (2) defines ZS
A ψ ij
where the scheme S depends on the choice of kinematics L−2 (cid:28)µ2 <µ2 (cid:28)(π/a)2
i i−1 i i
and projectors of the four-quark vertex function and on
the choice of quark wave function renormalization [13]. is satisfied, compute the step scaling matrix by
Atfinitelatticespacingaandforagivenrenormalization evaluating eq. (4).
scale µ we consider the matrix
R (µ,a)= lim (cid:2)Λ2(µ,a,m)Λ−1(µ,a,m)(cid:3) , (3) • At a scale µn high enough to apply perturbation
S m→0 A theory, multiply by the perturbative matching and
running factors corresponding to the desired scale
and we define the step scaling matrix by
and scheme (typically MS at µ=2 or µ=3 GeV).
σS(µ,sµ)= limΣS(µ,sµ,a)= lim(cid:2)R (µ,a)R−1(sµ,a)(cid:3). In this volume the usual Rome-Southampton con-
a→0 a→0 S S dition holds
(4)
We also note that ZA(a) cancels out in the ratio since, Λ2QCD (cid:28)µ2n (cid:28)(π/an)2 .
toaverygoodapproximation, itdoesnotdependonthe
scaleµ. Oneimportantpointisthatalthoughthequanti-
tiesΛ,Z andRdependonthedetailsofthecomputation In summary, the general equation can be written as
(cid:104)OMS(µ)(cid:105)=CMS←S(µ)×US(µ,µ )×σS(µ ,µ )×σS(µ ,µ )×...×σS(µ ,µ )× (cid:104)OS(µ )(cid:105) , (5)
n n n−1 n−1 n−2 1 0 0
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
Fine lattices Coarse lattices
where CMS←S(µ) represents the matrix of matching factors which converts the Z matrix computed in the
3
scheme S to the scheme MS, and US(µ,µn) is the s W d
usual running matrix in the scheme S computed in
perturbation theory. In the previous equation we have
made explicit the fact that our method consists in t t
re-expressing the running matrix, usually computed
in perturbation theory, by a product of continuum
non-perturbative step scaling matrices. d W s
FIG. 1: Example of box diagram contributing to K0 −K0
Onenoticesthatinthefirstn−1steps,thelowerbound
mixing in the Standard model.
oftheRome-Southamptonwindow(Λ )isreplacedby
QCD
a more advantageous one (L−1). In other words: we do
i
not need to be in the perturbative regime. Moreover a
better control of the upper bound is achieved by tak-
Zs [14, 18, 19]. However, how to continuum extrapolate
ing the continuum limit of the step scaling function (in
the distinct Fourier modes with distinct lattice artefacts
particular there is no discretisation error of order a2µ2
0 n must be addressed. Some cases have model input like
in contrast to the “naive” RI-MOM implementation). A
perturbative subtraction of the lattice discretisation ef-
couple of remarks are in order:
fects, or rules of thumb such as use of sin(p) instead of
p. Instead here we follow [12] and implement twisted
• Usingtwistedboundaryconditionscircumventsthe
boundary conditions to keep the orientation of the mo-
Fourier mode constraints and allows us to fix the
mentafixedandgivetheobservablesasmootha2 depen-
orientation of the momentum while changing its
dence, and allow for a full non-perturbative continuum
magnitude. As emphasised in [12], thanks to this
extrapolation.
property we can compute the vertex functions for
an arbitrary number of points lying on the same
Renormalization of kaon weak matrix elements.
scaling trajectory. The continuum limit of the ver-
texfunctionasafunctionofµ2 isthenproperlyde- In this section we give the definitions of the kaon
finedandinparticularwedonotneedtoperturba- four-quark operators that we consider in this work. We
tivelysubtractanylatticeartefact. Thecontinuum refer the reader who would like to find more details
extrapolationisalsoeasiersince,ifweknowthelat- about this part to the recent reviews [20, 21].
tice spacings with sufficient precision, we can sim-
ulate any arbitrary momentum. This is also useful In the standard model, neutral kaon mixing is dom-
because our method requires that each momentum inated by box diagrams like the one shown in fig-
p=µ ,µ ,...µ hastobecommontodifferentsets ure 1. The non-perturbative contributions are given by
0 1 n
of lattices. (cid:104)K0|O∆s=2 |K0(cid:105), where O∆s=2 is the parity conserv-
VV+AA VV+AA
ingpartof(sγ (1−γ )d)(sγ (1−γ )d). Itiswellknown
µ 5 µ 5
• Sinceweimposeperiodicboundaryconditions,one thatthisoperatorbelongstothe(27,1)representationof
could worry about the use of perturbation theory SU(3) ×SU(3) and renormalizes multiplicatively. To
L R
in a small volume, where the space extent is of studyneutralkaonmixingbeyondthestandardmodelit
the order of (or smaller than) Λ−1 [15]. How- isusefultointroducetheso-calledSUSYbasisof∆s=2
QCD
everweclaimthatinanon-exceptionalgraphwith operators. In this basis O∆s=2 is the standard model
1
hardexternalmomenta, decouplingwillensure, for operator, O∆s=2, i > 1 are beyond the standard model
i
µ (cid:29) L−1, that our computation is free from fi- (BSM) operators. Denoting by α and β the colour in-
nite volume effects and finite volume perturbation dices, one has
theory is not needed. While this might need fur-
ther investigation, it is not relevant here since we (27,1) O∆s=2 = (s γ (1−γ )d )(s γ (1−γ )d ),
1 α µ 5 α β µ 5 β
consider only “infinite” volumes (of spatial extent (cid:26)
O∆s=2 = (s (1−γ )d )(s (1−γ )d ),
much larger than Λ−1 ). (6,6) 2 α 5 α β 5 β
QCD O∆s=2 = (s (1−γ )d )(s (1−γ )d ),
3 α 5 β β 5 α
(cid:26)
Before closing this section we wish to mention other O∆s=2 = (s (1−γ )d )(s (1+γ )d ),
(8,8) 4 α 5 α β 5 β
works on non-perturbative running in RI-MOM. Taking O∆s=2 = (s (1−γ )d )(s (1+γ )d ).
5 α 5 β β 5 α
the continuum limit of the ratio of Z factors at differ-
ent energies in an RI-MOM scheme was first proposed Theoperatorshavebeenstudiedwithvariouslatticefor-
in [16] but the authors did not address how to match mulations, see for example [22–24]. As we wrote explic-
the momenta computed with different lattice spacings itly in the previous equations, O∆s=2 and O∆s=2 trans-
2 3
such that the lattice artefacts have an a2 expansion. Zh- form like (6,6) under SU(3) ×SU(3) and then mix
L R
estkov [17] used fine tuning of β in the quenched ap- together under renormalization. Likewise O∆s=2 and
4
proximation to exactly match the Fourier modes but did O∆s=2 belong to (8,8) and also mix together. Thus in
5
not define a quantity which has a well-defined contin- a scheme which preserves chiral symmetry the five-by-
uum limit. More recent work has looked at the ratios of five renormalization matrix is block diagonal: the only
4
non-zero Z factors can be divided in three subgroups:
a single factor for the (27,1) operator and two, two-by- 1
two matrices for the BSM operators. In practice it is 0.99
convenient to work in another basis where all the opera-
0.98
tors are colour unmixed and we consider only the parity
even component of the four-quark operators. Using the 2A0.97 a=0.114 fm
notation Γ⊗Γ → (sαΓdα)(sβΓdβ) we define the renor- Z0.96 a=0.086 fm
/
malization basis by: 110.95
Z
(27,1) Q∆s=2 = γ ⊗γ +γ γ ⊗γ γ , 0.94
1 µ µ µ 5 µ 5
(cid:26)
Q∆s=2 = γ ⊗γ −γ γ ⊗γ γ , 0.93
(8,8) 2 µ µ µ 5 µ 5
Q3∆s=2 = I⊗I−γ5⊗γ5, 0.92
(cid:26)
(6,6) Q4∆s=2 = I⊗I+γ5⊗γ5, 0.910 2 4 6 8 10 12
Q5∆s=2 = σµν ⊗σµν. µ2[GeV2]
The explicit relations between the two bases are given
in the appendix. We denote by Z∆s=2 the (block diag- FIG. 2: Renormalization factor Z /Z2 = Z , obtained at
11 A BK
onal) renormalization matrix defined in the renormaliza- two different values of the lattice spacing. Except for a few
tion basis. points, the error bars are smaller than the symbols.
It is interesting to note that the renormalization factors
of some ∆s = 1 operators which appear in K → ππ de-
cays can be obtained from those ofthe ∆s=2 operators
mentionedabove. Atlowenergyinthe∆I =3/2channel isopsin channels) with a two-pion final state [25]. The
they are three operators that contribute: a (27,1) called physical matrix elements are computed from the Eu-
Q(cid:48)∆s=1,∆I=3/2 which renormalizes multiplicatively and clidean ones by using the Lellouch-Lu¨scher formula [26].
1 The results are very promising but, since it was the first
two (8,8) which mix together: the electroweak penguins
computation of its kind, an unphysical pion mass of
Q(cid:48)∆s=1,∆I=3/2. Wegivetheexplicitformoftheseopera-
7,8 mπ ∼422MeV was used. In order to simulate the phys-
tors in the appendix. Denoting by Zi∆js=1, i,j =1,7,8 ical kinematics the collaboration is currently repeating
the corresponding renormalization factors, the relation the computation on a much larger volume - of spatial
to the Zi∆js=2 reads extent L0 ∼4.6fm - with a nearly physical pion mass of
m ∼ 140 MeV. Some promising preliminary results of
π
Z∆s=1 = Z∆s=2, the K → ππ matrix elements in the ∆I = 3/2 channel
11 11
Z∆s=1 = Z∆s=2, Z∆s=1 = −1Z∆s=2, have been reported in [27]. The gauge action used is a
77 22 78 2 23
Z∆s=1 = −2Z∆s=2, Z∆s=1 = Z∆s=2. modification of the Iwasaki gauge action following the
87 32 88 33
linesof[28,29],thatwecallthe“dislocationsuppressing
Inthisworkwegivesomeresultsforthenon-perturbative determinant ratio”. On the same lattice, a computation
running of Q∆i=s1=,.2..5 and Q(cid:48)1∆,7s=,81,∆I=3/2. A full computa- of the matrix element (cid:104)K0|OV∆Vs=+2AA|K0(cid:105) is also on the
tion of K →ππ decays requires also the renormalization way. With a lattice spacing a ∼ 0.14fm one might
0
of ∆I = 1/2 four-quark operators which transform worry about the size of the discretisation effects for a
like (8,1) under SU(3)L × SU(3)R. Because for these momentumoftwoorthreeGeV(indeedonemightdoubt
operators one has to compute disconnected diagrams we the existence of the Rome-Southampton window on this
donotconsidertheminthisworkandleavethemforthe lattice). The way out is to follow the strategy explained
future. Although these operators are important in a full intheprevioussectionwithn=1(twodifferentphysical
computation of K → ππ amplitudes the main purpose volumes). Firstly, in the volume L we compute the
0
of this paper is to give a method for a computation of a bare matrix elements and the renormalization factors
non-perturbative running in the operators mixing case. at a low energy µ , where we use µ ∼ 1.5GeV. Then
0 0
Furthermoretheoperatorsthatweconsiderherealready the continuum limit of the step scaling matrix σ(µ ,µ ),
1 0
have an important phenomenological relevance since with µ ∈ [µ , 3GeV] is obtained from another set
1 0
they allow for the computation of standard model and of simulations. There we use L ∼ 2.7fm and two
1
beyond the standard model neutral kaon mixing matrix finer lattice spacing a ∼ 0.086fm and 0.114fm (more
1
elements and of K → ππ amplitude in the ∆I = 3/2 details about these simulations can be found in [30, 31]).
channel. Finally we plan to combine the two results in order
to compute the renormalized matrix element at 2 or
3GeV and then apply the perturbative matching to
Numerical application MS. As mentioned above the two different volumes
The RBC-UKQCD collaboration has recently performed use two different gauge actions but the results can be
a computation of K → ππ decay amplitudes (in both combined together because we extrapolate the step
5
scaling matrix to the continuum. Although in principle olated to zero. As a consequence our results are affected
we would like to have a third lattice spacing in order to by a small systematic error, which was evaluated in [13]
have a better control on the continuum extrapolations for the (27,1) operator. We have checked that taking
in the volume L , in practice the discretisation effects the chiral extrapolation of the ratio of Z gives the same
1
on the step scaling matrix elements appear to be small. result as taking the ratio of the chiral extrapolation of
Obviously this method can be applied to all sort of Z. In figure 3 (right) we compare our non-perturbative
different quantities that one would like to extract from resultwiththenext-to-leadingorder(NLO)running[13].
our large lattice L . In the near future, when the next We note that the running is quite small (∼ 4% between
0
generation of supercomputers will be available, we plan µ ∼ 1.5GeV and ∼ 3.5GeV), while NLO perturbation
to add a finer lattice on the large volume L . theory predicts ∼ 7%. With our very small error bars
0
such a the difference is clearly visible. In figure 4, we
plot our results for the two (continuum) two-by-two step
Intherestofthissectionwepresentourresultsforthe
scaling matrices σ(µ,µ ) computed in the renormaliza-
step scaling matrices σ(µ ,µ) with µ varying in a range 0
0 tion basis. By definition, at the matching point µ = µ
[µ ,∼ 3GeV]. The computation of the renormalization 0
0 the matrices are equal to the identity. In figure 5 we
factors on the fine lattice is done using the same setup
compare our results for the electroweak penguins in the
as in [13, 32], to which we refer the reader for a more
∆s = 1 basis to the next-to-leading order (NLO) run-
detailed explanation. The computation of the Z factors
ning recently computed in [37]. In Figure 6 and 7 we
has been already presented in [32]. One of the main fea-
set µ = 3GeV and plot the non-perturbative running
tures of our computation is the use of the Domain-Wall 0
divided by the NLO prediction. In general we find that
fermion [33–35] which exhibits an almost exact chiral-
when NLO perturbation theory is available the results
flavoursymmetry. Asaconsequencetherenormalization
agree qualitatively with the non-perturbative ones. It is
pattern is the same as in the continuum (up to some interesting to note that for both the (8,8) and the (6,¯6),
numerically irrelevant lattice artefacts). Some other in-
wefoundthatoneoff-diagonalmatrixelementhasavery
teresting aspects of our calculation are the use of the
smallnonperturbativerunning,andthatthecorrespond-
volume sources [36] giving us a very good statistical pre-
ingoneloopanomalousdimensioniseitherzero(forσ )
cision, non-exceptional kinematics [10, 11] to suppress 32
or a very small number (for σ ).
the Goldstone pole contributions: here we use a scheme 45
which is called (γ ,γ )-scheme in [13]. In figure 2 we
µ µ
Conclusion
show the Z factor of the (27,1) operator, normalized by
Z2 and extrapolated to the chiral limit, obtained on the We have presented a general method to compute the
A non-perturbative continuum running in the operator
twodifferentlatticesatthesimulatedmomenta(covering
mixing case. In particular this method allows the use
a range from ∼1.1GeV to ∼3.5GeV). The correspond-
of an RI-MOM type scheme on a rather coarse lattice.
ing step scaling function is computed according to the
We have computed this running between ∼1.5GeV and
definitioneq(4). Infigure3(left)weplottheresultatfi-
∼ 3.5GeV in the case of four-quark operators which
nitelatticespacing,togetherwiththecontinuumextrap-
occur in neutral kaon mixing (including the BSM ones)
olation. Inordertodeterminethevertexfunctionatany
and ∆I = 3/2 K → ππ decays. Although the strategy
given momentum for each ensemble we fit our data as a
functionofp2andinterpolate. Asexplainedearlierinthe we have presented is very general we have shown why it
is important for the light quark physics program of the
text, because we are using twisted boundary conditions
RBC-UKQCD collaboration and in particular for a full
that keep the momentum orientation fixed with respect
computation of K →ππ amplitudes.
to the lattice axes this is a smooth interpolation. Hav-
ingobtainedinthiswaythevertexfunctiononanumber
of lattice spacings at a fixed physical p2 we extrapolate
this to the continuum limit using a constant plus O(a2) Acknowledgements
ansatz for each value of the momentum. This ansatz is WethankourcolleaguesintheRBCandUKQCDcollab-
justified in our approach because as we pick a fixed mo- orations,inparticularNormanChrist,ChristophLehner,
mentumorientationwithrespecttothelatticeaxes,O(4) and Chris Sachrajda for suggestions and stimulating dis-
breaking lattice artefacts are well parametrised [12]. As cussions. The calculations reported here were performed
one can see on the plot, the lattice spacing dependence on the QCDOC computers [38, 39] at Columbia Univer-
is very well under control. In [32] we have shown that sity, Edinburgh University, and at Brookhaven National
the renormalization pattern is the same as in the contin- Laboratory (BNL), Argonne Leadership Class Facility
uum,thechirallyforbiddenrenormalizationfactorsbeing (ALCF)BlueGene/PresourcesatArgonneNationalLab-
zero within error. The chiral extrapolation is done using oratory (ANL), and also the resources of the STFC-
three different quark masses and we found a very mild fundedDiRACfacility. Wewishtoacknowledgesupport
quark mass dependence for all our quantities. In our fromSTFCgrantST/H008845/1, DOEgrantDE-FG02-
setup the sea quark mass of the strange is fixed (to its 92ER40699 and to EU grant 238353 (STRONGnet).
physical value), whereas the valence sea quarks masses R.A. is supported by SUPA prize studentship.
are equal to the sea light quark masses and then extrap-
6
1.05
1
1.04
0.99
1.03
0.98
)1.02 )
µ01.01 µ00.97
µ, 1 a=0.114 fm µ,0.96
( a=0.086 fm (
110.99 cont extr 110.95
σ σ
0.98 0.94
0.97
0.93
0.96
0.92
0.95
0 2 4 6 8 10 12 2 4 6 8 10 12
µ2[GeV2] µ2[GeV2]
FIG. 3: Left: step scaling function (non-perturbative running) of the (27,1) operator at finite lattice spacings and in the
continuum in the lattice SMOM (γ ,γ )-scheme . The energy scale µ is fixed to µ ∼ 1.5GeV, and µ varies in the range
µ µ 0 0
[1.GeV, ∼3.5GeV]. Theerrorsbarsaresmallerthanthesymbols. Right: thesamequantityiscomparedtoperturbationtheory
(dashed black curve : one loop, solid black curve: two loops, solid blue curve: continuum extrapolation of non-perturbative
running with its error) in the range [µ ,∼3.5GeV].
0
1.8 1.8
1.6 s 1.6 s
88 44
1.4 1.4
1.2 1.2
) )
0 0
µ 1 s µ 1 s
, 77 , 55
µ 0.8 µ 0.8
( (
ij 0.6 ij 0.6
σ σ
s
0.4 s 0.4
54
78
0.2 0.2
s
45
0 s 0
87
−0.2 −0.2
2 4 6 8 10 12 2 4 6 8 10 12
µ2[GeV2] µ2[GeV2]
FIG.4: Stepscalingmatrixσ(µ,µ )giveninthe∆s=2renormalizationbasis. Ontheleftpanelweshowthematrixelements
0
of the (8,8) operators and on the right the (6,¯6) operators. Like in figure 3, the energy scale µ is fixed to µ ∼1.5GeV, and
0 0
µ varies in the range [µ , ∼3.5GeV].
0
APPENDIX (cid:40)(cid:2)O∆s=2(cid:3)+ = Q∆s=2,
(6,6) 2 4
(cid:2)O∆s=2(cid:3)+ = −1(Q∆s=2−Q∆s=2),
3 2 4 5
1. Denoting by the superscript “+” the parity even (cid:40)(cid:2)O∆s=2(cid:3)+ = Q∆s=2,
component,theexplicitrelationbetweenthetwo∆s=2 (8,8) (cid:2)O4∆s=2(cid:3)+ = −31Q∆s=2.
basis is 5 2 2
(27,1) (cid:2)O∆s=2(cid:3)+ = Q∆s=2,
1 1
7
2. To define the ∆I =3/2 part of the ∆s=1 operators, we follow the conventions of [40]
Q(cid:48) = (s¯ γ (1−γ )d )(cid:2)(u¯ γ (1−γ )u )−(d¯ γ (1−γ )d ))(cid:3)+(s¯ γ (1−γ )u )(u¯ γ (1−γ )d )) (A.1)
1 α µ 5 α β µ 5 β β µ 5 β α µ 5 α β µ 5 β
Q(cid:48) = (s¯ γ (1−γ )d )(cid:2)(u¯ γ (1+γ )u )−(s¯ γ (1+γ )s ))(cid:3)+(s¯ γ (1−γ )u )(u¯ γ (1+γ )d )) (A.2)
7 α µ 5 α β µ 5 β β µ 5 β α µ 5 α β µ 5 β
Q(cid:48) = (s¯ γ (1−γ )d )(cid:2)(u¯ γ (1+γ )u )−(s¯ γ (1+γ )s ))(cid:3)+(s¯ γ (1−γ )u )(u¯ γ (1+γ )d )) (A.3)
8 α µ 5 β β µ 5 α β µ 5 α α µ 5 β β µ 5 α
0.25
1
0.995 0.2
0.99
) )
00.985 00.15
µ µ
, 0.98 ,
µ µ
(0.975 ( 0.1
7 8
7 7
σ 0.97 σ
0.965 0.05
0.96
0.955 0
0.95
2 4 6 8 10 12 2 4 6 8 10 12
µ2[GeV2] µ2[GeV2]
0.035
1.7
0.03
1.6
0.025
1.5
) )
0 0
µ 0.02 µ
1.4
, ,
µ µ
(0.015 (1.3
7 8
8 8
σ σ
0.01 1.2
1.1
0.005
1
0
2 4 6 8 10 12 2 4 6 8 10 12
µ2[GeV2] µ2[GeV2]
FIG. 5: Step scaling matrix σ(µ,µ ) of the (8,8) electroweak penguins in the ∆s=1 renormalization basis. For each matrix
0
element we compare to perturbation theory (dashed black curve : one loop, solid black curve: two loops, solid coloured curve:
continuum extrapolation of non-perturbative running with its error).
8
1.01
1
1 1
]
)
0
µ
0.99
,
µ
(
U
0.98
/
)
0 s /LO
µ
0.97 s /NLO
,
µ
(
σ0.96
[
0.95
2 3 4 5 6 7 8 9 10
µ2[GeV2]
FIG. 6: Step scaling function σ (µ,µ ) of the (27,1) operator divided by the corresponding perturbative running U (µ,µ )
11 0 11 0
(dashedline: oneloop,solidcolouredcurve: twoloops). Foreaseofcomparisonwithperturbationtheoryandincontradiction
to the previous plots, µ is fixed at the conventional scale of 3GeV and µ varies in the range [1.5GeV, µ ].
0 0
1.005 0.015
7 8 s /LO
]7 1 ]7 0.01 s /NLO
) )
0 0
µ µ
0.005
, ,
µ0.995 µ
( (
U U
0
/ /
) )
0 0.99 s /LO 0
µ s /NLO µ−0.005
, ,
µ µ
(0.985 (
σ σ −0.01
[ [
0.98 −0.015
2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10
µ2[GeV2] µ2[GeV2]
0.01 1.02
7 0.008 s /LO 81.01
8 s /NLO 8
] 0.006 ]
) ) 1
0 0
µ 0.004 µ
0.99
, ,
µ 0.002 µ
( (0.98
U U
0
/ /
0.97
) )
0−0.002 0 s /LO
µ µ0.96 s /NLO
,−0.004 ,
µ µ
(−0.006 (0.95
σ σ
[−0.008 [0.94
−0.01 0.93
2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10
µ2[GeV2] µ2[GeV2]
FIG. 7: Same as figure 5 for the Step scaling matrix σ(µ,µ ) of the (8,8) electroweak penguins.
0
9
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