# K^{*} vector and tensor couplings from N_ {f}= 2 twisted mass QCD

**ABSTRACT** The mass mK* and vector coupling fK* of the K* meson, as well as the ratio of the tensor to vector couplings fT/fV|K*, are computed in lattice QCD. Our simulations are performed in a partially quenched setup, with two dynamical (sea) Wilson quark flavors, having a maximally twisted mass term. Valence quarks are either of the standard or the Osterwalder-Seiler maximally twisted variety. Results obtained at three values of the lattice spacing are extrapolated to the continuum, giving mK*=981(33) MeV, fK*=240(18) MeV, and fT(2 GeV)/fV|K*=0.704(41).

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**ABSTRACT:**We compute the vector meson decay constants fD*, fDs* from the simulation of twisted mass QCD on the lattice with Nf = 2 dynamical quarks. When combining their values with the pseudoscalar D(s)-meson decay constants, we were able (i) to show that the heavy quark spin symmetry breaking effects with the charm quark are large, fDs*/fDs = 1.26(3), and (ii) to check the factorization approximation in a few specific B-meson non-leptonic decay modes. Besides our main results, fD* = 278 \pm 13 \pm 10 MeV, and fDs* = 311 \pm 9 MeV, other phenomenologically interesting results of this paper are: fDs*/fD* = 1.16 \pm 0.02 \pm 0.06, fDs*/fD = 1.46 \pm 0.05 \pm 0.06, and fDs/fD* = 0.89 \pm 0.02 \pm 0.03. Finally, we correct the value for B(B0 \rightarrow D+ pi-) quoted by PDG, and find B(B0 \rightarrow D+ pi-) = (7.8 \pm 1.4) \times 10-7. Alternatively, by using the ratios discussed in this paper, we obtain B(B0 \rightarrow D+ pi-) = (8.3 \pm 1.0 \pm 0.8)\times10-7.Journal of High Energy Physics 01/2012; 2012(2). · 5.62 Impact Factor

Page 1

arXiv:1104.0188v1 [hep-lat] 1 Apr 2011

ROM2F/2011/04,

ICCUB-11-129, UB-ECM-PF-11-48

April 4, 2011

K∗vector and tensor couplings from Nf= 2 tmQCD

ETMC

P. Dimopoulosa, F. Mesciac, and A. Vladikasd

aDipartimento di Fisica, Universit` a di Roma “Tor Vergata”

Via della Ricerca Scientifica 1, I-00133 Rome, Italy

cDepartament dEstructura i Constituents de la Mat´ eria and Institut de Ci´ encies del Cosmos,

Universitat de Barcelona, Diagonal 647,

E-08028 Barcelona, Spain

dINFN, Sezione di “Tor Vergata”

c/o Dipartimento di Fisica, Universit` a di Roma “Tor Vergata”

Via della Ricerca Scientifica 1, I-00133 Rome, Italy

Abstract

The mass mK∗ and vector coupling fK∗ of the K∗-meson, as well as the ratio of the tensor

to vector couplingsfT

fV

in a partially quenched setup, with two dynamical (sea) Wilson quark flavours, having a

maximally twisted mass term. Valence quarks are either of the standard or the Osterwalder-

Seiler maximally twisted variety. Results obtained at three values of the lattice spacing are

extrapolated to the continuum, giving mK∗ = 981(33)MeV, fK∗ = 240(18)MeV and

fT(2GeV)

fV

???K∗, are computed in lattice QCD. Our simulations are performed

???K∗= 0.704(41).

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1 Basics

The aim of the present letter is to present novel lattice results for the mass of the K∗-

meson, as well as its vector and tensor couplings (fV and fT respectively), defined

in Euclidean space-time as follows:

?0|Vj|K∗;λ?

?0|T0j|K∗;λ?

=−ifVǫλ

−ifTǫλ

jmK∗ , (1.1)

=

jmK∗ .(1.2)

In the above expressions, Vj= ¯ sγjd is the vector current (spatial components only;

j = 1,2,3), T0j= i¯ sσ0jd is the tensor bilinear operator (temporal component), and

ǫλ

jdenotes the polarization vector.

Our results are based on simulations of the ETM Collaboration (ETMC) [1],

with Nf = 2 dynamical flavours (sea quarks) and “lightish” pseudoscalar meson

masses in the range 280MeV < mPS< 550MeV. With three lattice spacings (a =

0.065 fm, 0.085 fm and 0.1 fm) we are able to extrapolate our results to the continuum

limit. Our simulations are performed with the tree-level Symanzik improved gauge

action. For the quark fields we adopt a somewhat different regularization for sea

and valence quarks. The sea quark lattice action is the so-called maximally twisted

standard tmQCD (referred to as “standard tmQCD case”) [2]. The Nf= 2 light sea

quark flavours form a flavour doublet ¯ χ = (¯ u ,¯d) and the fermion lattice Lagrangian

in the so-called “twisted basis” is given by

Ltm = ¯ χ

?

DW + iµqγ5τ3?

χ , (1.3)

where τ3is the isospin Pauli matrix and DW denotes the critical Wilson-Dirac

operator. By “critical” we mean that, besides the standard kinetic and Wilson

terms, the operator also includes a standard, non-twisted mass term, tuned at the

critical value of the quark mass (κcrin the language of the hopping parameter), so

as to ensure maximal twist. With only two light dynamical flavours, strangeness

clearly enters the game in a partially quenched context. For the valence quarks we

use the so-called Osterwalder-Seiler variant of tmQCD, which consists in maximally

twisted flavours which, unlike the standard tmQCD case, are not combined into

isospin doublets:

LOS =

?

f=d,s

¯ qf

?

DW + iµfγ5

?

qf, (1.4)

with sign(µf) = ±1 (see below for details). This action, introduced in ref. [3] and

implicitly used in [4], has been studied in detail in ref. [5]. For the case in hand

(i.e. K∗-related quantities) we only need down- and strange-quark flavours in the

valence sector. Note that the choice of maximally twisted sea and valence quarks

implies O(a)-improvement of the physical quantities (i.e. the so-called automatic

1

Page 3

improvement of masses, correlation functions and matrix elements) [6]. Thus unitar-

ity violation, which plagues any partially quenched theory at finite lattice spacing,

is an O(a2) effect.

The sign of µsmay be that of µdor its opposite. We conventionally refer to the

setup in which sign(µd) = −sign(µs) as the “standard twisted mass regularization”

(denoted by tm) and the setup with sign(µd) = sign(µs) as the “Osterwalder-Seiler

regularization” (denoted as OS). Quenched pseudoscalar masses and decay constants

in tm- and OS-setups have already been studied [7,8].

The continuum operators of interest are expressed, in terms of their lattice

counterparts, as follows:

Vcont

µ

Tcont

µν

=ZAAtm

ZTTtm

µ

µν+ O(a2) = ZT˜TOS

+ O(a2) = ZVVOS

µ

µν + O(a2) ,

+ O(a2) ,(1.5)

=(1.6)

where˜Tµν= ǫµνρσTρσ. The vector and axial currents are normalized by the scale

independent factors ZV and ZA, while ZT ≡ ZT(µ) runs with a renormalization

scale µ (i.e. it is defined in a given renormalization scheme).

The vector boson mass, mV, as well as fV and fT, are obtained form two-point

correlation functions at zero spatial momenta and large time separations. These are

defined in the continuum (Euclidean space-time) as

Ccont

V

(x0)≡

1

3

?

j

f2

VmV

2

1

3

j

f2

TmV

2

?

d3x ?Vj(x) V†

j(0)?cont

→

exp[−mVT/2] cosh?mV(T/2 − x0)?

d3x ?T0j(x) T†

,(1.7)

Ccont

T

(x0)≡

?

?

0j(0)?cont

→

exp[−mVT/2] cosh?mV(T/2 − x0)?

. (1.8)

The asymptotic expressions of the above equations correspond to the large time

limit of the correlation functions (symmetrized in time), with periodic boundary

conditions for the gauge fields and (anti)periodic ones for the fermion fields in the

(time)space directions (i.e. 0 ≪ x0≪ T/2). These are actually the boundary

conditions of our lattice simulations. The lattice correlation functions are related to

the continuum ones as suggested by eqs. (1.5),(1.6):

Ccont

V

Ccont

T

(x0)

(x0)

=Z2

Z2

ACtm

TCtm

A(x0) + O(a2) = Z2

T(x0) + O(a2) = Z2

VCOS

TCOS

V(x0) + O(a2) ,

˜T(x0) + O(a2) .

(1.9)

=(1.10)

The meaning of the notation Ctm

The ratio fT/fV is computed from the square root of the ratio of correlations func-

tions Ccont

T

/Ccont

V

, in which many systematic effects cancel. We compute the vector

A, COS

˜T, etc. should be transparent to the reader.

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Page 4

meson mass and decay constant from Ccont

of correlation functions Ccont

multiplying fT/fV by fV.

Note that fV is a scale independent quantity, while fT(µ) depends on the renor-

malization scale µ, as well as the renormalization scheme. The scale and scheme

dependence of the latter quantity is carried by the renormalization factor ZT(µ); we

opt for the MS-scheme and for µ = 2 GeV.

V

and the ratio fT/fV from the ratio

. The tensor coupling fT is then obtained by

T

/Ccont

V

2 Results

ETMC has generated Nf= 2 configuration ensembles at four values of the inverse

gauge coupling; in this work we make use of only three of them. Light mesons consist

of a valence quark doublet, with twisted mass aµℓequal to that of the sea quarks;

aµℓ= aµsea. Heavy-light mesons consist of a valence quark pair (aµℓ= aµsea,aµh).

As already stated, these bare quark mass parameters are chosen so as to have light

pseudoscalar mesons (“pions”) in the range of 280 ≤ mPS≤ 550 MeV and heavy-

light pseudoscalar mesons (“Kaons”) in the range 450 ≤ mPS ≤ 650 MeV. The

simulation parameters are gathered in Table 1.

βa−4(L3× T)

243× 48

aµℓ = aµsea

0.0080, 0.0110

aµh

0.0165, 0.0200

0.0250

0.0150, 0.0220

0.0270

0.0150, 0.0220

0.0270

0.0150, 0.0220

0.0270

0.0120, 0.0150

0.0180

Nmeas

180 3.80

(a ∼ 0.1 fm)

3.90243× 48 0.0040 400

243× 480.0064, 0.0085,

0.0100

0.0030, 0.0040

200

3.90323× 64270/170

(a ∼ 0.085 fm)

4.05

(a ∼ 0.065 fm)

323× 64 0.0030, 0.0060,

0.0080

200

Table 1: Simulation details

Our calibrations are based on earlier collaboration results. The ratio r0/a,

known at each value of the gauge coupling β from ref. [9], allows to express our raw

dimensionless data (quark masses, meson masses and decay constants) in units of

r0. Knowledge of the renormalization constant ZP in the MS scheme at 2 GeV (see

ref. [10]) enables us to pass from bare quark masses to renormalized ones (again in r0

units). Using only data with light valence quarks in the tm-setup, we have applied

the procedure described in refs. [1,9] for the determination of the physical continuum

3

Page 5

light quark mass µMS

masses in the tm-setup [9], we determine the physical continuum strange quark mass

µMS

s (2GeV). These quark mass values are listed in Table 3. The Sommer scale we

use, based on an analysis with three values of the lattice spacing, is r0= 0.448(5) fm.

This updates our previous r0computation, derived with two β’s, cf. ref. [1].

We see from eqs. (1.9) and (1.10) that we need to know the renormalization

parameters ZV, ZA, and ZT. These quantities, as well as ZP, have been computed

in ref. [10], in the RI/MOM scheme; ZPand ZTare perturbatively converted to MS.

In the same work a ZV estimate, obtained from a Ward identity, is also provided.

In Table 2 we gather the most reliable estimates of ref. [10], which we have used in

the present analysis, as well as our estimates of the r0/a ratio.

u/d. From the data concerning light and heavy valence quark

βZV

ZA

ZMS

T(2 GeV)ZMS

P(2 GeV)r0/a

3.80

3.90

4.05

0.5816(02)

0.6103(03)

0.6451(03)

0.746(11)

0.746(06)

0.772(06)

0.733(09)

0.743(05)

0.777(06)

0.411(12)

0.437(07)

0.477(06)

4.54(07)

5.35(04)

6.71(04)

Table 2: The renormalization parameters used in our analysis and the r0/a values at

each gauge coupling. ZV is obtained from a lattice vector Ward identity, while the

other renormalization constants are obtained from the RI/MOM scheme; for details

see ref. [10].

µMS

u/d(2 GeV)µMS

s (2 GeV)

3.6(2) MeV95(6) MeV

Table 3: The quark mass values (in the MS scheme), used in our analysis; see ref. [9].

As can be seen in Table 1, at β = 3.90 we have performed more extensive

simulations, which enable us to check in some detail the quality and stability of the

measured physical quantities. We wish to highlight straightaway the two problems

we have encountered in these tests, performed for the tm-setup: (i) For all sea quark

4

Page 6

masses, when the valence quark attains its lightest value aµℓ= 0.0040, the vector

meson effective mass has a poor plateau. The situation already improves at the

next quark mass aµℓ= 0.0064. Nevertheless, since the signal-to-noise ratio behaves

as expected (i.e. it drops like exp[−(mV− mPS)x0]) the ρ-meson mass and decay

constant can still be extracted (see results presented in ref. [11]). (ii) A poor quality

vector meson effective mass is also seen when µℓ< µsea. This problem is absent in

the pseudoscalar channel.

The above problems are easily avoided in the present work, since the quantities

of interest are related to the K∗-meson, consisting of a down and a strange valence

quark mass (µu/d< µs). We thus proceed as follows: at each β value, we compute

the necessary observables (vector meson mass mV, vector decay constant fV, and

the ratio fT/fV), for all combinations of aµℓ= aµseaand aµh(with µℓ< µh). In

this way unitarity holds in the light quark sector, while the heavy valence quark

mass, in a partially quenched rationale, spans a range around the physical value µs.

Examples of the quality of our signal are given in Figs. 1 and 2; the lightest mass

is aµminand the heavy mass, corresponding to the physical strange value aµs, is

obtained by interpolation, as will be explained below.

= 4.05, = 0.0150

= 3.90, = 0.0220

β = 3.80, aµh= 0.0200

2x0/T

r0mtm

V

(ℓ,h)

0.80 0.70 0.600.500.40 0.300.200.10 0.00

1.80

1.60

1.40

1.20

1.00

0.80

0.60

0.40

0.20

(a)

= 4.05, = 0.0150

= 3.90, = 0.0220

β = 3.80, aµh= 0.0200

2x0/T

r0mOS

V

(ℓ,h)

0.80 0.700.60 0.500.40 0.300.20 0.100.00

1.80

1.60

1.40

1.20

1.00

0.80

0.60

0.40

0.20

(b)

Figure 1: Effective vector meson mass r0mV at three values of the lattice spacing.

The light quark mass is aµmin(see Table 1) and the heavy quark mass aµhis close

to that of the physical strange quark. (a) tm-setup ; (b) OS-setup. Plateau intervals

are indicated by straight lines.

Statistical errors are estimated with the bootstrap method, employing 1000

bootstrap samples. A reliable direct determination of the ratio fT/fV in the OS-

setup is not possible, because the ratio of correlation functions COS

display satisfactory plateaux, due to big statistical fluctuations of the tensor cor-

relator COS

˜T. We only present fT/fV results in the tm-setup, obtained from the

better-behaved correlation function Ctm

T. In Fig. 3 we show results for this ratio at

˜T/COS

V

do not

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Page 7

aµminand also at a heavier light quark mass.

= 4.05, = 0.0150

= 3.90, = 0.0220

β = 3.80, aµh= 0.0200

2x0/T

r0ftm

V

(ℓ,h)

0.80 0.700.600.500.40 0.300.20 0.10 0.00

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

(a)

= 4.05, = 0.0150

= 3.90,= 0.0220

β = 3.80, aµh= 0.0200

2x0/T

r0fOS

V

(ℓ,h)

0.80 0.700.60 0.500.40 0.300.200.10 0.00

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

(b)

Figure 2: Vector decay constant r0fV at three values of the lattice spacing. The

light quark mass is aµmin(see Table 1) and the heavy quark mass aµhis close to

that of the physical strange quark. (a) tm-setup ; (b) OS-setup. Plateaux intervals

are indicated by straight lines.

= 4.05, = 0.0150

= 3.90, = 0.0220

β = 3.80, aµh= 0.0200

2x0/T

[fT/fV]tm(ℓ,h)

0.80 0.700.600.50 0.400.30 0.20 0.100.00

1.20

1.10

1.00

0.90

0.80

0.70

0.60

0.50

0.40

(a)

= 4.05, = 0.0150

= 3.90, = 0.0220

β = 3.80, aµh= 0.0200

2x0/T

[fT/fV]tm(ℓ,h)

0.800.700.60 0.500.40 0.300.20 0.10 0.00

1.20

1.10

1.00

0.90

0.80

0.70

0.60

0.50

0.40

(b)

Figure 3: The ratio fT/fV in the tm-setup, at three values of the lattice spacing

and heavy quark mass aµh, close to that of the physical strange quark. (a) For

the lightest quark mass aµmin; (b) for the next-to-lightest quark mass. Plateaux

intervals are indicated by straight lines.

Regarding vector meson masses mV and couplings fV, both tm- and OS-results

display similar plateau quality and statistical accuracy. At finite lattice spacing and

for equal bare quark masses, tm- and OS-estimates of mV are compatible within

errors. Agreement is also very good for fV, with occasional discrepancies, interpreted

6

Page 8

as cutoff effects, showing up at the coarsest lattice1. Contrary to the well known

large O(a2) isospin breaking effects in the neutral to charged pion splitting mass,

no numerically large differences are observed between tm and OS results for fV and

mV. This fact is in agreement with theoretical expectations, see ref. [12].

β aµl

r0mtm

V(ℓ,s)r0mOS

V(ℓ,s)r0ftm

V(ℓ,s)r0fOS

V(ℓ,s)[fT/fV]tm(ℓ,s)

3.800.0080

0.0110

2.443(41)

2.508(32)

2.471(30)

2.500(23)

0.642(18)

0.651(14)

0.700(13)

0.706(15)

0.764(38)

0.792(35)

3.900.0040

0.0064

0.0085

0.0100

0.0030(L=32)

0.0040(L=32)

2.410(41)

2.441(32)

2.484(48)

2.468(54)

2.259(75)

2.364(32)

2.381(38)

2.427(35)

2.441(33)

2.481(32)

2.335(45)

2.371(50)

0.610(21)

0.626(22)

0.628(16)

0.619(20)

0.577(20)

0.599(22)

0.643(17)

0.659(12)

0.652(16)

0.657(16)

0.639(16)

0.640(21)

0.755(19)

0.726(20)

0.776(27)

0.774(31)

0.714(20)

0.722(19)

4.05 0.0030

0.0060

0.0080

2.305(86)

2.439(67)

2.512(65)

2.263(80)

2.295(76)

2.427(48)

0.568(49)

0.618(41)

0.649(31)

0.588(40)

0.578(46)

0.648(27)

0.742(27)

0.768(30)

0.741(31)

CLµu/d

2.227(71)2.200(60) 0.545(41) 0.525(30) 0.701(46)

expt. 2.025 0.493

Table 4: Results for three values of lattice spacing and several light quark masses

aµℓ, interpolated to the physical strange mass µs. Vector mass and vector decay

constant results are presented for both tm- and OS-setups . The ratio fT/fV results

are given only in the tm-setup. Our extrapolations at the µu/dphysical point and

in the continuum limit are also shown. In the last row the experimental results for

the vector mass and the vector decay constant, in units of r0, have been added.

The extrapolation to the physical quark masses is carried out in two steps.

First, for fixed values of the gauge coupling β and light quark mass aµℓ= aµsea,

we perform linear interpolations of r0mV, r0fV and fT/fV to the physical strange

quark mass µs. The second step consists in using these interpolated results for a

combined fit of our data at three lattice spacings and all available light quark masses,

in order to determine the continuum value of the quantity of interest (r0mV, r0fV

and fT/fV). The fitting function we use is

mVr0 = C0(µsr0) + C1(µsr0)µℓr0 + D(µsr0)a2

r2

0

, (2.1)

1Given the large fluctuations of fT/fV in the OS-setup at the finer lattice spacing, we only quote

results for this ratio in the tm-setup.

7

Page 9

and similarly for fVr0and fT/fV. The results of the interpolations in the heavy

quark mass µhto the physical value µs, at each β and aµℓ, are gathered in Table 4.

In the same Table we also display the results of the combined chiral and continuum

extrapolations. Note that for the three quantities of interest, mV,fV and fT/fV,

the value of χ2/d.o.f. is less than unity. The linear dependence of our data on the

light quark mass agrees with the predictions of chiral perturbation theory for the

ratio fT/fV in the K∗mass range; see refs. [13,14].

Our final results, extracted in the tm-setup, are

mK∗

fK∗

= 981(31)(10)[33]MeV ,(2.2)

= 240(18)(02)[18]MeV .(2.3)

The first error includes the statistical uncertainty and the systematic effects re-

lated to the simultaneous chiral and continuum fits, mass interpolations and ex-

trapolations, and uncertainties in the renormalization parameters. The second error

arises from that of r0. These two errors, combined in quadrature, give the to-

tal error in the square brackets. It is encouraging that these results agree with

the ones obtained in the OS-setup (which is a different regularization), namely

mK∗ = 969(27)(10)[29]MeV and fK∗ = 231(13)(02)[13]MeV. Compared to the ex-

perimentally known values, mK∗ = 892MeV and fK∗ = 217MeV, the vector meson

mass is 2-3 standard deviations off, while the decay constant is compatible within

about one standard deviation.

Our final estimate (tm-setup) for the ratio of vector meson couplings is

fT(2GeV)

fV

???K∗= 0.704(41) . (2.4)

This is compatible with the continuum limit quenched result [fT(2GeV)/fV]K∗ =

0.739(17)(3) of ref. [15]. We are also in agreement with the result of the RBC/UKQCD

collaboration [16]; using Nf= 2 + 1 dynamical fermions at a single lattice spacing,

they quote [fT(2GeV)/fV]K∗ = 0.712(12). The lattice results are also in agreement

with the sum rules’ estimate [fT(2GeV)/fV]K∗ = 0.73(4), quoted in [17].

Acknowledgements

We thank G.C. Rossi and C. Tarantino for having carefully read the manuscript

and for their useful comments and suggestions. We acknowledge fruitful collabo-

ration with all ETMC members. We have greatly benefited from discussions with

O. Cata, C. Michael, C. McNeile, S. Simula and N. Tantalo. F.M. acknowledges the

financial support from projects FPA2007-66665, 2009SGR502, Consolider CPAN,

and CSD2007-00042.

8

Page 10

expt. point

β = 4.05

β = 3.90

β = 3.80

r0ˆ µℓ

r0mtm

V(ℓ,s)

0.140.12 0.100.080.06 0.04 0.020.00

3.00

2.80

2.60

2.40

2.20

2.00

(a)

expt. point

β = 4.05

β = 3.90

β = 3.80

r0ˆ µℓ

r0mOS

V(ℓ,s)

0.14 0.120.10 0.080.060.04 0.020.00

3.00

2.80

2.60

2.40

2.20

2.00

(b)

Figure 4: r0mV plotted against the renormalized light quark mass r0ˆ µℓ; (a) tm-

setup; (b) OS-setup. The continuous lines are combined chiral and continuum ex-

trapolations to the physical point. The bottom (black) line corresponds to eq. (2.1)

at a = 0. The separation among the four lines in (a) is invisible to the naked eye

(i.e. small scaling violations).

expt. point

β = 4.05

β = 3.90

β = 3.80

r0ˆ µℓ

r0ftm

V(ℓ,s)

0.14 0.12 0.100.08 0.060.04 0.020.00

1.00

0.90

0.80

0.70

0.60

0.50

0.40

(a)

expt. point

β = 4.05

β = 3.90

β = 3.80

r0ˆ µℓ

r0fOS

V(ℓ,s)

0.14 0.120.100.08 0.060.040.02 0.00

1.00

0.90

0.80

0.70

0.60

0.50

0.40

(b)

Figure 5: r0fVplotted against the renormalized light quark mass r0ˆ µℓ; (a) tm-setup;

(b) OS-setup. The continuous lines are combined chiral and continuum extrapola-

tions to the physical point. The bottom (black) line corresponds to eq. (2.1) at

a = 0.

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β = 4.05

β = 3.90

β = 3.80

r0ˆ µℓ

[fT/fV]tm(ℓ,s)

0.140.12 0.10 0.080.06 0.040.020.00

0.95

0.90

0.85

0.80

0.75

0.70

0.65

0.60

0.55

Figure 6: fT/fV plotted against the renormalized light quark mass r0ˆ µℓin the tm-

setup. The continuous lines are combined chiral and continuum extrapolation to the

physical point. The bottom (black) line corresponds to eq. (2.1) at a = 0. The four

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