Moments of meson distribution functions with dynamical twisted mass fermions
Rémi Baron, Stefano Capitani, Jaume Carbonell, Karl Jansen, Zhaofeng Liu, Olivier Pène, Carsten Urbach
aa r X i v : . [ h e p - l a t ] O c t Moments of meson distribution functions withdynamical twisted mass fermions R ´ emi Baron SPhN-DAPNIA, CEA Saclay, 91191 Gif sur Yvette, FranceE-mail: [email protected]
Stefano Capitani
Fakultät für Physik, Universität Bielefeld, Universitätsstrasse, 33615 Bielefeld, GermanyE-mail: [email protected]
Jaume Carbonell
Laboratoire de Physique Subatomique et Cosmologie, 53 Av des martyrs, 38026 Grenoble,FranceE-mail: [email protected]
Karl Jansen
DESY, Platanenallee 6, 15738 Zeuthen, GermanyE-mail:
Zhaofeng Liu ∗ and Olivier P ` ene Laboratoire de Physique Th ´ eorique (B ˆ at. 210), Universit ´ e de Paris XI, Centre d’Orsay, 91405Orsay-Cedex, FranceE-mail: [email protected] , [email protected]
Carsten Urbach
Theoretical Physics Division, Dept. of Mathematical Sciences, University of Liverpool,Liverpool L69 7ZL, UKE-mail:
On behalf of the ETM Collaboration
We present our preliminary results on the lowest moment h x i of quark distribution functions ofthe pion using two flavor dynamical simulations with Wilson twisted mass fermions at maximaltwist. The calculation is done in a range of pion masses from 300 to 500 MeV. A stochasticsource method is used to reduce inversions in calculating propagators. Finite volume effects atthe lowest quark mass are examined by using two different lattice volumes. Our results show thatwe achieve statistical errors of only a few percent. We plan to compute renormalization constantsnon-perturbatively and extend the calculation to two more lattice spacings and to the nucleons. The XXV International Symposium on Lattice Field TheoryJuly 30 - August 4 2007Regensburg, Germany ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ oments of meson distribution functions with dynamical twisted mass fermions
Zhaofeng Liu
1. Introduction
Deep inelastic structure functions of mesons and nucleons are interesting since those functionsgive us information about the momentum and spin carried by quarks and gluons inside hadrons.Lattice QCD can calculate moments of structure functions from first principles. By doing an op-erator product expansion on the hadronic tensor, the moments of structure functions are related toreduced matrix elements of certain local operators. For example, for the case of scattering of a spinone particle, the details are given in Ref.[1]. On the lattice, calculations on moments of structurefunctions of mesons and nucleons can be found in, e.g., Ref.[2, 3, 4, 5].The Wilson twisted mass formulation of lattice QCD was introduced in Ref.[6, 7]. It providesautomatic O ( a ) improvement when tuned to maximal twist, which can be achieved by setting thePCAC mass to zero. Also, twisted mass fermions are protected from unphysical fermion zeromodes, thus the problem of exceptional configurations is avoided. Quenched simulations with Wil-son twisted mass fermions have been shown to be successful[8, 9, 10, 11]. Specifically, a quenchedcalculation of the lowest moment of the quark distribution function h x i in a pion was done inRef.[12]. Recently, two flavor dynamical simulations[13] with Wilson twisted mass fermions gaveprecise results on low energy constants in the chiral effective Lagrangian. Using those configura-tions, in this work we calculate h x i in a pion. The computation is performed at pion masses in therange of ∼ −
500 MeV. In Table 1, the parameters of our simulations are collected. The gaugeaction used in the simulations is the tree-level Symanzik improved gauge action. For more detailsof our data, please see Ref.[14]. b a (fm) a m m p (GeV) L × T N meas h x i bare3.9 0.0855(6) 0.0100 0.4839(12) 24 ×
48 170 0.295(3)0.0085 0.4470(12) 24 ×
48 167 0.279(4)0.0064 0.3903(9) 24 ×
48 250 0.279(5)0.0040 0.3131(16) 24 ×
48 230 0.268(8)0.0040 0.3082(55) 32 ×
64 316 0.251(8)
Table 1:
Simulation parameters, the number of measurements( N meas) and preliminary results of h x i bare.Measurements were done on gauge configurations separated by 20 HMC trajectories with trajectory length t = /
2. Methodology
By using the optical theorem, the cross section of deep inelastic scatterings can be related tothe hadronic tensor or the imaginary part of the forward current-hadron scattering amplitude W mn ,which is given by W mn ( p , q , l , l ′ ) = p Z d xe iq · x h p , l ′ | [ j m ( x ) , j n ( )] | p , l i . (2.1)Here l and l ′ are the polarization of the target hadron. p and q are the momenta of the hadron andthe virtual photon. Depending on the spin of the hadron, the above tensor W mn can be decomposed2 oments of meson distribution functions with dynamical twisted mass fermions Zhaofeng Liu into a number of independent structure functions. The spin-averaged structure functions F ( x , Q ) and F ( x , Q ) can tell us overall densities of quarks and gluons in a hadron. In the parton model,to leading order, the single flavor structure function F ( x ) is half the probability of finding a quarkwith momentum fraction x . If defining the n th moment of a function f ( x ) as M n ( f ) = Z x n − f ( x ) dx , (2.2)then for the pion, to leading twist order, we have from the operator product expansion of W mn M n ( F ) = C ( ) n v n , M n − ( F ) = C ( ) n v n , ( n ≥ , even) , (2.3)where C ( k ) n = + O ( a s ) are the Wilson coefficients, and the reduced matrix element v n is definedby h ~ p | O { m ··· m n } − traces | ~ p i = v n [ p m · · · p m n − traces ] . (2.4)Here {· · ·} means symmetrization on the Lorentz indices and the twist-2 operators are given by O m ··· m n = ( i ) n − G f f ′ ¯ y f g m ↔ D m · · · ↔ D m n y f ′ , (2.5)where ↔ D = → D − ← D and G f f ′ is a diagonal flavor matrix.To compute the lowest moment of the quark distribution function h x i , we need to consider n = O ( x ) =
12 ¯ u ( x )[ g ↔ D − (cid:229) k = g k ↔ D k ] u ( x ) , (2.6)where D m = ( ▽ m + ▽ ∗ m ) with ▽ m ( ▽ ∗ m ) being the usual forward (backward) derivative on thelattice. With the above operator, no external momentum is needed in our calculation, which isadvantageous since an external momentum increases the noise to signal ratio. One can also usean operator from the other representation, which needs an external momentum. In the continuumlimit, the two operators should give the same result on h x i .From Eq.(2.4) the bare moment h x i bare is given by h x i bare = v = m p h p ,~ | O | p ,~ i , (2.7)where the matrix element h p ,~ | O | p ,~ i between two pions at rest is calculated from the ratio ofthe following 3-point function and 2-point function with a source at t = T / h p ,~ | O | p ,~ i = m p C ( t ) C p ( T / ) ( ≪ t ≪ T / ) . (2.8)Here C ( t ) = (cid:229) ~ x ~ y h PS ( T / ,~ x ) O ( t ,~ y ) PS † ( , ) i , (2.9) C p ( T / ) = (cid:229) ~ x h PS ( T / ,~ x ) PS † ( , ) i , (2.10)3 oments of meson distribution functions with dynamical twisted mass fermions Zhaofeng Liu and PS ( x ) = ¯ u ( x ) g d ( x ) is the interpolating field for the pion. There are two contributions in theWick contractions of C ( t ) : one connected diagram and one disconnected diagram. The discon-nected contribution is ignored in our calculation at this moment, but will be computed by using astochastic source method.The above two and three point correlators are evaluated by using a stochastic time slice source(Z(2)-noise in both real and imaginary part) [16, 17, 18] for all color, spin and spatial indices. i.e.,the quark propagator X b b ( y ) is obtained by solving (cid:229) y D ab ab ( z , y ) X b b ( y ) = x ( ~ z ) a a d z , ( source at t = ) , (2.11)where the Z(2) random source x ( ~ z ) a a satisfies the random average condition h x ∗ ( ~ x ) a a x ( ~ y ) b b i = d ~ x ,~ y d a , b d a , b . (2.12)The generalized propagator[19] S b b ( y ) needed in computing C ( t ) is obtained by solving (cid:229) y D ab ab ( z , y ) S b b ( y ) = g X a a ( z ) d z , T / ( sink at t = T / ) . (2.13)The advantage of using the above stochastic source compared with a point source is that muchless inversions are needed. With a point source, 24 inversions per gauge configuration are needed:12 (3 colors × t = /
2. Statistical errors are from a Gamma-function analysis[20].
3. Preliminary results and outlook
At the smallest quark mass, a m = . h x i using both a point source methodand the above stochastic source method on the 24 ×
48 lattice. The results are in agreement withinerrors. Fig.1 shows the comparison of the pion effective masses obtained from the two methodson 230 gauge configurations. As we can see, the statistical errors from the two methods are of thesame size. In Fig. 2, we compare the results of h x i bare obtained by using the two sources. The twoplateaus of h x i bare around T / T / h x i bare are collected in the last column of Table 1. We only givebare quantities here since the renormalization constant of the matrix element has not been calcu-lated yet, which we plan to compute non-perturbatively. Fig. 3 shows the results of h x i bare againstthe pion mass squared. The cross in the graph at the lowest quark mass is from the 32 ×
64 lat-tice. It shows that the finite lattice volume effects are not big. A linear extrapolation in m p gives h x i bare=0.246(10) in the chiral limit (using only the points from the 24 ×
48 lattice).4 oments of meson distribution functions with dynamical twisted mass fermions
Zhaofeng Liu pointstochastic t/aam PS , eff Figure 1:
Comparison of the pion effective masses for a m = .
004 obtained from a point source and astochastic source on the 24 ×
48 lattice. They are from the same 230 gauge configurations. . . . . . .
230 gauges, point sources x < x > . . . . . .
230 gauges, stochastic sources x < x > Figure 2: h x i bare for a m = .
004 obtained from a point source and a stochastic source. We show the averageof the two plateaux around T / T /
4. The stochastic source method is much cheaper in computer time.
Using Wilson twisted mass fermions, our two flavor dynamical simulations go down to apion mass of around 300 MeV. Our first attempt in this work shows we can get h x i with smallstatistical errors of a few percent. This will enable us to compare numerical results of h x i inpions and nucleons with predictions from the chiral perturbation theory (see for example [21, 22,23]). With simulations at two more lattice spacings, we will be able to do an extrapolation tothe continuum limit (the data analysis is in progress). For the future, we plan to compute thedisconnected diagrams using a stochastic source method and calculate the renormalization constantnon-perturbatively. We also plan to compute h x i in nucleons. Acknowledgments
The numerical calculations were performed on the ZOO cluster at LPT in Orsay, the computers5 oments of meson distribution functions with dynamical twisted mass fermions
Zhaofeng Liu
Figure 3: h x i bare against the pion mass squared. Finite size effects at the lowest pion mass are not big.A linear extrapolation gives h x i bare=0.246(10) in the chiral limit (using only the points from the 24 × of the CCRT (Bruy`ere-le-Chatel) computing center and the Blue Gene/L in Jülich. We thank ChrisMichael for discussions on the stochastic source method. Zhaofeng Liu thanks Stefan Schaefer andAndrea Shindler for valuable discussions. References [1] P. Hoodbhoy, R. L. Jaffe and A. Manohar, Nucl. Phys. B (1989) 571.[2] M. Göckeler, R. Horsley, E. M. Ilgenfritz, H. Perlt, P. Rakow, G. Schierholz and A. Schiller, Phys.Rev. D (1996) 2317 [arXiv:hep-lat/9508004].[3] C. Best et al. , Phys. Rev. D (1997) 2743 [arXiv:hep-lat/9703014].[4] M. Guagnelli, K. Jansen, F. Palombi, R. Petronzio, A. Shindler and I. Wetzorke [Zeuthen-Rome(ZeRo) Collaboration], Eur. Phys. J. C (2005) 69 [arXiv:hep-lat/0405027].[5] D. Galletly et al. [QCDSF Collaboration], PoS LAT2005 (2006) 363 [arXiv:hep-lat/0510050].[6] R. Frezzotti, P. A. Grassi, S. Sint and P. Weisz [ALPHA collaboration], JHEP (2001) 058[arXiv:hep-lat/0101001].[7] R. Frezzotti and G. C. Rossi, JHEP (2004) 007 [arXiv:hep-lat/0306014].[8] K. Jansen, A. Shindler, C. Urbach and I. Wetzorke [XLF Collaboration], Phys. Lett. B (2004) 432[arXiv:hep-lat/0312013].[9] K. Jansen, M. Papinutto, A. Shindler, C. Urbach and I. Wetzorke [XLF Collaboration], Phys. Lett. B (2005) 184 [arXiv:hep-lat/0503031]. oments of meson distribution functions with dynamical twisted mass fermions Zhaofeng Liu[10] K. Jansen, M. Papinutto, A. Shindler, C. Urbach and I. Wetzorke [XLF Collaboration], JHEP (2005) 071 [arXiv:hep-lat/0507010].[11] A. M. Abdel-Rehim, R. Lewis and R. M. Woloshyn, Phys. Rev. D (2005) 094505[arXiv:hep-lat/0503007].[12] S. Capitani, K. Jansen, M. Papinutto, A. Shindler, C. Urbach and I. Wetzorke, Phys. Lett. B (2006) 520 [arXiv:hep-lat/0511013].[13] Ph. Boucaud et al. [ETM Collaboration], arXiv:hep-lat/0701012.[14] C. Urbach [ETM Collaboration], PoS ( LATTICE 2007 ) 022.[15] M. Göckeler, R. Horsley, E. M. Ilgenfritz, H. Perlt, P. Rakow, G. Schierholz and A. Schiller, Phys.Rev. D (1996) 5705 [arXiv:hep-lat/9602029].[16] S. J. Dong and K. F. Liu, Phys. Lett. B (1994) 130 [arXiv:hep-lat/9308015].[17] M. Foster and C. Michael [UKQCD Collaboration], Phys. Rev. D (1999) 074503[arXiv:hep-lat/9810021].[18] C. McNeile and C. Michael [UKQCD Collaboration], Phys. Rev. D (2006) 074506[arXiv:hep-lat/0603007].[19] G. Martinelli and C. T. Sachrajda, Phys. Lett. B (1987) 184.[20] U. Wolff [ALPHA collaboration], Comput. Phys. Commun. (2004) 143 [Erratum-ibid. (2007) 383] [arXiv:hep-lat/0306017].[21] D. Arndt and M. J. Savage, Nucl. Phys. A (2002) 429 [arXiv:nucl-th/0105045].[22] J. W. Chen and X. D. Ji, Phys. Lett. B (2001) 107 [arXiv:hep-ph/0105197].[23] W. Detmold and C. J. Lin, Phys. Rev. D (2005) 054510 [arXiv:hep-lat/0501007].(2005) 054510 [arXiv:hep-lat/0501007].