Aspects of quark mass generation on a torus

0.21 MB·English

by  C. S. Fischer

EPJ manuscript No.
(will be inserted by the editor)
Aspects of quark mass generation on a torus
C. S. Fischer1 and M. R. Pennington2 a
1 Institut fu¨r Kernphysik,Technical University of Darmstadt, Schlossgartenstraße 9, 64289 Darmstadt, Germany
7
2 IPPP, Durham University,Durham DH13LE, U.K.
0
0
2 Received: date/ Revised version: date

n
a Abstract. In this talk we report on recent results for the quark propagator on a compact manifold. The
J corresponding Dyson-Schwinger equations on a torus are solved on volumes similar to the ones used in
6 lattice calculations. The quark-gluon interaction is fixed such that the lattice results are reproduced. We
1 discussboththeeffectsintheinfinitevolume/continuumlimitaswellaseffectswhenthevolumeissmall.

1 PACS. 1 2.38.Aw, 12.38.Gc, 12.38.Lg, 14.65.Bt
v
3
2 1 Introduction
1
1 −1 −1 1
= -
0 InthelowenergysectorofQCDdynamicalchiralsymme- 2
7 try breaking is the dominant nonperturbative effect with
0 respect to hadron phenomenology. Thus any framework 1 1
- -
/ attempting to explainthe spectraanddecaypropertiesof 2 6
h
p light hadrons has to satisfy all constraints related to the
- correct pattern of chiral symmetry breaking in QCD. In
p thisrespectlatticesimulationshavetodealwithtwotypes 1
e ofproblems:onthe onehand,commutationrelationsbet- - 2 +
h
ween γ-matrices are affected by the finite lattice spacing.
:
v On the other hand, there is the finite volume. Since con- 1 1
Xi tinuous symmetries cannot be spontaneously broken at a − = − -
finite volume V, chiral symmetry is restored in the limit
r
a of zero current quark mass, m 0 (see e.g. [1]). Thus
one first has to perform both th→e continuum and infinite (cid:0)1 (cid:0)1
volume limit before one can investigate the chiral limit. = +
Inturn,studies withphysicalup-anddown-quarkmasses
necessarily require large volumes to avoid chiral restora- Fig.1.Dyson-Schwingerequationsforthegluon(curly),ghost
tion effects. (dashed) and quark (straight lines) propagators. Filled circles
denote dressed propagators and empty circles denote dressed
Itiscertainlydesirabletostudyvolumeeffectsinchiral
vertexfunctions.
symmetrybreakingindifferentframeworks.Chiralpertur-
bationtheoryhasturnedouttobeareliabletoolforboth
volumeandchiralextrapolations.Onthe otherhand,chi-
2 Quark-gluon interaction
ral perturbation theory has nothing to say about volume
effects in the underlying quark and gluon substructure.
For this the Green’s function approachemploying Dyson-
TheDyson-Schwingerequations(DSEs)fortheghost,gluon
Schwinger equations (DSEs) [2,3] provides a suitable al-
and quark propagatorsD ,D ,S(p)
ternative. In this talk we present results for the quark G µ,ν
propagator in Landau gauge QCD. Based on ideas devel-
oped in [4,5], we use lattice results to fix the quark-gluon G(p2)
D (p)= , (1)
interaction at quark masses accessible on the lattice. To G − p2
this end we solve the DSEs on tori with similar volumes. p p Z(p2)
µ ν
We then change currentquark massesand volumes to ex- Dµν(p)=(cid:18)δµν − p2 (cid:19) p2 , (2)
plore the corresponding effects on the quark propagator.
Z (p2)
f
a Talk given by CF at QNP06, Madrid, June 2006. S(p)= ip+M(p2) , (3)
6
2 C. S.Fischer, M. R.Pennington: Aspects of quarkmass generation on a torus
in quenched approximation are given diagrammatically
h Λg ΛQCD a1 a2 a3
in Fig. 1. This system of equations is closed but for the (GeV) (GeV)
dressed vertices, which have to be approximated by suit-
able ansa¨tze. The truncation scheme for the Yang-Mills staggered 1.48 1.50 0.35 18.39 5.25 -0.21
sectorisdiscussede.g.in[3,6].Thecorrespondingnumer-
Table 1. Parameters used in thevertex model, Eqs. (7-10).
ical solutions for the ghost and gluon propagator can be
represented accurately by
c(p2/Λ2 )κ+d(p2/Λ2 )2κ
R(p2)= YM YM , In[5]theparametershavebeenfittedsuchthatlattice
1+c(p2/Λ2 )κ+d(p2/Λ2 )2κ
YM YM results for the quark propagator [8,9] are reproduced on
α(p2) 1+2δ similar compact manifolds. To this end, solutions for the
Z(p2)=(cid:18)α(µ2)(cid:19) R2(p2), (4) ghost and gluon propagators on similar manifolds have
been used [10]. These solutions have been improved in
G(p2)=(cid:18)αα((µp22))(cid:19)−δR−1(p2), (5) [c1a1n]bweitsheetnheonefftoercitwthitahtvtohleumcoersrelacrtgienrfitnhiatenvVolum(1e0lfimm)it4s.
≈
Here we use the improved solutions to update the results
with the scale ΛYM = 0.658GeV, the coupling α(µ2) = for the quark propagator reported in [5]. The resulting
0.97 and the parameters c = 1.269 and d = 2.105 in the modified values for the parameters (staggered quarks [8]
auxiliaryfunctionR(p2).Thequenchedanomalousdimen- only) are given in Fig. 1. The numerical results, shown
sion of the ghost is given by δ = 9/44 and related via below, are discussed in the next section.
−
γ = 1 2δ to the one of the gluon. The infrared ex-
− −
ponent κ is determined in an analytical infrared analysis
[7]: κ = (93 √1201)/98 0.595. The running coupling
α(p2),define−dviathe nonp≈erturbativeghost-gluonvertex 0.5 Bowman et al (2002)
α(p2)=α(µ2)G2(p2)Z(p2), can be represented by Torus
Infinite volume/continuum
0.4 Chiral limit
1
α(p2)= α(0)+p2/Λ2
1+p2/Λ2 (cid:20) YM × ]
YM V
4π 1 1 e0.3
G
. (6)
β0 (cid:18)ln(p2/Λ2YM) − p2/Λ2YM −1(cid:19)(cid:21) M [
0.2
where α(0) 8.915/N is also known analytically [7].
c
≈
Apart from the gluon propagator the other missing
0.1
piece in the quark-DSEis the quark-gluonvertex.We use
an ansatz of the form [5]
0
Γν(k,µ2) = γνΓ1(k2)Γ2(k2,µ2)Γ3(k2,µ2) (7) 0 1 2 3 4
p [GeV]
with the components
πγ 1.0
Γ1(k2) = ln(k2/Λ2m +τ), (8)
QCD
0.9
Γ2(k2,µ2) =G(k2,µ2) G(ζ2,µ2) Z3(µ2) ]
(9) V
× h [ln(k2/Λ2ge+τ)]1+δ Ge 0.8
[
a(M)+k2/Λ2 Z f
Γ3(k2,µ2) =Z2(µ2) 1+k2/Λ2 QCD , (10) 0.7 BToorwumsan et al. (2002)
QCD
Infinite volume/continuum
Chiral limit
where γ = 12/33 is the quenched anomalous dimension
m 0.6
of the quark and τ = e 1 acts as a convenient infrared
−
cutoff for the logarithms. The quark mass dependence of
0 1 2 3 4
the vertex is parametrised by q [GeV]
a(M) = a1 , Fig.2.ThequarkmassfunctionM(p2)andthewavefunction
1+a2M(ζ2)/ΛQCD+a3M2(ζ2)/Λ2QCD Zf(p2)forrenormalisedcurrentquarkmassesofm(2.9GeV)=
(11) 44, 80, 151MeV.Comparedarelatticeresultsof[8]withDSE-
where M(ζ2) is determined during the iteration process solutions on a torus (straight lines with filled circles) and in
at ζ =2.9 GeV. theinfinitevolume/continuum limit (dashed lines).
C. S.Fischer, M. R.Pennington: Aspects of quarkmass generation on a torus 3
3 Numerical results
80
Our results for the quark mass function M(p2) and the ]
wavefunctionZ (p2)forrenormalisedcurrentquarkmasses eV 60
f M
of m(2.9GeV) = 44,80,151MeV are displayed in Fig. 2. [
The data for the quark mass function from the staggered GeV)40
latticeactioncanbeverywellreproducedwiththesimple =1
p
vertex ansatz specified in the last section. The results for M(
20
the wave functions agree less with each other: the lattice
dataforthethreedifferentmassesaresmallerspreadthan
theDSE-results.Thismaybeanindicationfortheimpor- 0
0.8 1.0 1.2 1.4 1.6 1.8 2.0
tanceoftensorstructuresinthequark-gluonvertexdiffer-
L [fm]
entfromthe simple γ -termusedin Eqs.(7-10).InFig. 2
µ
we also show results in the infinite volume/continuum Fig. 3. The quark mass function for an up/down quark at a
limit. For the quark mass function these are very close given momentum p = 1GeV plotted as a function of the box
length L=V1/4.
to the results on the compact manifold, indicating that
the used lattice volume, V = (2.04 fm)4 is indeed large
enough s.t. the correct pattern of chiral symmetry break-
manifold is at least
ing can be observed for the quark masses shown. Small
effects are only seen in the renormalisation point depen- L 1.6 fm. (12)
χSB
dentandthereforeunphysicalquarkwavefunctionZ (p2). ≃
f
Thisresultagreeswithconclusionsfromvolumestudieson Inparticularthisgivesa(surprisinglysmall)miminalbox
the lattice [12].1 lengthbelowwhichchiralperturbationtheorycannotsafely
Theresultingquarkmassandwavefunctionsinthein- be applied.
finitevolume,continuumandchirallimitarealsoshownin
Fig. 2. The correspondingquark mass at zero momentum
Acknowledgements
isM(0)=400MeV.Inthe complexmomentumplane the
propagatorhasaleadinganalyticstructuregivenbyapair
We thank the organisers of QNP06 for all their efforts.
of conjugate poles at m = 416(20) i304(20) MeV. The
± CF has been supported by the Deutsche Forschungsge-
corresponding spectral function violates positivity and is
meinschaft (DFG) under contract Fi 970/7-1.
therefore characteristical for a confined quark. The chi-
ral condensate is q¯q 2GeV = (235 5MeV)3, using the
|h i|MS ±
quenched scale Λ = 225(21)MeV in the MS scheme.
MS References
The pion decay constant, as determined from solutions of
the pion Bethe-Salpeter equation, is f = 72MeV. Both
π 1. H.LeutwylerandA.Smilga,Phys.Rev.D46(1992) 5607.
values are considerably smaller than the experimental re-
2. R.Alkofer and L. von Smekal,Phys. Rept. 353 (2001) 281
sult.Takenatfacevaluethismeansthatthequenchedthe-
[arXiv:hep-ph/0007355].
oryunderestimatesthe amountofchiralsymmetrybreak- 3. C. S. Fischer, J. Phys. G 32 (2006) R253
ingbyabout10-20percent.Asimilarconclusionhasbeen [arXiv:hep-ph/0605173].
drawn in [4]. 4. M. S. Bhagwat et al., Phys. Rev. C 68, 015203 (2003)
Finallywestudythebehaviourofthequarkmassfunc- [arXiv:nucl-th/0304003].
tion at smallvolumes.In Fig. 3 we plotM(p=1GeV) as 5. C.S.FischerandM.R.Pennington,Phys.Rev.D73(2006)
a function of the box length L. The corresponding cur- 034029 [arXiv:hep-ph/0512233].
rent quark mass is of the order of a typical up/down- 6. C. S. Fischer and R. Alkofer, Phys. Lett. B 536 (2002)
quark mass, M(p = 2.9GeV) = 10MeV. We clearly see 177[arXiv:hep-ph/0202202];Phys.Rev.D67(2003)094020
that the quark mass function grows rapidly in the range [arXiv:hep-ph/0301094].
7. C. Lerche and L. von Smekal, Phys. Rev. D 65, 125006
1.0 < L < 1.6 fm signalling the onset of dynamical chi-
(2002) [arXiv:hep-ph/0202194].
ral symmetry breaking. Above L = 1.6 fm, a plateau is
8. P.O.Bowman,U.M.HellerandA.G.Williams,Phys.Rev.
reached.Thispicturedoesnotchangewhenweextractthe
D 66 (2002) 014505 [arXiv:hep-lat/0203001].
mass function M(p2) at smaller momenta p2 or when we 9. J. B. Zhang et al., Phys. Rev. D 71, 014501 (2005)
employ even smaller quark masses. The value is also not
[arXiv:hep-lat/0410045].
dependent on the type of quark-gluon interaction used.
10. C. S. Fischer, R. Alkofer and H. Reinhardt, Phys. Rev.
The result indicates that a safe volume to reproduce the D 65 (2002) 094008 [arXiv:hep-ph/0202195]; C. S. Fischer,
correctpatternofchiralsymmetrybreakingonacompact B. Gruter and R. Alkofer, Annals Phys. 321 (2006) 1918
[arXiv:hep-ph/0506053].
1 The larger volume effects found in [5] also for the mass 11. C.S.Fischer,A.Maas,J.M.PawlowskiandL.vonSmekal,
functionareduetoanoverestimationoftheeffectsintheYang- arXiv:hep-ph/0701050.
Mills sector. An improved analysis of volume effects in the 12. J.B.Zhangetal.,Nucl.Phys.Proc.Suppl.141,15(2005).
ghost and gluon propagators can be found in [11].