Disconnected contributions to hadronic structure
Sara Collins, Gunnar Bali, Andrea Nobile, Andreas Schafer, Yoshifumi Nakamura, James Zanotti
aa r X i v : . [ h e p - l a t ] N ov Disconnected contributions to hadronic structure
Sara Collins ∗ , Gunnar Bali, Andrea Nobile, Andreas Schäfer Institut für Theoretische Physik, Universität Regensburg,93040 Regensburg, GermanyE-mail: [email protected]@physik.uni-regensburg.deandrea.nobile@physik.uni-regensburg.deandreas.schaefer@physik.uni-regensburg.de
Yoshifumi Nakamura
Center for Computational Sciences, University of Tsukuba,Tsukuba, Ibaraki 305-8577, JapanE-mail: [email protected]
James Zanotti
School of Physics, University of Edinburgh,Edinburgh EH9 3JZ, UKE-mail: [email protected] (QCDSF Collaboration)
We present an update of an on-going project to determine the disconnected contributions tohadronic structure, specifically, the scalar matrix element, h N | ¯ qq | N i , and the quark contributionto the spin of the nucleon D q = h N | ¯ q g m g q | N i / m N . The XXVIII International Symposium on Lattice Field Theory, Lattice2010June 14-19, 2010Villasimius, Italy ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ isconnected contributions to hadronic structure
Sara Collins
1. Introduction
In the last few years there has been an upsurge in interest in calculating the scalar matrixelement on the lattice [1 – 5] either directly, by calculating the corresponding connected and dis-connected terms, or indirectly, via the Feynman-Hellman theorem. High statistics for dynamicalsimulations mean that reasonable signals can be obtained for disconnected terms and similarly thesmall statistical uncertainty on nucleon mass as a function of the quark masses enable reasonablefits to be made. Ideally, the results of both approaches should agree.Such calculations have also become particularly timely since the advent of the LHC becausethe scalar coupling f T q = m q h N | ¯ qq | N i / m N determines the fraction of the proton mass m N that iscarried by quarks of flavour q . The strength of the coupling of the Standard Model (SM) Higgsboson or of any similar scalar particle to the proton is mainly determined by (cid:229) q f T q for q ∈ { u , d , s } .Therefore, an accurate calculation of these quantities will help to increase the precision of SMphenomenology and to shed light on non-SM processes.The spin of the nucleon can be decomposed into a quark spin contribution DS = D u + D d + D s + . . . , a quark angular momentum contribution L q and a gluonic contribution (spin and angularmomentum) D G : 12 = DS + L q + D G . (1.1)Experimentally, D s is not well determined: HERMES obtained [6] D s = − . ( )( )( ) in the MS scheme. However, the signal is dominated by contributions in the small x region where modelsare used to extrapolate from the experimental results obtained at larger x .In these proceedings we present an update of an on-going project to calculate f T q and D q . Inparticular, the following improvements have been implemented since Lattice 2009 [7]: • Statistics on the 24 ×
48 and 32 ×
64 lattices have been significantly increased, by factorsof roughly two and three, respectively, using the SFB/TR55 QPACE computers [8, 9] (detailsare given in the next section). • An additional volume of 40 ×
64 has been analyzed.The analysis is not yet finalized and we plan to further increase the statistics for the larger twovolumes. The results presented here are preliminary.
2. Simulation details
The simulations were performed on n f = b = .
29 and k sea = . k sea value is around 270 MeV, using an inverse lattice spacing of 2 .
59 GeV determined from r ( b , k ) = .
467 fm.The scalar and axial-vector matrix elements we are interested in are extracted on the latticefrom the three-point functions corresponding to the diagrams given in fig. 1. The axial-vectormatrix element is related to D q through, h N , s | ¯ q g m g q | N , s i = m N s m D q , (2.1)2 isconnected contributions to hadronic structure Sara Collins GG q q q q Figure 1:
The connected (left) and disconnected (right) diagrams associated with the scalar ( G = ) andaxial vector matrix elements ( G = g g i ). in Minkowski space notation, where m N is the nucleon mass and s m its spin ( s m = − D s and h N | ¯ ss | N i only these terms contribute.We varied the quark mass of the current in- b = . n f = k sea = . ×
48 32 ×
64 40 × ≈ ≈ ≈ m PS L Table 1:
Details of the configurations used. sertion (disconnected loop), as well as the mass ofthe valence quarks in the nucleon. In the follow-ing, we denote the k value corresponding to thequark loop and the valence quarks in the nucleonas k loop and k val , respectively. All combinations k loop , k val ∈ { . , . , . } were used.These values correspond to the pseudoscalar masses, m PS ≈ k val = . D q and h N | ¯ qq | N i were extracted from the ratios of three-point functions to two-point functions (at zero momentum), R dis ( t f , t , t i ) = − Re D G ab C ba ( t f , t i ) (cid:229) x Tr ( M − ( x , t ; x , t ) G loop ) ED G ab unpol C ba ( t f , t i ) E , (2.2)where the nucleon source and sink are at t i and t f respectively, and the current is inserted at t . Thethree-point function is simply the combination of the nucleon two-point function, C ( t f , t i ) , andthe disconnected loop, (cid:229) x Tr [ M − ( x , t ; x , t ) G loop ] . For the scalar matrix element we used, G = G unpol : = ( + g ) / G loop = . For D q we calculated the difference between two polarizations: G = g j g G unpol and G loop = g j g , where we average over all three possible j -orientations.In the limit of large times, t f ≫ t ≫ t i , depending on the G -combination used, R dis will eitherapproach the disconnected axial matrix element D q dis or the disconnected scalar matrix element h N | ¯ qq | N i dis (once the vacuum contribution is subtracted). However, the statistical noise increasesrapidly with increasing t − t i and this time difference needs to be minimized, using smeared sourcesand sinks for the nucleon, in order to obtain a reasonable signal. A smearing study on a limitednumber of configurations indicated that the nucleon plateaued around t ≥ a ≈ . t i = t ≥ a only. At zero momentum, the excited state contribution to R dis is governed by the time difference, t f − t i , and we must be careful to choose t f large enough.3 isconnected contributions to hadronic structure Sara Collins t f < N | qq | N > d i s t f -0,1-0,09-0,08-0,07-0,06-0,05-0,04-0,03-0,02-0,0100,010,020,030,040,05 D q d i s Figure 2:
The results for h N | ¯ qq | N i dis and D q dis extracted from the corresponding R dis as a function of thesink timeslice, t f , for all three volumes used and the heaviest k val = k loop = . In fig. 2 we show the results for R dis as a function of t f > t for all three volumes studied andthe heaviest k val = k loop = . t f = a or 9 a based onthe quality of the plateau within given statistical errors. This depends on the observable and latticevolume. Once final statistics are reached we will fit the three- and two-point functions within R dis of eq. (2.2) separately, as functions of t f , in order to extract the asymptotic values.The disconnected loop, (cid:229) x Tr [ M − ( x , t ; x , t ) G loop ] , was calculated using stochastic estimates,together with several noise reduction techniques: • Partitioning [12, 13]: the stochastic source has support on eight timeslices. Additional two-point functions were generated for four time separated source points on each configuration.The forward and backward propagation from these four source points was combined withthe loop to give us eight measurements of R dis per configuration. • Hopping parameter expansion [14]: for the clover action the first two terms in the expansionof the disconnected loop,Tr ( M − G loop ) = k Tr [( − k D ) − G loop ] = Tr [( k + k D + k D M − ) G loop ] , (2.3)vanish and hence only contribute to the noise. This means Tr [ k D M − ( x , t ; x , t ) G loop ] canbe used as an improved estimate of the loop. (In the case of G loop = the non-vanishing firstterm (cid:229) x k Tr = k L can easily be corrected for.) • Truncated solver method [15]: 730 conjugate gradient solves were used, where the solverwas truncated after 40 iterations. 50 BiCGStab solves running to full convergence weregenerated to correct for the truncation error.The noise reduction techniques other than time partitioning are only necessary for determining D q ;for the scalar matrix element the gauge noise dominates.
3. The scalar matrix element: f T s and m q h N | ¯ qq | N i dis The results for f T s are presented in fig. 3 as functions of m corresponding to the mass ofthe (valence) quarks in the nucleon. No renormalization is required as the combination, m q h N | ¯ qq | N i ,4 isconnected contributions to hadronic structure Sara Collins (m PS ) (val) (GeV) f T s = m s < N | ss | N > / m N P r e li m i n a r y (m PS ) (loop) (GeV) m q < N | qq | N > d i s ( G e V ) m p phys. k val =0.13550 (m PS =690 MeV) k val =0.13609 (m PS =440 MeV) k val =0.13632 (m PS =270 MeV) P r e li m i n a r y Figure 3:
Results for the scalar matrix element: (left) f T s as a function of m for the valence quark mass (i.e.the mass of the quarks in the nucleon) for the three volumes studied. (right) m q h N | ¯ qq | N i dis as a function of m for the loop quark mass on the 32 ×
64 volume, for the three valence quark masses. is scale and scheme independent. There is consistency between the values obtained on the differentvolumes for the heaviest valence quark mass, however, the spread between the results increaseswhen the quark mass is reduced. Whether this is an indication of finite size effects will be clarifiedonce the statistics for the 40 volume is increased.In fig. 3 we also display the results for the disconnected scalar matrix element, m q h N | ¯ qq | N i ,for the 32 volume as a function of m corresponding to the loop quark mass. This combinationis relevant for extracting the sigma term, s N = m q h N | ¯ uu + ¯ dd | N i . (3.1)We found 2 m q h N | ¯ uu | N i dis for k val = k loop = k sea = . s N for this volume. However, a more sophisticated method of extractingthe matrix element from R dis is required in order to make a firm comparison.
4. The spin contribution: D s and D q The results for D s on all three volumes are shown in fig. 4. No significant dependence onthe valence quark mass nor on the lattice size is seen in the data. Neither is there any significantvariation in the results if the loop quark mass is reduced. These numbers will have to be multipliedby a renormalization constant of approximately 0 . MS scheme [16]. In contrast to thescalar case, the disconnected contributions are much smaller than the connected terms, at around10% for D d and 5% for D u .
5. Outlook
In the short term, the aim is to reach our target statistics of 2000 trajectories for each volume.We then plan to begin an analysis close to the physical sea quark mass. The nonperturbative renor-malization for D q needs to be calculated while for the scalar strangeness matrix element mixingwith the light flavours needs to be considered. 5 isconnected contributions to hadronic structure Sara Collins (m PS ) (val)(GeV) -0.1-0.09-0.08-0.07-0.06-0.05-0.04-0.03-0.02-0.010 D s P r e l i m i n a r y , n o r e n o r m a l i z a t i o n (m PS ) (val)(GeV) -0.1-0.09-0.08-0.07-0.06-0.05-0.04-0.03-0.02-0.010 D q d i s k loop =0.13550 (m PS =690 MeV) k loop =0.13609 (m PS =440 MeV) k loop =0.13632 (m PS =270 MeV) P r e l i m i n a r y , n o r e n o r m a l i z a t i o n Figure 4:
Results for D q : (left) D s as a function of m for the valence quark mass (i.e. the mass of thequarks in the proton) for the three volumes studied. (right) D q dis from the 32 ×
64 volume for different loopquark masses, again as a function of m for the valence quark mass. Acknowledgments
This work was supported by the EU ITN STRONGnet, the I3 HadronPhysics2 and the DFGSFB/Transregio 55. Sara Collins acknowledges support from the Claussen-Simon-Foundation(Stifterband für die Deutsche Wissenschaft). Computations were performed on the IBMBlueGene/L at EPCC (Edinburgh,UK), Regensburg’s Athene HPC cluster, the BlueGene/P(JuGene) and the Nehalem Cluster (JuRoPA) of the Jülich Supercomputer Center and theSFB/TR55 QPACE supercomputers. The Chroma software suite [17] was used extensively in thiswork.
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