# Strangeness Contribution to the Proton Spin from Lattice QCD

###### Abstract

We compute the strangeness and light-quark contributions , and to the proton spin in lattice QCD at a pion mass of about 285 MeV and at a lattice spacing fm, using the non-perturbatively improved Sheikholeslami-Wohlert Wilson action. We carry out the renormalization of these matrix elements which involves mixing between contributions from different quark flavours. Our main result is the small negative value of the strangeness contribution to the nucleon spin. The second error is an estimate of the uncertainty, due to the missing extrapolation to the physical point.

###### pacs:

12.38.Gc,14.20.Dh,13.88.+e,13.85.Hd^{†}

^{†}preprint: Adelaide ADP-11-43/T765, Edinburgh 2011/39, Liverpool LTH 934

QCDSF Collaboration Introduction.—The proton spin can be split into a quark spin contribution , a quark angular momentum contribution and a gluonic contribution (including spin and angular momentum) hep-ph/9603249 :

(1) |

In the naïve non-relativistic SU(6) quark model, , with vanishing and . In this case there will be no strangeness contribution to , where, in our notation, contains both, the spin of the quarks and of the antiquarks .

Experimentally, is obtained by integrating the strangeness contribution to the spin structure function over the momentum fraction . The integral over the range in which data exist agrees with zero; see, e.g., new COMPASS data arXiv:1001.4654 ; arXiv:1007.4061 for or HERMES data hep-ex/0609039 for , while global analyses give values arXiv:1010.0574 ; arXiv:1007.0351 ; arXiv:0904.3821 , suggesting a large negative at very small . Pioneering lattice simulations of disconnected matrix elements also indicated values hep-ph/9502334 ; 498992 . However, the errors given in these studies are quite optimistic while the global fits rely on an extrapolation of the integrated experimental to small and constrain the axial octet charge to a value, obtained from hyperon -decays, assuming SU(3) flavour symmetry. Some time ago, employing heavy baryon chiral perturbation theory, Savage and Walden hep-ph/9611210 pointed out that SU(3) symmetry in weak baryonic decays may be violated by as much as 25 % and hence could remain close to zero also for ; see also Bass:2009ed . SU(3) symmetry is however supported by lattice simulations of hyperon axial couplings hep-lat/0208017 ; arXiv:0712.1214 ; arXiv:0911.2447 ; arXiv:1102.3407 , albeit within non-negligible errors.

In this Letter, we directly compute the matrix elements that contribute to the , including quark line disconnected diagrams. Preliminary results were presented at conferences arXiv:1112.0024 ; arXiv:1011.2194 ; arXiv:0911.2407 .

Simulation details and methods.—We simulate non-perturbatively improved Sheikholeslami-Wohlert Fermions, using the Wilson gauge action, at and . Setting the scale from the chirally extrapolated nucleon mass sternbeck , we obtain the lattice spacing , where the errors are statistical and from the extrapolation, respectively.

We realize two additional valence values, and . The corresponding pion masses are , and . was fixed so that the value is close to the mass of a hypothetical strange-antistrange pseudoscalar meson: MeV. We investigate volumes of and lattice points, i.e., and 4.20, respectively, where the largest spatial lattice extent is fm.

The quark polarizations are extracted from the large-time behaviour of ratios of three-point over two-point functions. We create a polarized proton at a time , probe it with an axial current at a time and destroy the zero momentum proton at . Quark line connected and disconnected terms contribute:

(2) | ||||

Here is the lattice Dirac operator, is a parity projector and projects out the difference between the two polarizations (in direction . We average over to increase statistics. For the up and down quark matrix elements we compute the sum of connected and disconnected terms while only contributes to .

For disconnected contributions we fix the time distance between the source and the current insertion fm and vary . Both and the distance between current and sink should be taken large, to suppress excited state contributions. Using the sink and source smearing described in arXiv:1111.1600 , we find the asymptotic limit to be effectively reached for –; see Fig. 1 for an example. The saturation into a plateau at and the convergence of the point sink data towards the same value demonstrate that was reasonably chosen. To be on the safe side, we only fit the fm smeared-smeared ratios. Building upon previous experience Khan:2006de , the connected part, for which the statistical accuracy is less of an issue, is obtained at the larger, fixed value , varying .

The disconnected contribution is computed with the stochastic estimator methods described in arXiv:0910.3970 ; arXiv:1011.2194 , employing time partitioning, a second order hopping parameter expansion and the truncated solver method. We compute the Green functions for four equidistant source times on each gauge configuration. We also construct backwardly propagating nucleons, replacing the positive parity projector by , seeding the noise vectors on eight (four times two) timeslices. In addition to the 48 (four times spin times colour) solves for smeared conventional sources, that are necessary to construct the two-point functions, we run the Conjugate Gradient (CG) algorithm on complex noise sources for iterations. The bias from this truncation is corrected for arXiv:0910.3970 by BiCGstab solves that are run to convergence. We analyse a total of 2024 thermalized trajectories on each of the two volumes where we bin the data to eliminate autocorrelations.

Renormalization.—Non-singlet axial currents renormalize with a renormalization factor that only depends on the lattice spacing. This was determined non-perturbatively for the action and lattice spacing in use Gockeler:2010yr : .

However, due to the axial anomaly, the renormalization constant of singlet currents, , acquires an anomalous dimension. To first non-trivial order this reads Kodaira:1979pa ; Larin:1993tq . deviates from starting at in perturbation theory. Both factors have been calculated to this order, with the result for the conversion into the scheme at a scale Skouroupathis:2008mf

(3) |

where we have set the Sheikholeslami-Wohlert parameter to be consistent to this order in perturbation theory. To this first non-trivial order no scale enters the coupling parameter . Since perturbation theory in terms of the bare lattice parameter is known to converge poorly, we substitute by a coupling defined from the measured average plaquette , where we have used the chirally extrapolated value Gockeler:2005rv .

No dimension-four operator can be constructed that mixes with the relevant forward matrix element of and that cannot be removed, using the equations of motion Capitani:2000xi . This also holds for the singlet case hep-lat/0511014 , such that we only need to replace

(4) |

to achieve full improvement. The factor is known to Capitani:2000xi : . We obtain the values

(5) |

where the first error is due to the uncertainty in the quark mass and the second error corresponds to 50 % of the one-loop correction. Considering the small size of this correction it is unlikely that the (two-loop) difference between singlet and non-singlet -factors will result in any noticeable effect, and in particular not at the light-quark mass , where it will be needed [see Eq. (11) below].

For we get

(6) |

at the renormalization scale . We again include a 50 % systematic error to allow for higher order corrections. Due to the small anomalous dimension that only sets in at , the difference between singlet and non-singlet renormalization constants remains small, also at other scales. For instance, we obtain and .

In the theory the matrix elements renormalize as follows:

(7) | ||||

(8) | ||||

(9) |

We remark that for non-equal quark masses the non-singlet combinations Eqs. (7) and (8) also receive contributions from disconnected quark line diagrams.

We employ sea quarks so that our singlet current is rather than the of Eq. (9). This modifies the renormalization pattern:

(10) |

receives light-quark contributions but the and remain unaffected by the (quenched) strange quark. Obviously, unitarity is violated, due to this quenching. The combination still transforms with [Eq. (8)] while Eq. (9) is violated, as it should be; instead, the singlet operator renormalizes with . We remark that the above renormalization pattern is similar to that of the scalar matrix element in the theory Gockeler:2004rp ; arXiv:1111.1600 ; Babich:2010at . Note that in spite of the quenched strange quark the mismatch between directly converting the result into the scheme at a scale , using , and first converting into the scheme at another scale and subsequently running within the scheme with to the scale is tiny.

Results and systematics.—In Fig. 2 we display the volume and (light) valence quark mass dependence of our unrenormalized . There are no statistically significant finite size or mass effects.

1.065(22) | -0.034(16) | 0.794(21)(2) | ||

-0.344(14) | -0.034(16) | -0.289(16)(1) | ||

0 | -0.031(12) | -0.023(10)(1) | ||

fm | 1.409(24) | 0 | 1.082(18)(2) | |

0.721(26) | -0.006(18) | 0.550(24)(1) | ||

0.721(26) | -0.098(42) | 0.482(38)(2) | ||

1.071(15) | -0.049(17) | 0.787(18)(2) | ||

-0.369( 9) | -0.049(17) | -0.319(15)(1) | ||

0 | -0.027(12) | -0.020(10)(1) | ||

fm | 1.439(17) | 0 | 1.105(13)(2) | |

0.702(18) | -0.044(19) | 0.507(20)(1) | ||

0.702(18) | -0.124(44) | 0.448(37)(2) |

Using Eqs. (10) and (4) we can renormalize

(11) |

for . As discussed above, we omit the improvement factor of the term. This is of and numerically negligible. We display the bare lattice numbers for the connected and disconnected contributions to the proton spin and the renormalized improved values in Table 1, for the two volumes. The and values are reduced by about 0.035, due to the sea quark contributions while increases by 0.002 ( %), due to the mixing with light-quark flavours.

The uncertainties associated to the renormalization are much smaller than the statistical errors. Below we will only quote large volume results, with statistical and renormalization errors added in quadrature. Error sources that have so far not been accounted for are the missing continuum limit extrapolation, the quenching of the strange quark and simulating at a light sea quark mass value that is four times bigger than the physical one. There are no indications of radical quark mass effects: the flavour mixing effects within the renormalization are small in spite of the comparatively large and values. The dependence on the valence quark mass is small too; see Fig. 2.

Nevertheless, having simulated only at one lattice spacing and sea quark mass, we cannot extrapolate our results to the physical point. Consequently, we underestimate the value arXiv:0812.3535 from neutron -decays by and find instead. Our prediction differs by the same from the phenomenological estimate arXiv:0812.3535 . We take this as an indication of the size of the remaining systematics and add an additional 20 % error to all our results.

Conclusions.—We determined the first moments of proton flavour singlet and non-singlet polarized parton distributions from lattice QCD, at a pion mass of 285 MeV, at a single lattice spacing fm. We found and a small negative , in the scheme, at a scale GeV. We underestimated both and by similar factors and this may suggest that some of the systematics cancel when considering ratios of matrix elements. Nevertheless, we emphasize that there is a considerable uncertainty in the value hep-ph/9611210 and our is already relatively large, due to the small difference .

Interestingly, our results are in remarkable agreement with the cloudy bag model prediction of Bass:2009ed . The small (unrenormalized) value obtained recently in Babich:2010at is also consistent with our study. Our value is larger than previously expected, however, it is compatible with the latest COMPASS number arXiv:1001.4654 . The experimental number may increase further once smaller -values become accessible. We suggest relaxing the weak hyperon decay SU(3) constraint on in determinations of polarized parton distribution functions arXiv:1010.0574 ; arXiv:1007.0351 ; arXiv:0904.3821 , and including our prediction instead.

###### Acknowledgements.

Acknowledgments.— This work is supported by the EU (Grant No. 238353, ITN STRONGnet) and by the DFG Grant No. SFB/TR 55. S.C. acknowledges support from the Claussen-Simon-Foundation (Stifterverband für die Deutsche Wissenschaft) and J.Z. from the Australian Research Council (Grant No. FT100100005). Computations were performed on the SFB/TR55 QPACE supercomputers, the BlueGene/P (JuGene) and the Nehalem Cluster (JuRoPA) of the Jülich Supercomputer Center, the IBM BlueGene/L at the EPCC (Edinburgh), the SGI Altix ICE machines at HLRN (Berlin/Hannover) and Regensburg’s Athene HPC cluster. The Chroma software suite Edwards:2004sx was used and gauge configurations were generated with the BQCD code Nakamura:2010qh .## References

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