#### Non-perturbative renormalisation and running of BSM four-quark operators in \(N_\mathrm {\scriptstyle f}=2\) QCD

Eur. Phys. J. C
Non-perturbative renormalisation and running of BSM four-quark operators in Nf = 2 QCD
ALPHA Collaboration
P. Dimopoulos 1 2
G. Herdoíza 0 6
M. Papinutto 5
C. Pena 0 6
D. Preti 0 4
A. Vladikas 3
0 Instituto de Física Teórica UAM/CSIC, c/Nicolás Cabrera 13-15, Universidad Autónoma de Madrid , Cantoblanco, 28049 Madrid , Spain
1 Dipartimento di Fisica, Università di Roma Tor Vergata , Via della Ricerca Scientifica 1, 00133 Rome , Italy
2 Centro Fermi-Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi” , Compendio del Viminale, Piazza del Viminale 1, 00184 Rome , Italy
3 INFN-Sezione di Tor Vergata, c/o Dipartimento di Fisica, Università di Roma Tor Vergata , Via della Ricerca Scientifica 1, 00133 Rome , Italy
4 INFN Sezione di Torino , Via Pietro Giuria 1, 10125 Turin , Italy
5 Dipartimento di Fisica, “Sapienza” Università di Roma, and INFN, Sezione di Roma , Piazzale A. Moro 2, 00185 Rome , Italy
6 Departamento de Física Teórica, Universidad Autónoma de Madrid , Cantoblanco, 28049 Madrid , Spain
We perform a non-perturbative study of the scaledependent renormalisation factors of a complete set of dimension-six four-fermion operators without power subtractions. The renormalisation-group (RG) running is determined in the continuum limit for a specific Schrödinger Functional (SF) renormalisation scheme in the framework of lattice QCD with two dynamical flavours (Nf = 2). The theory is regularised on a lattice with a plaquette Wilson action and O(a)-improved Wilson fermions. For one of these operators, the computation had been performed in Dimopoulos et al. (JHEP 0805, 065 (2008). arXiv:0712.2429); the present work completes the study for the rest of the operator basis, on the same simulations (configuration ensembles). The related weak matrix elements arise in several operator product expansions; in F = 2 transitions they contain the QCD long-distance effects, including contributions from beyond-Standard Model (BSM) processes. Some of these operators mix under renormalisation and their RG-running is governed by anomalous dimension matrices. In Papinutto et al. (Eur Phys J C 77(6), 376 (2017). arXiv:1612.06461) the RG formalism for the operator basis has been worked out in full generality and the anomalous dimension matrix has been calculated in NLO perturbation theory. Here the discussion is extended to the matrix step-scaling functions, which are used in finite-size recursive techniques. We rely on these matrix-SSFs to obtain non-perturbative estimates of the operator anomalous dimensions for scales ranging from O( QCD) to O(MW).
1 Introduction
In lattice QCD, the renormalisation of composite operators
is an important step towards obtaining estimates of hadronic
low-energy quantities in the continuum limit. Quark masses,
decay constants, form factors, etc. are extracted from matrix
elements of such operators; see Ref. [
3
] for a recent review
of lattice flavour phenomenology. Of interest to the present
work is the class of dimension-six, four-fermion composite
fields, arising in operator product expansions (OPE), in which
the heavier quark degrees of freedom are integrated out. For
F = 2 and many F = 1 transitions (F stands for flavour
here), the resulting weak matrix elements of these operators
govern long-distance QCD effects. They can be reliably
evaluated by applying an intrinsically non-perturbative approach.
Lattice QCD is our regularisation of choice which, by
combining theoretical and computational methods, allows for an
evaluation of these quantities with errors that can be reliably
estimated and systematically improved.
Here we address the problem of calculating the
renormalisation parameters and their RG-running for the operators
defined in Eq. (2.1) below. We opt for the lattice
regularisation consisting in the Wilson plaquette gauge action and
the O(a)-improved Wilson quark action. We renormalise the
bare operators in the Schrödinger Functional
renormalisation scheme. This problem has first been studied with Wilson
fermions for the relatively simple case of the multiplicatively
renormalisable operators Q1± of Eq. (2.1), both
perturbatively [
4
] and non-perturbatively in the quenched
approximation [
5
]. Subsequently results for Q1± have also been obtained
with Nf = 2 dynamical sea quarks [
1
]. (An analogous study
with quenched Neuberger fermions may be found in Ref. [
6
].)
Recently the perturbative calculations have been extended in
Ref. [
2
] for the rest of the operator basis Qk±(k = 2, . . . , 5)
of Eq. (2.1). The present is a companion paper of this work,
complementing it by providing non-perturbative results for
the renormalisation and RG-running of Q2±, Q3±, Q4±, Q5±,
computed in Nf = 2 lattice QCD.
As stressed in Refs. [
2,7
], these operators, treated here
in full generality, become relevant for a number of
interesting processes, once specific physical flavours are assigned
to their fermion fields. For example, with ψ1 = ψ3 = s
and ψ2 = ψ4 = d (cf. Eq. (2.2)), the weak matrix
element K¯ 0|Q1+|K 0 comprises leading long-distance
contributions in the effective Hamiltonian formalism for
neutral K -meson oscillations in the Standard Model (SM).
Allowing for beyond-Standard-Model (BSM) interactions
introduces similar matrix elements of the remaining
operators Q2+, . . . , Q5+. In some lattice regularisations, the
corresponding bare matrix elements are expressed in terms of
the operators Q1+, . . . , Q5+, with some important
simplifications in their renormalisation properties [
7–9
]. In Ref. [2]
other flavour assignments are listed, leading to four-fermion
operators related to the low-energy effects of B = 2
transitions (B0–B¯ 0 and Bs0–B¯s0 mixing) and to the S = 1
effective weak Hamiltonian with an active charm quark. More
specifically, the operators O3/2, O3/2, O93/2, O130/2 (defined
7 8
for instance in Ref. [
7
]), characterising K → π π
transitions with I = 3/2, renormalise without power
subtractions and can be obtained from the general basis of Eq. (2.1)
with the proper flavour assignments. Operators of the same
basis, again with proper flavour assignments, also
characterise the I = 1/2 channel of K → π π decays. The
I = 1/2 operators, besides their logarithmic
renormalisation, also require power subtractions with coefficients
proportional to the quark masses,1 which vanish in the chiral
limit. Consequently, the determination of the logarithmic
renormalisation parameters in a mass-independent
renormalisation scheme is unaffected by the power subtractions and
our results for the logarithmic renormalisation parameters
also apply in this case. Other interesting applications of the
same four-fermion operators can be found in nuclear parity
violation processes (for instance see [
10,11
] and references
therein).
Several other approaches, based on different lattice
regularisation and renormalisation schemes, have addressed the
problem of non-perturbative renormalisation of four fermion
operators. Limiting ourselves to S = 2 oscillations, we
note that Wilson fermion results (of the standard and
twistedmass variety) have been mostly based on the RI/MOM
renor1 Mass-independent power divergences cancel through the GIM
mechanism.
malisation scheme [
7,12–15
], with Ref. [
16
] using RI/MOM
and the update [
17
] adopting the RI/SMOM variant. The
RI/MOM scheme has also been used with overlap fermions
in Ref. [
18
], while more recent works adopting domain
wall fermions are based on both RI/MOM [
19,20
] and
RI/SMOM [
19,21
]. (On the other hand, staggered fermion
operators have so far only been renormalised perturbatively
at 1-loop [
22–24
].) In these works renormalisation is
performed non-perturbatively at a scale of about 2–3 GeV, while
RG-running to higher scales is done in perturbation theory. It
is currently assumed that uncertainties inherent in the
renormalisation and/or RG-running of the S = 2 operators may
be at the root of some discrepancies of certain operator matrix
elements; see Ref. [
3
] for a comparison of results and a review
of the methods used to obtain them. Our approach may offer
an explanation of these discrepancies as it differs in both the
choice of renormalisation scheme and the method of
RGrunning: the scheme is the Schrödinger Functional and the
renormalisation scale is of O( QCD). Moreover, the RG
evolution is obtained between the hadronic and the electroweak
scale of O(MW). Both renormalisation and RG-running are
non-perturbative in this scale range. The scope of the present
work is to present in full detail these renormalisation results;
a study of their influence on the B-parameters and a
comparison with physical results existing in the literature will be
presented in a future separate publication.
It is important to keep in mind that the sets {Q2±, . . . , Q5±}
and {Q2±, . . . , Q5 } are parity-even and parity-odd
compo±
nents of operators with chiral structures (such as “left-left”
or “left-right”) which ensure their transformation under
specific irreducible chiral representations. Chiral symmetry may
be broken by the regularisation (e.g. lattice Wilson fermions)
but it is recovered by the continuum theory. An important
consequence is that our results, obtained for the continuum
RG-evolution of the parity-odd bases {Q2±, . . . , Q5±}, are also
valid for the parity-even ones {Q2±, . . . , Q5±}.
The paper is organised as follows: in Sect. 2 we list the
operators we are studying and their basic renormalisation
pattern. We also derive their RG-equations, and define the
evolution matrices and the renormalisation-group invariant
operators, which are scale- and scheme-independent
quantities. This is an abbreviated version of section 2 of Ref. [
2
].
The interesting feature of the renormalisation pattern of
operators Qk±(k = 2, . . . , 5) is that they mix in pairs.2 In the case
of Qk±(k = 2, . . . , 5), mixing is not an artefact of the
lattice regularisation, as it also happens in schemes where all
symmetries of the continuum target theory (QCD) are
preserved; cf. Ref. [
7
]. An important consequence of this
prop2 This is not to be confused with the operator mixing of Q1± (the operator
arising in F = 2 transitions in the Standard Model), which mixes with
Qk±(k = 2, . . . , 5) when Wilson lattice fermions are used, and chiral
symmetry is broken by the regularisation.
flavours. A complete set of Lorentz-invariant operators is
Q1± = OVV+AA,
±
Q2± = OVV−AA,
±
Q3± = OSS−PP,
±
Q4± = OSS+PP,
±
Q5± = −2 OT±T,
where
Q1± = OVA+AV,
±
Q2± = OVA−AV,
±
Q3± = OPS−SP,
±
Q4± = OPS+SP,
±
Q5± = −2 OTT˜
± ,
erty is that the RG-running of these operators is governed
by anomalous-dimension and RG-evolution matrices, rather
than scalar functions. The RG-evolution matrices are well
known in NLO perturbation theory; cf. Refs. [
25,26
]. Here,
following Ref. [2], we use them in closed form, suitable for
non-perturbative evaluations.
In Sect. 3 we outline our strategy. First we define the
SF renormalisation conditions for the operators Qk±(k =
1, . . . , 5); again this is an abridged version of Sect. 3.3 of
Ref. [
2
]. Next, we define in the SF scheme the matrix
stepscaling functions (matrix-SSFs) as the RG-evolution
matrices for a change of renormalisation scale by a fixed arbitrary
factor; this factor is 2 in the present work. These are our
basic lattice quantities, computed for a sequence of lattice
spacings, at fixed renormalised gauge coupling. They have a
well defined continuum limit, which is obtained by
extrapolation, as explained in the same section. Repeating the
calculation for a range of renormalisation scales (i.e. a range
of renormalised couplings) and interpolating our data points,
we finally have the matrix-SSFs as continuous polynomials
of the gauge coupling, from which we obtain the anomalous
dimension matrices, with NLO perturbation theory taking
over only at O(MW) scales.
In Sect. 4 we present our results. Besides the
aforementioned SSFs, RG-evolution matrices and
anomalousdimension matrices, we also compute the renormalisation
matrices for values of the gauge coupling corresponding to
low-energy scales. These renormalisation factors are needed,
in order to renormalise the corresponding bare matrix
elements at these hadronic scales. The computation of the latter
requires independent simulations on large physical lattices
(of about 3-5 fm), which is beyond the scope of this work.
Appendix A collects additional tests of the
comparison between perturbative and non-perturbative RG
evolution, including the specific renormalisation scale range,
[2 GeV, 3 GeV], considered in Ref. [
21
]. Further details about
one-loop cutoff effects in the matrix-SSFs are presented in
Appendix B.
2 Renormalisation of four-quark operators
This section is an abridged version of Sect. 2 of Ref. [
2
],
which we repeat here for completeness.
2.1 Renormalisation and mixing of four-quark operators
We recapitulate the main renormalisation properties of the
four-fermion operators under study. These results have been
obtained in full generality in Ref. [
7
]. The absence of
subtractions is elegantly implemented by using a formalism in
which the operators consist of quark fields with four distinct
(2.1)
(2.2)
(2.3)
O±1 2 = 21 (ψ¯ 1 1ψ2)(ψ¯ 3 2ψ4)
O 1 2± 2 1 ≡ O 1 2 ±O 2 1 . The operator subscripts
obvi± ± ±
ously correspond to the labelling V → γμ, A → γμγ5,
S → 1, P → γ5, T → σμν , T˜ → 21 εμνρτ σρτ , with
σμν ≡ 2i [γμ, γν ]. Repeated Lorentz indices, such as γμγμ
and σμν σμν are summed over. In the above expression round
parentheses indicate spin and colour traces and the subscripts
1, . . . , 4 of the fermion fields are flavour labels. Note that
operators Qk± are parity-even, and Qk± are parity-odd.
In the following we will assume a mass-independent
renormalisation scheme. Renormalised operators can be
written as
Q¯ k± = Zk±l (δlm +
l±m )Qm±,
Q¯ k± = Zk±l (δlm + Dl±m )Qm±
(summations over l, m are implied), where the
renormalisation matrices Z±, Z± are scale-dependent and reabsorb
logarithmic divergences, while ±, D± are (possible) matrices
of finite subtraction coefficients that only depend on the bare
coupling. Throughout this work we use boldface symbols for
the column vectors of four-fermion operator and the
matrices which act on these vectors (e.g. Q, Q, Z, , Z, D, etc.)
while their elements are indicated with explicit indices (e.g.
Qk , Qk , Zkl , kl , Zkl , Dkl , etc.). We also introduce a
simplification in our notation, by dropping the ± superscripts ,
wherever no ambiguity arises. This should not be a problem
as the symmetric operator bases {Qk+} and {Qk+} (symmetric
under flavour exchange 2 ↔ 4) never mix with the
antisymk } and {Qk−}, and thus equations are valid
metric ones {Q−
separately for each basis.
The renormalisation matrices have the generic structure
Z = ⎜⎜⎜
⎜
⎝
⎛ Z11
0
0
0
0
0
Z22
Z32
0
0
0
Z23
Z33
0
0
15 ⎞
Analogous expressions hold for Z and D. If chiral
symmetry is preserved by the regularisation, both and D
vanish. In the case of Wilson fermions, with chiral symmetry
explicitly broken, we have = 0, whereas due to residual
discrete flavour symmetries D = 0; this is the main result
of Ref. [
7
]. Therefore the left-left operators Q1 = OVA+AV,
which mediate Standard Model-allowed transitions,
renormalise multiplicatively, while operators Q2, . . . , Q5, which
appear as effective interactions in extensions of the Standard
Model, mix in pairs: {Q2, Q3} and {Q4, Q5}.
In conclusion, with Wilson fermions the parity-odd basis
{Qk } renormalises in a pattern analogous to that of a
chirally symmetric regularisation, while the parity-even one
{Qk } has a more complicated renormalisation pattern due to
the non-vanishing of . We will henceforth concentrate on
the non-perturbative renormalisation of the parity-odd basis
{Qk } with Wilson fermions.
2.2 Renormalisation group equations
The scale dependence of renormalised quantities is governed
by renormalisation group evolution. Denoting as μ the
running momentum scale and μ the renormalisation scale where
mass-independent renormalisation conditions are imposed,
we have the following Callan–Symanzik equations for the
gauge coupling and quark masses respectively:
μ
d
dμ
d
dμ
μ
g (μ) = β(g (μ)),
mf (μ) = τ (g (μ)) mf (μ),
where f is a flavour label. The scheme mass-independence
implies that the Callan–Symanzik function β and the mass
anomalous dimension τ depend only on the coupling.
Asymptotic perturbative expansions read
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
β(g) g≈∼0 −g3(b0 + b1g2 + · · · ),
τ (g) g≈∼0 −g2(d0 + d1g2 + · · · ).
Let us now turn to Euclidean correlation functions of
gauge-invariant composite operators, of the form3
Gk (x ; y1, . . . , yn ) = Qk (x )O1(y1) · · · On(yn)) ,
(2.9)
3 To simplify the notation, we have omitted the dependence of Gk on
the coupling, the masses and the UV cutoff (e.g. the lattice spacing).
with x = y j ∀ j, y j = yk ∀ j = k. For concreteness we have
opted for correlation functions of the parity-odd operators
Qk , which are the subject of the present work. Nevertheless,
the results of this section apply to any set of operators that mix
under renormalisation. The operators Ol (l = 1, · · · , n) may
be any convenient, multiplicatively renormalisable source
field. For example they could be currents or densities (e.g.
Vμ(y), Aμ(y), S(y) and/or P(y)), or Schrödinger functional
sources at the time-boundaries. The latter will be explicitly
discussed in Sect. 3. Renormalised correlation functions
satisfy the system of Callan–Symanzik equations
γ jk (gR) +
γ˜l (gR) δ jk G¯ k
(2.10)
γ jk (g) g≈∼0 −g2(γ j(k0) + γ j(k1)g2 + · · · ).
In standard fashion we can then derive
μ
5
l=1
d
dμ Z jl (Z−1)lk = γ jk .
This result implies that the block-diagonal form of the
renormalisation matrices Z (and Z) of Eq. (2.4) induces the
same block-diagonal structure for the anomalous dimension
matrix γ . Thus the sums in Eqs. (2.12) and (2.14) simplify:
for operator Q¯ 1 and its anomalous dimension γ11 there is
no summation; for operators Q¯ 2, Q¯ 3 summations run over
indices 2 and 3 only, and similarly for the operator sub-basis
Q¯ 4, Q¯ 5 .
μ
d
dμ G¯ j =
k
or, expanding the total derivative,
n
l=1
⎩
−
n
l=1
⎨⎧ μ ∂∂μ + β(gR) ∂g∂R +
k
Nf
f=1
γ˜l (gR) G¯ j =
γ jk (gR) G¯ k ,
(2.11)
where γ is a matrix of anomalous dimensions describing the
mixing of {Qk } (cf. Eq. (2.12) below), and γ˜l is the anomalous
dimension of Ol (defined in a way analogous to Eq. (2.12)).
A possible term arising from the running of the gauge
parameter λ of the action is omitted here, for reasons explained in
Ref. [
2
]. A convenient shorthand notation for the anomalous
dimension matrix of the operators Q¯ k is thus
d
μ dμ Q¯ j (μ) =
5
k=1
γ jk (g (μ))Q¯ k (μ).
The operator anomalous dimensions admit perturbative
expansions of the form
τ (gR)mR,f
∂
∂mR,f
(2.12)
(2.13)
(2.14)
2.3 Evolution matrices and renormalisation group
invariants
One can then easily integrate the exponent in Eq. (2.17) and
obtain the LO approximation of the evolution matrix:
In order to obtain a solution of Eq. (2.12) in closed form, it
is convenient to introduce the renormalisation group
evolution matrix U(μ2, μ1) that evolves renormalised operators
between scales4 μ1 and μ2 < μ1:
Q i (μ2) = Ui j (μ2, μ1) Q j (μ1).
Substituting the above into Eq. (2.12) we obtain for the
running of U(μ2, μ1)
μ2
d
dμ2
U(μ2, μ1) = γ [g (μ2)]U(μ2, μ1),
with initial condition U(μ1, μ1) = 1. Note that the r.h.s.
is a matrix product. Following a standard procedure, the
above expression can be converted into a Volterra-type
integral equation and solved iteratively, viz.
g (μ2)
dg
g (μ1)
1
β(g)
U(μ2, μ1) = Texp
γ (g) ,
(2.17)
where as usual the notation Texp denotes a Taylor expansion
of the exponent, in which each term is an ordered (here
gordered) product. Explicitly, for a generic matrix function
M(x ), we have
(2.15)
(2.16)
Texp
≡ 1 +
+
+
= 1 +
x+
dx M(x )
x−
x+
dx M(x )
x−
x+
dx1 M(x1)
x−
x+
dx1 M(x1)
x−
+ · · ·x+
dx M(x )
1
+ 2
x− x+
dx1
! x−
x−
x1
dx2 M(x2)
x−
x1
dx2 M(x2)
x−
x2
dx3 M(x3)
x−
x+
dx2 θ (x1 − x2)M(x1)M(x2)
+θ (x2 − x1)M(x2)M(x1)
+ · · ·
(2.18)
In the specific case of interest, M(g) = γ (g)/β(g), with
γ (g) a matrix function and β(g) a real function. To leading
order (LO) we have that M(g) = γ (0)/(b0g) and the
independence of the matrix γ (0) from the coupling g simplifies
Eq. (2.18), so that the Texp becomes a standard exponential.
4 Restricting the evolution operator to run towards the IR avoids
unessential algebraic technicalities below. The running towards the UV
can be trivially obtained by taking [U (μ2, μ1)]−1.
γ (0)
g 2(μ2) 2b0
U(μ2, μ1) L=O g 2(μ1)
≡ ULO(μ2, μ1).
(2.19)
When next-to-leading order corrections are included, the
Texponential becomes non-trivial. Further insight is gained
upon realising that the associativity property of the evolution
matrix U(μ3, μ1) = U(μ3, μ2)U(μ2, μ1) implies that it can
actually be factorised in full generality as
U(μ2, μ1) =
U˜ (μ2)
−1
U˜ (μ1),
and the matrix U˜ (μ) can be expressed in terms of a matrix
W(μ), defined through
U˜ (μ) ≡
The matrix W can be interpreted as the piece of the
evolution operator containing contributions beyond the leading
perturbative order. Putting everything together, we see that
U(μ2, μ1) ≡ [W(μ2)]−1 ULO(μ2, μ1)W(μ1),
and thus we make contact with the literature (see e.g. [
25,
26
]).
Upon inserting Eq. (2.22) in Eq. (2.16) we obtain for W
the RG equation
μ
d
dμ
W(μ) = −W(μ)γ (g (μ)) + β(g (μ))
γ (0)
b0g (μ)
W(μ)
= [γ (g (μ)), W(μ)]
− β(g (μ))
γ (g (μ)) γ (0)
β(g (μ)) − b0g (μ)
Expanding perturbatively we can check [
2
] that W is regular
in the UV, and all the logarithmic divergences in the evolution
operator are contained in ULO; in particular,
W(μ)
μ→=∞
1.
Rewriting Eq. (2.15) as
γ (0)
g 2(μ2) − 2b0
4π
=
γ (0)
g 2(μ1) − 2b0
4π
W(μ2)Q(μ2)
W(μ1)Q(μ1),
and observing that the l.h.s. (respectively r.h.s.) is obviously
independent of μ1 (respectively μ2), we conclude that these
(2.20)
(2.21)
(2.22)
W(μ).
(2.23)
(2.24)
(2.25)
are scale-independent expressions. Thus we can define the
vector of RGI operators as
Qˆ ≡
= μl→im∞
where in the last step we use Eq. (2.24).
the above observation about boundary fields, and the need to
saturate flavour indices, the minimal structure involves three
boundary bilinear operators and the introduction of an extra,
“spectator” flavour (labeled as number 5, keeping with the
notation in Eq. (2.2)). We thus end up with correlation
functions of the generic form
Fk;s (x0) ≡ Qk (x )Ss ,
Gk;s (T − x0) ≡ ηk Qk (x )Ss ,
where Ss is one of the five source operators
3 Schrödinger functional renormalisation setup
In this section we introduce the finite volume Schrödinger
Functional renormalisation schemes and the RG evolution
matrix between scales separated by a fixed factor (i.e. the
matrix-step-scaling function).
3.1 Renormalisation conditions
We first define Schrödinger Functional renormalisation
schemes for the operator basis of Eq. (2.1). This section is an
abridged version of sec. 3.3 of Ref. [
2
]. We use the standard
SF setup as described in [
27
], where the reader is referred for
full details including unexplained notation.
We work with lattices of spatial extent L and time extent
T ; here we opt for T = L. Source fields are made up of
boundary quarks and antiquarks,
Oαβ [ ] ≡ a
Oαβ [ ] ≡ a
6
6
y,z
y,z
ζ¯α(y) ζβ (z),
ζ¯α(y) ζβ (z),
where α, β are flavour indices, unprimed (primed) fields live
at the x0 = 0 (x0 = T ) boundary, and is a Dirac matrix. The
boundary fields ζ, ζ¯ are constrained to satisfy the conditions
ζ (x) = 21 (1 − γ0)ζ (x),
ζ¯ (x) = ζ¯ (x) 21 (1 + γ0), (3.3)
and similarly for primed fields. This implies that the Dirac
matrices must anticommute with γ0, otherwise the
boundary operators Oαβ [ ] and Oαβ [ ] vanish; thus may be
either γ5 or γk (k = 1, 2, 3).
Renormalisation conditions are imposed in the massless
theory, in order to obtain a mass-independent scheme by
construction. They are furthermore imposed on the parity-odd
four-quark operators {Qk±} of Eq. (2.1), since working in the
parity-even {Qk±} sector would entail dealing with the extra
mixing due to explicit chiral symmetry breaking with
Wilson fermions, cf. Eq. (2.4). In order to obtain non-vanishing
SF correlation functions, we then need a product of source
operators with overall negative parity; taking into account
1
S2 ≡ 6
k,l,m=1
1 3
S3 ≡ 3
k=1
1 3
S4 ≡ 3
k=1
1 3
S5 ≡ 3
k=1
with
W[γ5, γk , γk ],
W[γk , γ5, γk ],
W[γk , γk , γ5]
W[
1, 2, 3
] ≡ L−3O21[
1
]O45[
2
]O53[
3
],
and similarly for Ss , which is defined with the boundary fields
exchanged between time boundaries; e.g O53 ↔ O53 etc.
The constant ηk is a sign that ensures Fk;s (x0) = Gk;s (x0)
for all possible indices;5 it is easy to check that η2 =
− 1, ηs=2 = +1.We also use the two-point functions of
boundary sources
(2.26)
(3.1)
(3.2)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
,
where α is an arbitrary real parameter. The geometry of
Fk;s , f1, and k1 is illustrated in Fig. 1.
5 This time reversal property, besides being a useful numerical cross
check of our codes, allows taking the average of Fk;s (x0) and Gk;s (x0)
so as to reduce statistical fluctuations. From now on Fk;s (x0) denotes
this average.
the four-quark operator insertion, and dashed lines indicate the explicit
time-like link variable involved in boundary-to-boundary quark
propagators
(3.15)
(3.16)
We can now impose Schrödinger functional
renormalisation conditions on the ratio of correlation functions defined in
Eq. (3.14), at fixed bare coupling g0, vanishing quark mass,
and scale μ = 1/L . For the renormalisable multiplicative
operators Q1 we set
Z11;s,α A1;s,α = A1;s,α g02=0 .
For operators that mix in doublets, we impose6
Z22;s1,s2,α
Z32;s1,s2,α
=
A2;s1,α
A3;s1,α
Z23;s1,s2,α !
Z33;s1,s2,α
A2;s2,α !
A3;s2,α
g02=0
,
A2;s1,α
A3;s1,α
A2;s2,α !
A3;s2,α
and similarly for Q4,5. The product of boundary-to-boundary
correlators in the denominator of Eq. (3.14) cancels the
renormalisation of the boundary operators in Fk;s , and
therefore Z j k;s1,s2,α only contains anomalous dimensions of
four-fermion operators. Following [
1, 5, 28
], conditions are
imposed on renormalisation functions evaluated at x0 =
T /2, and the phase that parameterises spatial boundary
conditions on fermion fields is fixed to θ = 0.5. Together with
the L = T geometry of our finite box, this fixes the
renormalisation scheme completely, up to the choice of boundary
source, indicated by the index s, and the parameter α. The
latter can in principle take any value, but we restrict our choice
to α = 0, 1, 3/2.
One still has to check that the above renormalisation
conditions are well-defined at tree-level. This is straightforward
for Eq. (3.15), but not for Eq. (3.16): it is still possible that
the matrix of ratios A has zero determinant at tree-level,
rendering the system of equations for the renormalisation matrix
ill-conditioned. This is indeed obviously the case for s1 = s2,
but the determinant vanishes also for other non-trivial choices
of s1 = s2. In practice, out of the ten possible schemes one
6 S. Sint, unpublished notes (2001).
is only left with six, viz.7
(s1, s2) ∈ {(1, 2), (1, 4), (1, 5), (2, 3), (3, 4), (3, 5)}.
(3.17)
This property is independent of the choice of θ and α. Thus,
we are left with a total of 15 schemes for Q1, and 18 for each
of the pairs (Q2, Q3) and (Q4, Q5).
Given the strong scheme dependence of the matrices
γ (1);SF (cf. Eq. (2.13)), a criterion has been devised in
Ref. [
2
] in order to single out the scheme with the smallest
NLO anomalous dimension. This consists in choosing the
scheme with the smallest determinant and trace of the matrix
16π 2γ (1);SF[γ (0)]−1 for each non-trivial 2 × 2 anomalous
dimension matrix. It turns out that the scheme defined by
α = 3/2 and (s1, s2) = (3, 5) satisfies these requirements
in all cases (i.e. for the matrices related to (Q2, Q3) and
(Q4, Q5)). In the following we will present non-perturbative
results for this scheme only.8
3.2 Matrix-step-scaling functions and non-perturbative
computation of RGI operators
In order to trace the RG evolution non-perturbatively,
we introduce matrix-step-scaling functions (matrix-SSFs),
defined as9
σ (u) ≡ U(μ/2, μ)|g 2(μ)=u
= [W(μ/2)]−1 ULO(μ/2, μ)W(μ).
(3.18)
7 Note that schemes obtained by exchanging s1 ↔ s2 are trivially
related to each other.
8 Although we have completed our analyses in all schemes discussed
here, for reasons of economy of presentation we will not show these
results. In any case, the α = 3/2 and (s1, s2) = (3, 5) scheme displays
the most reliable matching to perturbative RG-running at the
electorweak scale.
9 The relative factor between the scales is arbitrary; one could introduce
a σ (s, u) that evolves from scale μ to scale μ/s. In this notation, our
choice corresponds to s = 2.
The above definition generalises the step-scaling functions
defined for quark masses [
28
] and multiplicatively
renormalisable four-fermion operators [
5
] such as Q1±. Just like
the anomalous dimension matrix γ , the matrix-SSF σ has a
block-diagonal structure. So the above definition either refers
to one of the two multiplicative operators Q1±, or to one of
the four pairs of operators that mix under renormalisation;
i.e. (Q2±, Q3±) or (Q4±, Q5±). In the former cases σ is a real
function, whereas in the latter cases it is a 2 × 2 matrix of
real functions. Again in what follows the ± superscripts will
be suppressed.
The advantage of working with step-scaling functions is
that they can be computed on the lattice with all systematic
uncertainties under control. More concretely, we define the
lattice matrix-SSF in a finite (L/a)3 × (T /a) lattice; as
repeatedly stated previously, in this work we set L = T .
Working in the chiral limit, at a given bare coupling g0 (i.e. at
a given finite UV cutoff a−1) , is defined as the following
“ratio” of renormalisation matrices at two renormalisation
scales μ = 1/L and μ/2 = 1/(2L):
(g02, a/L) ≡ Z "g02, 2aL #
Z "g02, La # −1 .
(3.19)
This quantity has a well defined continuum limit. For a
sequence of lattice sizes L/a, we tune the bare coupling
g0(a) (and thus the corresponding lattice spacing a) to a
sequence of values which correspond to a constant
renormalised squared coupling g¯2(1/L) = u. Keeping u fixed
implies that the renormalisation scale μ = 1/L is also held
fixed. It is then straightforward to check that satisfies
(3.20)
σ (u) = al→im0
(g02, a/L) g¯2(1/L)=u .
Thus, the computation of the renormalisation matrices Z at
a fixed value of the renormalised squared coupling u and
various values of the lattice sizes L/a and 2L/a, allows for a
controlled extrapolation of the matrix-SSFs to the continuum
limit.
The strategy for obtaining non-perturbative estimates of
RGI operators proceeds in standard fashion: We start from
a low-energy scale μhad = 1/Lmax, implicitly defined by
g¯2(1/Lmax) = u0. The SSF σ (u) for the coupling, defined
as σ (g¯2(1/L)) = g2(1/(2L)), is known for Nf = 2 from
Ref. [
29
]. Thus we generate a sequence of squared couplings
(u1, . . . , u N ) through the recursion σ −1(un−1) = un, and
compute recursively the matrix-SSFs (σ (u1), . . . , σ (u N ))
which correspond to a sequence of physical lattice lengths
(inverse renormalisation scales) (Lmax/2, . . . , Lmax/2N ).
This is followed by the computation of
U(μhad, μpt) = σ (u1) · · · σ (u N ),
with μhad = 2−N μpt = Lm−a1x. Here μpt ∼ O(MW) is
thought of as a high-energy scale, safely into the
perturbative regime, and μhad ∼ O( QCD) as a low-energy scale,
(3.21)
characteristic of hadronic physics. The RGI operators of Eq.
(2.26) can finally be constructed as follows:
Qˆ =
In other words, once we know the column of renormalised
operators Q(μhad) at a hadronic scale from a standard
computation on a lattice of “infinite” physical volume (which
is beyond the scope of the present paper), we can combine
it with the non-perturbative evolution matrix [U (μhad, μpt)
(which is the result of this work) and the remaining
μptdependent factors at scale μpt (known in NLO perturbation
theory from Ref. [
2
]), to obtain the RGI operators.10 All
factors on the r.h.s. must be known in the same
renormalisation scheme, which here is the SF. The scheme dependence
should cancel in the product of the r.h.s., since Qˆ is
schemeindependent. In practice a residual dependence remains due
to the fact that W(μpt) is only known in perturbation theory
(typically to NLO). Finally we stress that Qˆ depends, through
the operators Q(μ), on the values of the quark masses; of
course the result also depends on the flavour content of the
QCD model under scrutiny (i.e. Nf ).
We mentioned above that the matrix W(μpt) is known
in NLO perturbation theory from Ref. [
2
]. This statement
requires a brief elucidation: W(μpt) is obtained by
numerically integrating Eq. (2.23), using the NLO (2-loop)
perturbative result for γ and the NNLO (3-loop) perturbative result
for β. In what follows this will be abbreviated as NLO-2/3PT.
In line with Ref. [
2
], also the present work devotes
considerable effort to the investigation of the reliability of NLO-2/3PT
at the scale μpt.
3.3 Matrix-step-scaling functions and continuum
extrapolations
We now turn to some practical considerations concerning the
extrapolation of (u, a/L) to the continuum limit a/L →
0, from which we obtain σ (u); cf. Eq. (3.20). We stress
that although fermionic and gauge actions are
Symanzikimproved by the presence of bulk and boundary
counterterms, correlation functions with dimension-six operators in
the bulk of the lattice, such as those defined in Eqs. (3.4)
and (3.14), are subject to linear discretisation errors. Their
removal could be achieved in principle by the subtraction
of dimension-7 counter-terms, but their coefficients are not
easy to determine in practice. We therefore expect linear
cutoff effects and consequently fit with the Ansatz
10 The computation of operators Q(μhad) (i.e. their physical matrix
elements) must be known with a precision similar to that of the evolution
matrix.
(u, a/L) = σ (u) + ρ(u)(a/L) .
In analogy to Ref. [
30
], we explore the reliability of the
above extrapolations with the help of the lowest-order
perturbative expression for i j , which includes O(ag02) terms.
In general the perturbative series for the operator
renormalisation matrices has the form [
30
]
Z(g0, L/a) = 1 +
Z(l)(L/a)g02l ,
∞
l=1
where in the limit a/L → 0 the coefficients Z(l) are
ldegree polynomials in ln(L/a) up to corrections of O(a/L).
In particular the coefficient of the logarithmic divergence in
Z(1) is given by the one-loop anomalous dimension γ (0), and
thus we parametrise Z(1) as
Z
(1) = CF z(θ , T /L) + γ (0) ln(L/a) + O(a/L) , (3.25)
with θ = 0.5 and T /L = 1. It is now easy to see that the
one-loop perturbative expression for the matrix-SSF is given
by
(gR2, a/L) = 1 + k(L/a)gR2 + O(gR2) ,
with
k(L/a) = Z(1)(2L/a) − Z(1)(L/a) .
In the continuum limit (a/L → 0 with g¯2 = u fixed) we
have
k(∞) = γ (0) log(2).
The quantity
δk (L/a) ≡ k(L/a)[k(∞)]−1 − 1.
contains all lattice artefacts at O(g02). Results for δk (L/a)
are reported in Appendix B.
The “subtracted” matrix-SSF, defined as
˜ (u, a/L) ≡
(u, a/L)
1 + u log(2)δk (a/L)γ (0) −1
u=g¯2(L)
(3.30)
also tends to σ in the continuum limit, but has the O(ag¯2)
effects removed. We will also use this quantity when studying
the reliability of the linear continuum extrapolations below.
3.4 Perturbative expansion of matrix-step-scaling functions
Once the continuum matrix-SSF σ (u) has been computed for
N discrete values of the renormalised coupling g¯2(1/L) = u,
it is useful to interpolate the data so as to obtain σ (u) as a
continuous function. This is done by fitting the N points by
a suitably truncated polynomial
σ (u) = 1 + r1u + r2u2 + r3u3 + · · · .
(3.31)
With only a few (N ) points at our disposal, the fit stability is
greatly facilitated by fixing the first two coefficients
(matrices) r1 and r2 respectively to their LO and NLO perturbative
values, leaving r3 as the only free fit parameter. We will now
derive the perturbative coefficients r1 and r2.
Since the operator RG-running is coupled to that of the
strong coupling, we also need the LO and NLO coefficients
of its SSF; i.e.
Given the strong coupling value g¯2(1/L) = u at a
renormalisation scale μ = 1/L, its SSF is defined as σ (u) =
g¯2(1/2L); cf. Ref. [
31
]. Combining this definition with that
of the Callan–Symanzik β-function of Eq. (2.5), we find that
− ln 2 =
√σ (u) dg
√u
β(g)
.
Plugging the NLO expansion of Eq. (2.7) in the above and
taking Eq. (3.32) into account, we obtain the coefficients of
the coupling SSF
(3.23)
(3.24)
(3.26)
(3.27)
(3.28)
(3.29)
Matrix-SSFs for four-quark operators have been
introduced in Eq. (3.18). In order to calculate the coefficients
r1 and r2 of its perturbative expansion Eq. (3.31), we first
write down the LO evolution matrix as
(3.32)
(3.33)
(3.34)
(3.35)
+ . . .
(3.36)
(3.37)
ULO(μ/2, μ)|g 2(μ)=u =
= exp
+u2
= 1 + uγ (0) ln 2
γ (0)
2b0
ln
σ (u)
u
γ (0)
σ (u) 2b0
u
b1 ! γ (0) ln 2 + ln2 2 "γ (0)#2
b0 ln 2 + b0 2
Furthermore, the matrix W(μ) of Eq. (3.18) has the NLO
perturbative expansion (cf. Ref. [
2
] and references therein)
W(μ) = 1 + uJ1 + u2J2 + . . . ,
from which the inverse matrix is readily obtained:
[W(μ/2)]−1 = 1 − σ (u)J1 + (J12 − J2)σ (u)2 + · · ·
= 1 − uJ1 + u2(J12 − s1J1 − J2) + · · · .
(3.38)
We arrive at the last expression on the rhs by inserting the
power-series expansion of σ (u) form Eq. (3.32).
Substituting the various terms in Eq. (3.18) by the perturbative
series (3.36), (3.37) and (3.38), we find
r1 = γ (0) ln 2,
From the first expression obtained for r2 we see explicitly
that O(u2) corrections to W do not contribute (i.e. terms
with J2 are absent), in accordance with the fact that the O(u)
term of W already contains all NLO contributions. The
second expression for r2 is obtained by using the property (cf.
Ref. [
2
] and references therein)
2b0J1 − [γ 0, J1] = bb01 γ (0) − γ (1) .
Remarkably, the final result for r2 is the exact analogue of
the one found for operators that renormalise multiplicatively,
cf. e.g. Eq. (6.6) in [
4
].
4 Non-perturbative computations
Our simulations are performed using the lattice regularisation
of QCD consisting of the standard plaquette Wilson action for
the gauge fields and the non-perturbatively O(a) improved
Wilson action for Nf = 2 dynamical fermions. The fermion
action is Clover-improved with the Sheikoleslami–Wohlert
(SW) coefficient csw determined in [
32
]. The matrix-SSFs
are computed at six different values of the SF renormalised
coupling, corresponding to six physical lattice extensions L
(i.e six values of the renormalisation scale μ). For each
physical volume three different values of the lattice spacing a are
simulated, corresponding to lattices with L/a = 6, 8, 12;
this is achieved by tuning the bare coupling g0(a) so that the
renormalised coupling (and thus L) is approximately fixed.
At the same g0(a) we also generate configuration ensembles
at twice the lattice volume; i.e. 2L/a = 12, 16, 24
respectively. We compute Z(g0, a/L) and Z(g0, a/(2L)) and thus
(g02, a/L); cf. Eq. (3.19). The gauge configuration
ensembles used in the present work and the tuning of the lattice
parameters (β, κ) are taken over from Ref. [
33
] where all
technical details concerning these dynamical fermion
simulations are discussed. As pointed out in [
33
], the gauge
configurations at the three weakest couplings have been produced
using the one-loop perturbative estimate of ct [
34
], except
for (L/a = 6, β = 7.5420) and (L/a = 8, β = 7.7206).
For these two cases and for the three strongest couplings the
two-loop value of ct [
35
] has been used.
(3.40)
(3.41)
Statistical errors are computed by blocking (binning) the
measurements of each renormalisation parameter and
calculating the bootstrap error on the binned averages. In order to
take their autocorrelation length into account, we determine
the block-size for which the bootstrap error of a given
renormalisation parameter reaches a plateau. This varies for each
of the four matrix elements of a given 2 × 2 renormalisation
matrix. We conservatively fix our preferred block-size to the
maximum of all four cases, and estimate our statistical error
accordingly. We crosscheck our results by also applying the
Gamma method error analysis of Ref. [
36
], and by varying
the summation-window size. The results from the two
methods agree within the (relevant) uncertainties.
Numerical results for [Z(g0, a/L)]−1 and Z(g0, a/(2L)),
computed from Eq. (3.16), are collected in Tables 7 and 8.
The reason we prefer quoting the inverse of Z(g0, a/L) is
that it is this quantity which is required for the computation
of the matrix-SSFs; cf. Eq. (3.19).
4.1 Lattice computation of matrix-functions
We perform linear extrapolations in a/L of both and ˜ (cf.
Eqs. (3.23) and (3.30)), so as to crosscheck the reliability of
the continuum value σ (u). The extrapolation results can be
found in Tables 1 and 2, as well as in Figs. 2, 3, 4, and 5. In
most cases both extrapolations agree; at worst the agreement
is within two standard deviations (e.g. in Fig. 2 the difference
between off-diagonal elements of the matrices and ˜ is
sizeable). We quote, as our best results, those obtained from
linear extrapolations in a/L, involving all three data-points
of the “subtracted” matrix-SSFs. We estimate the systematic
error as the difference between the value of σ obtained by
extrapolating and ˜ . This error is added in quadrature to
the one from the fit.
Similar checks with another two definitions of
“subtracted” matrix-SSFs, namely:
(u, a/L) ≡
1 + u log(2)δk (a/L)γ (0) −1
(u, a/L) ,
(u, a/L) ≡
(u, a/L) − u log(2)δk (a/L)γ (0) ,
which differ at O(u2) have not revealed any substantial
differences in the results.
4.2 RG running in the continuum
In order to compute the RG running of the operators in the
continuum limit, matrix-SSFs have to be fit to the functional
form shown in Eq. (3.31). Several fits have been tried out,
with different orders in the polynomial expansion and r2
either kept fixed to its perturbative value or allowed to be
a free fit parameter. Fits with r1 fixed by perturbation theory
(4.1)
(4.2)
Table 1 Continuum
matrix-SSFs for the operator
bases {Q2±, Q3 }
±
Table 2 Continuum
matrix-SSFs for the operator
bases {Q4±, Q5 }
±
u
σ (−2,3)(u)
1.0003(74)
− 0.0094(41)
1.0098(83)
− 0.0055(40)
1.007(12)
− 0.0106(60)
0.9952(85)
− 0.0213(55)
0.986(14)
− 0.0200(75)
0.950(19)
− 0.0500(95)
σ (−4,5)(u)
and r2 the only free fit parameter do not describe the data
well. This is understandable, as deviations from LO are large
for some matrix elements (for σ 5+4 in particular) and
knowledge of the NLO anomalous dimension γ (1) (and therefore
r2; cf. Eq. (3.40)) is necessary for a well-converging fit. It is
however an encouraging crosscheck that the r2 value returned
by the fit is close to the perturbative prediction of Eq. (3.40).
If, besides r2, we also include r3 as a free fit parameter, the
results have large errors. The best option turns out to be the
one with the polynomial expansion of Eq. (3.31) truncated
at O(u4), r1 and r2 fixed to their perturbative values and r3
left as free fit parameter. The plots of the matrix-SSFs are
collected in Figs. 6 and 7.
In the same Figures we also show the LO and NLO
perturbative results, calculated from Eq. (3.31), truncated at O(u)
and O(u2) respectively. The comparison between the
nonperturbative, the LO, and the NLO results provides a
useful assessment of the reliability of the perturbative series.
There is coincidence of all three curves at very small
(perturbative) values of the squared gauge coupling u, but this
is obviously guaranteed by the form of our fit function, as
described above. At larger u-values one would ideally hope
to see the NLO curves lying closer to the non-perturbative
ones, compared to the LO curves. For σ + this is mostly the
case, as shown in Fig. 6, the only exception being [σ +]23 and
[σ +]44. For the operator basis {Q2+, Q3+}, non-perturbative
and NLO curves seem in good agreement for the diagonal
elements [σ +]22 and [σ +]33. This is less so for the non-diagonal
[σ +]23 and [σ +]32. For the operator basis {Q4+, Q5+},
nonperturbative and NLO curves mostly agree, with the
exception of [σ +]44. We also note that the non-perturbative [σ +]23
tends to decrease at large u, unlike the monotonically
increasing perturbative predictions. For σ − the NLO curves lie
closer to the non-perturbative results compared to the LO
ones, in all cases but [σ −]23 and [σ −]55 (for [σ −]54 LO and
NLO are very close to each other). In several cases
nonperturbative and NLO curves are in fair, or even excellent,
agreement also at large u-values (cf. [σ3−2], [σ3−3], [σ4−4] and
[σ4−5]). In other cases this comparison in less satisfactory.
Note that the NLO [σ −]54 and [σ −]55 curves are
monotonically increasing, as opposed to the non-perturbative ones. In
conclusion the overall picture in the renormalisation scheme
under investigation is in accordance with our general
expec0
0.03
0.06
0.12
0.15
0.18
0
0.03
0.06
0.12
0.15
0.18
0.09
a/L
0.09
a/L
0
0.03
0.06
0.12
0.15
0.18
0
0.03
0.06
0.12
0.15
0.18
Fig. 2
Continuum limit extrapolation of
0.09
a/L
579
+(u, a/L ) in red and ˜ +
0
0.03
0.06
0.12
0.15
0.18
0
0.03
0.06
0.12
0.15
0.18
0
0.03
0.06
0.12
0.15
0.18
0
0.03
0.06
0.12
0.15
0.18
0.09
0.09
a/L
0.03
0.06
0.12
0.15
0.18
0.09
a/L
0
0.03
0.06
0.12
0.15
0.18
0.09
a/L
579
0
0.03
0.06
0.12
0.15
0.18
0
0.03
0.06
0.12
0.15
0.18
0.09
a/L
0.09
a/L
0
0.03
0.06
0.12
0.15
0.18
0
0.03
0.06
0.12
0.15
0.18
coupling, u = 0.9793, 1.1814, 1.5078, 2.0142, 2.4792, 3.3340, grow from top to bottom for each element of the matrix-SSFs
Fig. 5 Continuum limit extrapolation of
−(u, a/L ) in red and ˜ −(u, a/L ) in blue, of the operator basis {Q4−, Q5−}. The values of the renormalised
Fig. 6 Continuum matrix-SSFs
for operator bases {Q2+, Q3 }
+
(top) and {Q4+, Q5+} (bottom).
The LO perturbative result is
shown by the dotted black line,
while the NLO one by the dashed
blue line. The red line (with error
band) is the non-perturbative
result from the O(u3) fit
as described in the text. The two
error bars on each data point are
the statistical and total
uncertainties; the systematic error
contributing to the latter has been
estimated as explained in the text
1.10
1.05
0.15
0.10
0.05
LO
NLO
3
LO
NLO
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2
2.5
3.5
0
0.5
1
1.5
2
2.5
3.5
0
0.5
1
1.5
2
2.5
3
3.5
LO
NLO
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2
2.5
3
3.5
LO
NLO
LO
NLO
3
3.5
Fig. 7 Continuum matrix-SSFs
for operator bases {Q2−, Q3 }
−
(top) and {Q4−, Q5−} (bottom).
The LO perturbative result is
shown by the dotted black line,
while the NLO one by the
dashed blue line. The red line
(with error band) is the
non-perturbative result from the
O(u3) fit as described in the
text. The two error bars on each
data point are the statistical and
total uncertainties; the
systematic error contributing to
the latter has been estimated as
explained in the text
1.10
LO
NLO
u
u
u
u
LO
NLO
LO
NLO
LO
NLO
3
Table 3 The matrix
U˜ (±2,3)(μhad), corresponding to
the operator bases {Q2±, Q3±}. It
is computed for a fixed
low-energy scale μhad and
varying higher-scales 2nμhad.
For sufficiently large n, the
results should not depend on the
higher-energy scale
n
0
1
2
3
4
5
6
7
8
U˜ (+2,3)(μhad)
U˜ (−2,3)(μhad)
tations, although there are signs of slow or bad convergence
of the perturbative results to the non-perturbative ones.
Once the matrix-SSFs are known as continuum functions
of the renormalised coupling, we can obtain the RG-running
matrix U(μhad, 2nμhad) = σ (u1) . . . σ (un); cf. Eq. (3.21).
We check the reliability of our results by writing Eq. (2.20)
as
perturbative running at μhad are obtained from Eq. (4.3) and
for n = 8. They are:
U˜ (+2,3)(μhad) =
U˜ (+4,5)(μhad) =
(4.4)
0.5657(158)(2) 0.0224(11)(0) !
1.7245(4070)(627) 2.1317(679)(25) ,
W(2nμhad) U(μhad, 2nμhad) −1 .
for the operator bases {Q2+, Q3+},{Q4+, Q5+} and
(4.3)
U˜ (−2,3)(μhad)
U˜ (μhad) = U˜ (2nμhad) U(μhad, 2nμhad) −1
γ(0)
g 2(2nμhad) − 2b0
= 4π
The matrix U˜ (μhad) does not depend on the higher-energy
scale 2nμhad, so the n-dependence on the rhs should in
principle cancel out. We check this by computing the
second line for varying n, using our non-perturbative result for
U(μhad, 2nμhad) and the perturbative one for U˜ (2nμhad). As
explained in the comments following Eq. (3.22), the latter
is obtained as the NLO-2/3PT W(2nμhad), multiplied by
g 2(2nμhad)/(4π ) −(γ (0)/2b0). The scale μhad is held fixed
through g 2(μhad) = 4.61, which defines Lmax; see the
Nf = 2 running coupling computation of Ref. [
33
] for
details. The higher-energy scale 2nμhad is varied over a range
of values n = 0, . . . , 8; for each of these U˜ (μhad) is
computed. Our results are shown in Tables 3 and 4. As expected,
2nμhad-independence sets in with increasing n.
More specifically, taking log( SF/μhad) = − 1.298(58)
from Ref. [
33
] and r0 SF = 0.30(3) from Ref. [
29
] with
r0 = 0.50 fm, we obtain the hadronic matching energy
scale μhad ≈ 432(50) MeV. Our final results for the
non(4.6)
=
=
U˜ (−4,5)(μhad)
0.4297(195)(5) − 0.03145(88)(1)!
, (4.7)
− 1.6825(2182)(387) 0.8976(176)(29)
for {Q2−, Q3−},{Q4−, Q5−}. The first error refers to the
statistical uncertainty, while the second is the systematic one
due to the use of NLO-2/3PT at the higher scale 2nμhad. We
estimate the systematic error as the difference between the
final result, obtained with perturbation theory setting in at
scale 28μhad, and the one where perturbation theory sets in
at 27μhad (cf. Tables 3,4).
We note that systematic errors are almost negligible
compared to statistical ones, the latter being the result of error
propagation in the product of matrix-SSFs from μhad to
28μhad. This however does not tell us much about the
accuracy of NLO-2/3PT around the scale μpt = 2nμhad. We
investigate this issue in Appendix A, where we compare
Table 4 The matrix
U˜ (±4,5)(μhad), corresponding to
the operator bases {Q4±, Q5±}. It
is computed for a fixed
low-energy scale μhad and
varying higher-scales 2nμhad.
For sufficiently large n, the
results should not depend on the
higher-energy scale
n
0
1
2
3
4
5
6
7
8
U˜ (+4,5)(μhad)
σ (un), calculated in NLO-2/3PT and non-perturbatively. For
several matrix elements of σ (un) we see that NLO-2/3PT is
not precise enough, even at the largest scale we can reach
(corresponding to n = 8).
We now play the inverse game, keeping fixed μpt =
28μhad and calculating
U˜ (μ) =
for decreasing μ. The results for U˜ (μ) are shown in Figs. 8
and 9. They are the first non-perturbative computation of the
RG-evolution of operators which mix under renormalisation
in the continuum. We stress that these results are scheme
dependent. Note that the computation thus described enforces
the coincidence of our most perturbative point to the
perturbative prediction, which we assume to describe accurately
the running from μpt ∼ O(MW) to infinity. The
discrepancies between perturbation theory and our results are evident
at ever decreasing scales μ. These discrepancies are
sometimes dramatic; e.g. [U˜ −]55(μ). This is related to the
discussion of Figs. 6 and 7 above, concerning disagreements
between non-perturbative and NLO behaviour of several σ
matrix elements. Since U(μ, μpt) −1 in Eq. (4.8) is a
product of several σ matrices, these disagreements accumulate,
becoming very sizeable as μ/ SF decreases (Figs. 8, 9).
Finally, we compare the perturbative (NLO-2/3PT) to the
non-perturbative RG evolution U(μ, μ∗) between scales μ
and μ∗, where μ∗ = 3.46 GeV is kept fixed and μ is varied in
the range [0.43 GeV, 110 GeV]. The comparison is described
Appendix A and confirms the unreliability of the perturbative
computation of the RG running at scales of about 3 GeV.
4.3 Matching to hadronic observables with
non-perturbatively O(a) improved Wilson fermions
Having computed the non-perturbative evolution matrices
U˜ (μhad) as in Eq. (4.8), which provide the RG-running at
the low energy scale μhad, we proceed to establish the
connection between bare lattice operators and their RGI
counterparts. Starting from the definition of Eq. (2.26), we write
the RGI operator as
Qˆ ≡
=
Qˆ is independent of any renormalisation scheme or scale;
of course it is also independent of the regularisation. It is a
product of several quantities:
γ (0)
• The factors [g 2(μpt)/(4π )]− 2b0 and W(μpt) depend on
a high-energy scale μpt and are calculated in NLO
perturbation theory. This was one of the main objectives of
Ref. [
2
].
Fig. 8 Non-perturbative
running U˜ (+2,3)(μ) for the
operator basis {Q2+, Q3+} (top)
and U˜ (+4,5)(μ) of the operator
basis {Q4+, Q5+} (bottom).
Results are compared to the
perturbative predictions,
obtained by numerically
integrating Eq. (2.23), with the
NLO result for γ and the NNLO
one for β, in the SF scheme
1.32
1.3
2/3 PT
NP
101
101
Fig. 9 Non-perturbative
running U˜ (−2,3)(μ) of the
operator basis {Q2−, Q3−} (top)
and U˜ (−4,5)(μ) of the operator
basis {Q4−, Q5−} (bottom).
Results are compared to the
perturbative predictions,
obtained by numerically
integrating Eq. (2.23), with the
NLO result for γ and the NNLO
one for β, in the SF scheme
1.1100
101
5.20
Although the last item in the above list is beyond the
scope of this paper, we have computed Z (g02, aμhad )
following [
33
], at three values of the lattice spacing, namely
β = 6/g02 = {5.20, 5.29, 5.40}, which are in the range
commonly used for simulations of Nf = 2 QCD in
physically large volumes. The results are listed in Tables 5, 6.
In order to interpolate to the target renormalized coupling
u(μhad) = 4.61, the data can be fitted with a polynomial. Our
numerical studies reveal that additional values of β would be
needed to improve the quality of the interpolation to the target
value of the coupling.
5 Conclusions
In the present work we have studied the non-perturbative
RGrunning of the parity-odd, dimension-six, four-fermion
operators Q2±, . . . , Q5±, defined in Eqs. (2.1) and (2.2). Assigning
physical flavours to the generic fermion fields ψ1, . . . , ψ4,
the above operators describe four-quark effective
interactions for various physical processes at low energies. Under
renormalisation, these operators mix in pairs, as discussed
in Sect. 2. This mixing is not an artefact of the eventual loss
of symmetry due to the (lattice) regularisation; rather it is
a general property of operators belonging to the same
representations of their symmetry groups. It follows that also
the RG-running of each operator is governed by two
anomalous dimensions, and the corresponding RG-equations are
imposed on 2 × 2 evolution matrices. This makes the
problem of RG-running more complicated than the cases of
multiplicatively renormalised quantities, such as the quark masses
or BK.
The innovation of the present work is that, using
longestablished finite-size scaling techniques and the Schrödinger
Functional renormalisation conditions described in Sect. 3,
we have computed the non-perturbative evolution matrices
of these operators between widely varying low- and
highenergy scales μhad ∼ O( QCD) and μpt ∼ O(MW ) for
QCD with two dynamical flavours. Our results are shown
in Figs. 8 and 9 and Eqs. (4.6) and (4.7). The accuracy of
our results for the diagonal matrix elements ranges from 3 to
5%. The accuracy on the determination of the non-diagonal
matrix elements ranges from as high as 3% to as poor as 60%.
Clearly there is room for improvement. In our next project
concerning the renormalisation and RG-running of the same
operators for QCD with three dynamical flavours, we plan to
introduce several ameliorations, which ought to improve the
precision of our results significantly.
Perturbation theory is to be used for the RG-running
for scales above μpt ∼ O(MW ). In our SF scheme the
perturbative results at our disposal are NNLO (3-loops)
for the Callan–Symanzik β-function and NLO (2-loops)
for the four-fermion operator anomalous dimensions. In
Figs. 8 and 9 we see the presence of possibly relevant
nonperturbative effects already at scales of about 3 GeV, where
it is often assumed that beyond-LO perturbation theory
converges well.11 We have also performed some checks by
computing the RG-evolution matrix from a generic scale to a scale
of about 3 GeV and found some matrix elements where the
NLO perturbative result significantly differs from the
nonperturbative one (see Appendix A). This should serve as a
warning for other non-perturbative approaches which assume
that perturbation theory is convergent at such scales.
Finally, at a fixed hadronic scale and for three values of the
bare gauge coupling, we have computed the renormalisation
constants (again in 2 × 2 matrix form) of our four-fermion
operators.
As a closing remark we wish to point out that the
non-perturbative evolution matrices computed in this work
describe not only the RG-running of the parity-odd operators
Q2±, . . . , Q5±, but also that of their parity-even counterparts
Q2±, . . . , Q±. This is because evolution matrices are
contin5
uum quantities: in the continuum, each parity-odd operator
combines with its parity-even counterpart to form an
operator which transforms in a given chiral representation, both
parts having consequently the same anomalous dimension
matrices.
In the case, for instance, of S = 2 transitions, we are
dealing with operator matrix elements between two neutral
K -meson states and therefore only the parity-even operators
(Q1+ in the SM and Q2+,...,5 for BSM) contribute. Our results
for the continuum RG-evolution, obtained for the parity-odd
basis, can be used in this case. The renormalization of the
bare operators, however, depends on the details of the
lattice action. If the lattice regularisation respects chiral
symmetry (e.g. lattice QCD with Ginsparg–Wilson fermions),
then the parity-even and parity-odd parts of a given basis
of chiral operators renormalise with the same
renormalisation constants. Consequently they also have the same
matrixSSFs and evolution matrices. All results obtained for the
parity-odd operators Q2±, . . . , Q5± are then also valid for the
Q2±, . . . , Q±, without further ado.
5
Things are somewhat more complicated if the
regularisation breaks chiral symmetry (e.g. lattice QCD with Wilson
fermions). Then parity-even and parity-odd operators again
have the same anomalous dimensions, as these are
continuum quantities, but the “ratio” of their renormalisation
matrices {Z−1 Z} is a finite (scale-independent) matrix which is
a function of the bare gauge coupling; it becomes the unit
matrix in the continuum limit. This “ratio” is fixed by
lattice Ward identities, as discussed for example in Ref. [
7
].
So the subtlety here is that once the renormalisation
condition has been fixed for say, the parity-odd operator bases
at a value g02 of the squared gauge coupling, the
condition for the parity-even counterparts is also fixed through
{Z−1 Z}. Consequently, renormalisation matrix “ratios” like
Z $g02, 2aL % Z $g02, La % −1 are equal to their parity-even
counterparts Z $g02, 2aL % Z $g02, La % −1. Thus matrix-SSFs
(g02, a/L) and evolution matrices are the same for
parityodd and parity-even cases; cf. Eq. (3.19). But if we wish to use
the evolution matrices of the present work also for the
parityeven operators, we must ensure that these are renormalised in
the “same” SF scheme employed for their parity-odd
counterparts. This is ensured by writing the RGI parity-even operator
column (in analogy to Eq. (4.9)) as:
11 We have checked that other SF schemes, with different choices of α
and (s1, s2) (see Sect. 3.1), display similar overall behaviour.
Qˆ ≡
4π
Z−1 Z
(5.1)
where Qsub ≡ (1 + ) Q is the “subtracted” bare operator,
as suggested by Eq. (2.3). The term in square brackets of
the last expression is the renormalised parity-even operator
Z Q. It is computed however in a way that ensures that the
bare operator Q(g02) is renormalised in our SF scheme: the
SF renormalisation parameter Z(g02, aμhad) (which removes
the logarithmic divergences) is multiplied by the
schemeindependent, scale-independent “ratio” {Z−1 Z}.
Clearly, the procedure sketched above for the
renormalisation of parity-even operators is fairly cumbersome. It is also
prone to enhanced statistical uncertainties, as it involves
subtracted operators Qsub with non-zero . Fortunately, there
is a way to circumvent the problem: it is well known that,
using chiral (axial) transformations of the quark fields, we
can obtain continuum correlation functions of specific
parityeven composite operators in terms of bare correlation
functions of parity-odd operators of the same chiral multiplet,
regularised with twisted-mass (tmQCD) Wilson fermions [
37
].
The prototype example is the one expressing renormalized
correlation functions of the axial current in terms of bare
twisted-mass Wilson-fermion correlation functions of the
properly renormalised vector current. The situation is more
complicated with four-fermion operators: in Ref. [
8
] it was
shown that such chiral rotations do indeed relate parity-even
to parity-odd 4-fermion operators, but the resulting tmQCD
Wilson-fermion determinant is not real, and thus unsuitable
for numerical simulations. This problem is circumvented by
working with a lattice theory with sea- and valence-quarks
regularised with different lattice actions [
9
]. The valence
action is the so-called Osterwalder–Seiler [
38
] variety of
tmQCD, with valence twisted-mass fermion fields suitably
chosen so as to enable the mapping of correlation functions
involving parity-even operators {Qk } to those of the
parityodd basis {Qk }. The sea-quark action may be any tenable
lattice fermion action. While the price to pay is the loss of
unitarity at finite values of the lattice spacing, this is,
however, outweighed by the advantage of vanishing finite
subtractions (D = 0 in Eq. (2.3)). This approach has been put
to practice in Refs. [
14,15
]. Alternatively, the problems
arising from chiral symmetry breaking by the regularisation can
be avoided altogether by using domain wall fermions, as in
Refs. [
19–21
].
Acknowledgements The present work is an extension of previous
efforts dedicated to the SF renormalisation and running of the
BKparameter in the Standard Model. We are indebted to our collaborators
at the time, namely M. Guagnelli, J. Heitger, F. Palombi, and S. Sint,
for their early contributions. We owe a lot to their participation in the
defining phase of this project. G.H., C.P. and D.P. acknowledge support
by the Spanish MINECO grant FPA2015-68541-P (MINECO/FEDER),
the MINECO’s Centro de Excelencia Severo Ochoa Programme under
grant SEV-2016-0597 and the Ramón y Cajal Programme
RYC-201210819. M.P. acknowledges partial support by the MIUR-PRIN grant
2010YJ2NYW and by the INFN SUMA project.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Non-perturbative vs perturbative behaviour of the RG evolution
In analogy to Appendix C of Ref. [
21
] we construct the
quantity:
D(n) ≡ [U˜ (2nμhad)U(μhad, 2nμhad)−1]
Once again U˜ (2nμhad), U˜ (2n+1μhad), and
U(2nμhad, 2n+1μhad) are perturbative quantities known in
NLO-2/3PT , while σ (un+1) is a single non-perturbative
matrix-SSF. In the last line of Eq. (A.1), the product
[U(2nμhad, 2n+1μhad)−1σ (un +1)] is the ratio of the
nonperturbative over the perturbative RG evolution between
scales 2nμhad and 2n+1μhad. If perturbation theory were
reliable at these high scales, D(n) would vanish at large n. The
results for the D(n) matrix elements are shown in Figs. 10,
11. At the largest n values some of them are compatible with
0 while others are not. The latter case signals that due to large
anomalous dimensions, NLO-2/3PT performs poorly even at
scales as high as 2nμhad and 2n+1μhad (Tables 7, 8).
Moreover, in Appendix C of Ref. [
21
] the non-perturbative
RG evolution U(μ, μ∗) between scales μ and μ∗ has been
compared to the result from NLO-2/3PT. In Ref. [
21
], μ
is kept fixed to 2 GeV while μ∗ is varied in the range [2
GeV, 3 GeV]. We perform a similar study by fixing the
refFig. 10
n
The quantity D(n) Eq. (A.1) for the operator bases {Q2+, Q3 } (top) and {Q4+, Q5 } (bottom)
+ +
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
n
Page 26 of 40
Fig. 11 The quantity D(n) Eq.
(A.1) for the operator bases
{Q2− , Q3 } (top) and {Q4− , Q5 }
− −
(bottom)
0
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
n
n
n
n
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
1
2
3
4
5
6
7
0
10-3
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
R
lttiacem κrc .1031532 .1031305 .1031069 .1032509 .1032291 .1031975 .1033705 .1033497 .1033063 .1035260 .1034891 .1034432 .1036110 .1035767 .1035227 .1036665 .1036608 .1036139
itilsaaonm 26g/β0= 005900. 437910. 7500551. 005800. 227830. 399866. 245700. 027760. 520899. 806650. 128670. 390700. 331600. 223690. 136664. 126550. 908570. 811616.
7
leb L() 793
a 2 .9
T g¯ 0
Z
2 5 9 9 1 5 5 7 3 0 1 2 0 7 7 5 8 9
3 0 6 0 9 7 0 9 6 6 9 3 1 6 2 6 0 3
5 3 0 5 2 9 7 4 0 2 8 4 1 7 2 6 6 1
1 1 1 2 2 1 3 3 3 5 4 4 6 5 5 6 6 6
rc .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13
κ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
d β
26g/0 0000 4310 7555 0000 2230 3966 2400 0260 5299 8050 1270 3900 3300 2290 1364 1250 9070 8116
= 5. 7. 0. 5. 7. 9. 5. 7. 0. 6. 8. 0. 1. 3. 6. 6. 8. 1.
9 9 0 8 8 8 7 7 8 6 6 7 6 6 6 5 5 6
1
e
u
n
i
t
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c
7
leb L)( 397
a 2 .9
T g¯ 0
4
1
8
1
.
1
41( 7) 35( 9)
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)
1− !678051() 27851() !863081() 87166() !060081() 36135() !896081() 91105() !199092() 45274() !976022() 18371() !992191() 41418() !822194() 83587() !041115() 24445() !888132() 22530() !528561() 63898() !005951() 935530() !421442() 35434() !789312() 29329() !575691() 33303() 12530046.() 7536110.() !448023() 74219() !2157)( 0754)(
00. 10. 2 1
lttiacem κrc .0113532 .0113305 .0113069 .0123509 .1032291 .1039175 .1037305 .1034397 .1030363 .1032560 .1038491 .1034432 .3161100 .3156770 .3152270 .3166650 .3160680 .3163190
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8
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l L 9
b ( 7
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3( 4(
3000. 5990. 2000. 6990. 1000. 2001. 3000. 6990. 3000. 1001. 2000. 1001. 4000. 5990. 0300. 0201. 2000. 7990. 0400. 9290. 2000. 0999. 3000. 3990. 0400. 9090. 0400. 7990. 0300. 8390. 0100. 7390. 0000. 5890. 500 969
− −
00. 0.
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L
! !
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26g/0= 000059. 104739. 557050. 000058. 230278. 369698. 204057. 062077. 592908. 850066. 172086. 309007. 303016. 292036. 163466. 152065. 970085. 618161.
1
Fig. 12 Non-perturbative
evolution factor U+(μ, μ∗) =
[U˜ +(μ)]−1U˜ +(μ∗), where
μ∗ = 3.46 GeV, for the operator
bases {Q2+, Q3+} (top) and for
{Q4+, Q5+} (bottom). Results are
compared to the perturbative
prediction, obtained by
numerically integrating Eq.
(2.23), with γ (at NLO) and β
(at NNLO) in the SF scheme
1.05
1
2/3 PT
NP
2/3 PT
NP
2/3 PT
NP
0.6
10 2
10 4
[MeV]
10 4
[MeV]
10 4
[MeV]
2/3 PT
NP
2/3 PT
NP
2/3 PT
NP
Page 32 of 40
Fig. 13 Non-perturbative
evolution factor U−(μ, μ∗) =
[U˜ −(μ)]−1U˜ −(μ∗), where
μ∗ = 3.46 GeV, for the operator
bases {Q2−, Q3−} (top) and for
{Q4−, Q5−} (bottom). Results are
compared to the perturbative
prediction, obtained by
numerically integrating Eq.
(2.23), with γ (at NLO) and β
(at NNLO) in the SF scheme
10 2
10 2
10 4
2/3 PT
NP
2/3 PT
NP
2/3 PT
NP
2/3 PT
NP
0.8
0.6
0.9
10 2
2/3 PT
NP
2/3 PT
NP
2/3 PT
NP
[δk ](2+3;csw=0)
[δk ](4+5;csw=0)
[δk ](2−3;csw=0)
erence scale μ∗ = 3.46 GeV = 23μhad, corresponding to
the squared coupling u3. This is the scale closest to the
interval [2 GeV, 3 GeV] of Ref. [
21
], for which we have directly
computed the matrix-SSFs non-perturbatively. The scale μ
is varied in the range [0.43 GeV, 110 GeV]. We compute
U(μ, μ∗) in the following way:
U(μ, μ∗) = [U˜ (μ)]−1U˜ (μ∗)
= U(μ, μpt) [W(μpt)]−1
4π
W(μpt) U(μ∗, μpt) −1
= U(μ, μpt)U(μ∗, μpt)−1.
U(μ, μ∗) can be evaluated in a purely non-perturbative way
for integer n1 = log2(μpt/μ) and n2 = log2(μpt/μ∗). The
results are presented Figs. 12, 13. Relevant non-perturbative
effects are clearly visible for the elements (2, 3), (3, 2), (4, 4)
and (4, 5) of the operators {Q2+, Q3+} and {Q4+, Q5 } while
+
much larger discrepancies can be seen for the elements (2, 2),
(3, 2), (4, 5), (5, 4) and (5, 5) of the operators {Q2−, Q3−} and
{Q4−, Q5−}. Given the large deviation from the NLO-2/3PT
running already seen in Fig. 9, these results are not
surprising and simply confirm the non-reliability of the NLO-2/3PT
computation of the RG running at scales around 3 GeV.
Notice that the scale interval where this comparison has been
performed in Ref. [
21
] is completely contained in our plots
between the third and the fourth point which correspond to
scales of 1.73 GeV and 3.46 GeV. We remind the reader that a
direct comparison between our results and those of Ref. [
21
]
is meaningless, the crucial differences being, among many
others, the renormalisation scheme and the Nf -value.
Appendix B: One-loop cutoff effects in the step scaling function
In Table 9 we gather numerical values for δk (L/a), defined in
Eq. (3.29). We have calculated this quantity for a fermionic
action with (csw = 1) and without (csw = 0) a Clover term.
These results are also displayed in Figs. 14, 15, 16, and 17
(the target scheme α = 3/2, (s1, s2) = (3, 5) is plotted
with a blue triangle). Notice that the element (3, 2) of δk is
independent from α due to an accidental cancellation. This
is why all data-points in the corresponding figures are not in
colour. As expected, the Clover term has an important effect
on the discretisation errors, which are significantly reduced
when csw = 1. The observed O(ag02) discretisation effects
in Figs. 14 and 15 are only due to the unimproved operators,
the action being tree-level improved.
Fig. 17 Matrix elements of
δk (L/a) with csw = 0 for the
operator bases {Q2−, Q3−} (top)
and {Q4−, Q5−} (bottom).
Different colours distinguish the
various choices of α and
different symbols the various
choices of (s1, s2)
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