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arXiv:1111.3596v1 [hep-lat] 15 Nov 2011

η and η′meson masses from Nf= 2+1+1 twisted

mass lattice QCD

Konstantin Ottnad∗, Carsten Urbach

Helmholtz Institut für Strahlen und Kernphysik and Bethe Center for Theoretical Physics,

Universität Bonn, Nussallee 14-16, 53115 Bonn, Germany

E-mail: ottnad,urbach@hiskp.uni-bonn.de

Chris Michael

Theoretical Physics Division, Department of Mathematical Sciences

The University of Liverpool, Liverpool, L69 3BX, UK

E-mail: c.michael@liv.ac.uk

Siebren Reker

Centre for Theoretical Physics

University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands

E-mail: s.f.reker@rug.nl

for the European Twisted Mass collaboration

We determine mass and flavour content of η and η′states using Nf= 2+1+1 Wilson twisted

mass lattice QCD. We describe how those flavour singlet states need to be treated in this lattice

formulation. Results are presented for two values of the lattice spacing, a ≈ 0.08 fm and a ≈

0.09 fm, with a range of light quark masses correspondingto values of the pion mass from 270 to

500 MeV and fixed bare strange and charm quark mass values.

The XXIX International Symposium on Lattice Field Theory, Lattice2011

July 11-16, 2011

Lake Tahoe, USA

∗Speaker.

c ? Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence.

http://pos.sissa.it/

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η and η′masses from Nf= 2+1+1 tmLQCD

Konstantin Ottnad

ensemble

A40.24

A60.24

A80.24

A80.24s

B25.32

B35.32

B85.24

β

aµℓ

0.0040

0.0060

0.0080

0.0080

0.0025

0.0035

0.0085

aµσ

0.150

0.150

0.150

0.150

0.135

0.135

0.135

aµδ

0.190

0.190

0.190

0.197

0.170

0.170

0.170

L/a

24

24

24

24

32

32

24

1.90

1.90

1.90

1.90

1.95

1.95

1.95

Table 1: Theensembles used in this investigation. The notationof ref. [3] is used forlabelingthe ensembles.

1. Introduction

From experiment it is known that the masses of the nine light pseudo-scalar mesons show an

interesting pattern. Taking the quark model point of view, the three lightest mesons, the pions,

contain only the two lightest quark flavours, the up- and down-quarks. The pion triplet has a mass

of Mπ≈140 MeV. For the other six, the strange quark contributes also, and hence they are heavier.

In contrast to what one might expect five of them, the four kaons and the η meson, have roughly

equal mass around 500 to 600 MeV, while the last one, the η′meson, is much heavier, with mass

of about 1 GeV. On the QCD level, the reason for this pattern is thought to be the breaking of

the UA(1) symmetry by quantum effects. The η′meson is, even in a world with three massless

quarks, not a Goldstone boson. In this proceeding contribution, we discuss the determination of η

and η′meson masses using twisted mass lattice QCD (tmLQCD) with Nf= 2+1+1 dynamical

quark flavours. This will not only allow a study of the dependence of the η,η′masses on the light

quark mass value, but also an investigation of the charm quark contribution to both of these states.

Moreover, the ηcmeson mass can be studied in principle. For recent lattice studies in Nf= 2+1

flavour QCD see [1, 2].

2. Lattice Action

Weusegauge configurations asproduced by theEuropean TwistedMass Collaboration (ETMC)

with Nf= 2+1+1 flavours of Wilson twisted mass quarks and Iwasaki gauge action [3, 4]. The

details are described in ref. [3] and the ensembles used in this investigation are summarised in

table 1. The twisted mass Dirac operator in the light – i.e. up/down – sector reads [5]

Dℓ= DW+m0+iµℓγ5τ3

(2.1)

and in the strange/charm sector [6]

Dh= DW+m0+iµσγ5τ1+µδτ3,

(2.2)

where DWis the Wilson Dirac operator. The value of m0was tuned to its critical value as discussed

in refs. [7, 3] in order to realise automatic O(a) improvement at maximal twist [8]. Note that the

bare twisted masses µσ,δare related to the bare strange and charm quark masses via the relation

mc,s= µσ ± (ZP/ZS) µδ

(2.3)

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Konstantin Ottnad

with pseudo-scalar and scalar renormalisation constants ZPand ZS. Quark fields in the twisted basis

are denoted by χℓ,hand in the physical basis by ψℓ,h. They are related via the axial rotations

χℓ= eiπγ5τ3/4ψℓ,

¯ χℓ= ¯ ψℓeiπγ5τ3/4,

χh= eiπγ5τ1/4ψh,

¯ χh= ¯ ψheiπγ5τ1/4.

(2.4)

With automatic O(a) improvement being the biggest advantage of tmLQCD at maximal twist,

the downside is that flavour symmetry is broken at finite values of the lattice spacing. This was

shown to affect mainly the mass value of the neutral pion mass [9, 10, 11], however, in the case of

Nf=2+1+1 dynamical quarks, it implies the complication of mixing between strange and charm

quarks.

3. Flavour Singlet Pseudo-Scalar Mesons in Nf= 2+1+1 tmLQCD

In order to compute masses of pseudo-scalar flavour singlet mesons we have to include light,

strange and charm contributions to build the appropriate correlation functions. In the light sector,

one appropriate operator is given by [12]

1

√2(ψuiγ5ψu+ψdiγ5ψd)

→

1

√2(¯ χuχu− ¯ χdχd) ≡ ℓ.

(3.1)

In the strange and charm sector, the corresponding operator reads

?

¯ ψc

¯ ψs

?

iγ51±τ3

2

?

ψc

ψs

?

→

?

¯ χc

¯ χs

?

−τ1±iγ5τ3

2

?

χc

χs

?

.

(3.2)

In practice we need to compute correlation functions of the following interpolating operators

Pss ≡ ( ¯ ψsiγ5ψs) = (¯ χciγ5χc− ¯ χsiγ5χs)/2−ZS

Pcc ≡ ( ¯ ψciγ5ψc) = (¯ χsiγ5χs− ¯ χciγ5χc)/2−ZS

ZP(¯ χsχc+ ¯ χcχs)/2,

ZP(¯ χsχc+ ¯ χcχs)/2.

(3.3)

Note that the sum of pseudo-scalar and scalar contributions appears with the ratio of renormal-

isation factors Z ≡ ZS/ZP, which needs to be taken into account properly. Z has not yet been

determined for all values of β non-perturbatively.

However, for the mass determination, we can avoid this complication by changing the basis

and compute the real and positive definite correlation matrix

C =

ηℓℓ ηℓPhηℓSh

ηPhℓηPhPhηPhSh

ηShℓηShPhηShSh

,

(3.4)

with the notation

Ph≡ (¯ χciγ5χc− ¯ χsiγ5χs)/2,

Sh≡ (¯ χsχc+ ¯ χcχs)/2 (3.5)

and ηXYdenoting the corresponding correlation function. Masses can determined by solving the

generalised eigenvalue problem [13, 14]

C(t) η(n)(t,t0) = λ(n)(t,t0) C(t0) η(n)(t,t0).

(3.6)

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Konstantin Ottnad

Taking into account the periodic boundary conditions for a meson, we can determine the effective

masses by solving

λ(n)(t,t0)

λ(n)(t +1,t0)=

e−m(n)(t+1)+e−m(n)(T−(t+1))

for m(n), where n counts the eigenvalues. The state with the lowest mass should correspond to the

η and the second state to the η′meson.

From the components η(n)

tents c(n)

e−m(n)t+e−m(n)(T−t)

(3.7)

0,1,2of the eigenvectors, we can reconstruct the physical flavour con-

ℓ,s,cfrom

c(n)

ℓ

=

1

N(n)(η(n)

1

N(n)(−Zη(n)

1

N(n)(−Zη(n)

0)

c(n)

s

=

1+η(n)

2)/√2

c(n)

c =

1−η(n)

2)/√2

(3.8)

with normalisation

N(n)=

?

(η(n)

0)2+(Zη(n)

1)2+(η(n)

2)2.

At this point the ratio Z ≡ ZS/ZPis needed again. Assuming for a moment that charm does not

contribute significantly to the η and η′states, one can extract the η-η′mixing angle φ from

cos(φ) = c(0)

ℓ

≈ c(1)

s ,

sin(φ) = −c(0)

s

= c(1)

ℓ

(3.9)

with(0)((1)) denoting the η (η′) state.

4. Results

We have computed all contractions needed for building the correlation matrix of eq. (3.4).

For the connected contributions, we used stochastic time-slice sources (the so called “one-end-

trick” [15]). For the disconnected contributions, we used stochastic volume sources with complex

Gaussian noise [15]. As discussed in ref. [12] one can estimate the light disconnected contributions

very efficiently using the identity

D−1

u−D−1

d= −2iµℓD−1

d

γ5D−1

u

.

For the heavy sector such a simple relation does not exist, but we can use the so called hopping

parameter variance reduction, which relies on the same equality as in the mass degenerate two

flavour case (see ref. [15] and references therein)

D−1

h= B−BHB+B(HB)2−B(HB)3+D−1

h(HB)4

with Dh=(1+HB)A,B=1/A and H the twoflavour hopping matrix. Weuse 24 stochastic volume

sources per gauge configuration in both the heavy and the light sector.

We use both local and fuzzed sources to enlarge our correlation matrix by a factor two. In

addition to the interpolating operator quoted in eqs. (3.1) and (3.2), we also plan to consider the

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Konstantin Ottnad

“ηc”

η′η

t/a

aM

151050

2

1.5

1

0.5

0

(a)

charm content

strange content

light content

t/a

flavour content

151050

1

0.5

0

(b)

Figure 1: (a) Effective masses in lattice units determined from solving the generalised eigenvalue problem

with t0/a = 1 for ensemble B25.32. We show the results extracted from a 3×3 matrix. (b) squared flavour

content of η for B25.32.

γ-matrix combination iγ0γ5, which will increase the correlation matrix by another factor of two.

The number of gauge configurations investigated per ensemble is in most cases around 1200, and

for ensemble B25.32 is 1500. Statistical errors are computed using the bootstrap method with 1000

samples.

In figure 1 we show the effective masses determined from solving the generalised eigenvalue

problem for ensemble B25.32 from a 3×3 matrix with local operators only. We keptt0/a=1 fixed.

One observes that the ground state is very well determined and it can be extracted from a plateau fit.

The second state, i.e. the η′, is much more noisy and a mass determination is questionable, at least

from a 3×3 matrix. Enlarging the matrix size significantly reduces the contributions of excited

states to the lowest states and, due to smaller statistical errors at smaller t values, a determination

becomes possible. The third state appears to be in the region where one would expect the ηcmass

value, however, the signal is lost at t/a = 5 already, which makes a reliable determination not

feasible.

In figure 2 we show the masses of the η and η′mesons for the various ensembles we used

as a function of the squared pion mass. In addition we show the corresponding physical values.

The scale was set from fπand mπusing the results of ref. [3]. It is clear that the η meson mass

can be extracted with high precision, while the η′meson mass requires a larger correlation matrix,

which is work in progress. The comparison with the corresponding physical values seems to point

towards good agreement.

We also determine the flavour content of the two states as explained above. It turns out that

the η has a dominant strange quark content (see right panel of figure 1), while the η′is dominated

by light quarks. For both the charm contribution is rather small, however, for the η it turns out

to be significantly non-zero. A preliminary determination of the mixing angle eq. (3.9) yields a

very stable value of about 60◦. Note that this is the mixing angle to the flavour eigenstates. The

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