The strange and light quark contributions to the nucleon mass from Lattice QCD
We determine the strangeness and light quark fractions of the nucleon mass by computing the quark line connected and disconnected contributions to the matrix elements in lattice QCD, using the non-perturbatively improved Sheikholeslami-Wohlert Wilson Fermionic action. We simulate mass degenerate sea quarks with a pion mass of about 285 MeV and a lattice spacing fm. The renormalization of the matrix elements involves mixing between contributions from different quark flavours. The pion-nucleon -term is extrapolated to physical quark masses exploiting the sea quark mass dependence of the nucleon mass. We obtain the renormalized values MeV at the physical point and for the strangeness contribution at our larger than physical sea quark mass.
Most of the nucleon’s mass is generated by the spontaneous breaking of chiral symmetry and only a small part can be attributed directly to the masses of its valence and sea quarks. The quantities
parameterize the fractions of the nucleon mass that are carried by quarks of flavour . Almost all visible matter of the universe is composed of nucleons and yet little is known experimentally about these quark contributions to their mass.
The scalar matrix elements
also determine the coupling strength of the Standard Model (SM) Higgs boson (or of any similar scalar particle) at zero recoil to the nucleon. This then might couple to heavy particles that could be discovered in LHC experiments, some of which are dark matter candidates Ellis:2009ai (). The combination , , will appear quadratically in this cross section that is proportional to where
with the couplings . Due to the trace anomaly of the energy momentum tensor one obtains Shifman:1978zn ()
The -terms are also needed for precision measurements of SM parameters in collisions at the LHC. For instance the resolution of a (hypothetical) mass difference between the and bosons is limited by our knowledge of the asymmetries between the up and down as well as between the strange and charm sea quark contents of the proton Aad:2009wy (). An accurate calculation of these quantities will help to increase the precision of SM phenomenology and to shed light on non-SM processes.
The light quark contribution, the pion-nucleon -term, is defined as
where denotes the pseudoscalar mass. Some time ago, employing dispersive analyses of pion-nucleon scattering data, the values Gasser:1990ce () MeV and Pavan:2001wz () MeV were obtained while a calculation in the framework of heavy baryon chiral perturbation theory resulted in Borasoy:1995zx () MeV. A recent covariant baryon chiral perturbation theory (BPT) analysis of the available pion-nucleon scattering data Alarcon:2011zs () resulted in the value MeV. Even less is known about the strangeness contribution . Since no elastic Higgs-nucleon scattering experiments exist all phenomenological estimates largely depend on modelling. Therefore input from lattice simulations is urgently required.
We have witnessed an upsurge of interest in calculating flavour singlet matrix elements recently, either directly Babich:2010at (); Doi:2010nw (); Alexandrou:2011ar (); Engelhardt:2010zr (); Takeda:2010cw (), by calculating the corresponding quark line connected and disconnected terms, or indirectly, via the Feynman-Hellmann theorem Durr:2011mp (); roger (); Takeda:2010cw (); Horsley:2010th (); Toussaint:2009pz (); Young:2009zb (); Young:2009ps (). High statistics simulations including light sea quarks mean that reasonable signals can be obtained for disconnected terms. Similarly, the small statistical uncertainty on baryon masses as functions of the quark masses enable reasonable fits to be made. Ideally, the results of both approaches should agree.
Preliminary results of this study were presented at past Lattice conferences Bali:2009dz (); Collins:2010gr (). This article is organised as follows. In Sec. II we detail the gauge configurations, simulation parameters and methods used. In Sec. III we then explain how the lattice results are renormalized and finally we present our results in Sec. IV, before we summarize.
Ii 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, we obtain sternbeck () the value fm for the Sommer scale, in the physical limit. This results in the lattice spacing
where the errors are statistical and from the scale setting, respectively. The ratio is obtained by chirally extrapolating the QCDSF simulation points najjar (). An extrapolation of the axial Takahashi-Ward identity (AWI) mass yields the critical hopping parameter value
In addition to , we realize the valence -values, and . The corresponding three pseudoscalar masses read
The strange quark mass was fixed so that the above value for 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. We analyse 2024 thermalized trajectories on each of the volumes. To effectively eliminate autocorrelations for the observables that we are interested in, bin sizes of eight are found to be sufficient.
The matrix element is extracted from the ratio of three-point functions, see Fig. 1, to two-point functions at zero momentum. Defining
we can write the disconnected part as
where is the lattice Dirac operator for the quark flavour of the current and denotes the two-point function of the zero momentum projected proton connecting the source time with . Note that, unlike its expectation value, computed on one configuration will in general also have an imaginary part. This means that we can reduce the variance of the above expression by explicitly setting this to zero, using the relation that follows from the -Hermiticity, .
In the limit of large times, ,
where the term only contributes for . Quark field smearing (see below) at the source and the sink significantly enhances the coupling of the nucleon creation and destruction operators to the physical ground state. Still, the time distances between the source and the current insertion as well as between the current and the sink need to be taken sufficiently large to suppress excited state contributions.
We find the nucleon smeared-smeared effective masses to be constant for . It suffices if excited state effects are much smaller than the statistical errors. These errors however are expected to be substantially larger for the disconnected three-point function than for the nucleon two-point function. Thus, we set the time of the current insertion to a smaller value fm. The method that we apply requires us to fix , but can be varied. If the data were not constant for then this would have implied that our choice of was too ambitious. Fortunately, employing sink and source smearing, we find the asymptotic limit to be effectively reached for and compute the matrix elements by fitting the above ratios for fm to constants.
As an example, in Fig. 2 we display the disconnected ratio for strange valence and current quark masses as a function of for smeared-smeared three- over two-point functions for lattices, together with this fit result. In addition we show the corresponding smeared-point ratio that converges towards the same value, giving us additional confidence that was chosen sufficiently large to warrant ground state dominance within our statistical errors.
to quark fields , where we set and use 400 iterations. We replace the spatial links above by APE-smeared Falcioni:1984ei () links
where . denotes a projection operator into the group and the sum is over the four spatial “staples”, surrounding . We employ 25 such gauge covariant smearing iterations and use the weight factor . For the projector we somewhat deviate from Ref. Bali:2005fu () and maximize , iterating over subgroups. The connected part, for which the statistical accuracy is less of an issue, is obtained with a less effective smearing at the larger, fixed value , varying .
We stochastically estimate . For this purpose we employ complex noise vectors, , , whose spacetime spin colour components carry uncorrelated random phases at the time and are set to zero elsewhere, to reduce the noise (partitioning Bernardson:1993yg ()).
Solving the linear systems
for we can then substitute,
The inner product is only taken over three-space, spin and colour indices. In the case of the scalar matrix element it is relatively easy to push the stochastic error below the level of the inherent error from fluctuations between gauge configurations111Note that both error sources will scale in proportion to . Bali:2009hu (). Therefore, here we do not need to employ the Truncated Solver Method (TSM) Bali:2009hu () and do not exploit the hopping parameter expansion. Instead, to reduce the dominant gauge error, we compute the nucleon two-point functions for four equidistant source times on each gauge configuration. In addition we exploit backwardly propagating nucleons, replacing the positive parity projector by within the nucleon two-point function, . Consequently, the noise vectors are seeded on 8 time slices simultaneously, reducing the degree of time partitioning. We find this not to have any adverse effect on the stochastic error. In addition to the 48 (4 timeslices times 4 spinor components times 3 colours) point-to-all sources necessary to compute the two-point functions we solve for noise vectors per configuration and current quark mass.
In the continuum, for light quark flavours , the -terms are invariant under renormalization group transformations. However, Wilson fermions explicitly break chiral symmetry and this enables mixing not only with gluonic contributions but also with other quark flavours. Note that due to the use of a quenched strange quark, the renormalization of the corresponding matrix element is particularly large and needs to be studied carefully. A consistent improvement of the quark scalar matrix elements requires the inclusion of the gluonic operator . We have not measured this as yet. Therefore we will neither include any improvement of the renormalization constants nor of the scalar current. However, we will account for the mixing between quark flavours.
We follow the procedure outlined in Ref. Gockeler:2004rp (), see also Sec. 6 of Ref. Bhattacharya:1997ht (). The same result can be obtained by taking derivatives of the nucleon mass with respect to the sea quark masses Bhattacharya:2005rb () via the Feynman-Hellmann theorem. This also holds for the case of a partially quenched strange quark Babich:2010at (); roger (). In the renormalization the strangeness matrix element will receive subtractions not only from light quark disconnected but also from the numerically larger connected diagrams. This has first been pointed out in Ref. Michael:2001bv ().
For Wilson actions the vector Takahashi-Ward identity (VWI) lattice quark mass is given by
In the partially quenched theory one can define a as the -value at which the valence pseudoscalar mass vanishes.
We distinguish between singlet and non-singlet quark masses that will renormalize differently. In the case of the theory with () sea quarks, we have a mass term
where ; the lattice singlet quark mass is given by the average of the sea quark masses.
A renormalized quark mass for the flavour at a scale is given by
where singlet quark masses renormalize with a renormalization constant and non-singlet combinations with . Notice that the ratio is independent of the scale but depends on the lattice spacing , through the coupling .
In the partially quenched theory with mass degenerate sea quarks Eq. (21) results in
where we introduce a VWI valence quark mass through
At , vanishes so that we obtain
The non-singlet mass above can also be obtained from the bare AWI mass,
that renormalizes with the ratio of the renormalization constants over of the non-singlet axial and pseudoscalar currents. This results in the alternative prescription to Eq. (25),
where is calculated at . The required combination of scalar, pseudoscalar and axial non-isosinglet renormalization factors for our simulation with sea quarks at reads Gockeler:2010yr ()
The two methods that differ by terms of are illustrated in Fig. 3 for our configurations with mass degenerate sea quark flavours.
The results read
Corrections to these values at a non-zero quark mass will be of ; the differences between the two definitions are indicative of effects. We remark that is of leading order in the strong coupling parameter and it will therefore decrease with the lattice spacing .
For sea quarks Eq. (21) amounts to
where is defined in Eq. (25). This can be written as
where the lattice quark masses on the right hand side are defined by the VWI, Eq. (18). For clarity we have included the lattice spacing dependence above, which below we will drop again. The sum is invariant under renormalization group transformations222 Note that these scalar matrix elements are differences of scalar currents within the nucleon, relative to their vacuum expectation values. Therefore, unlike the chiral condensates alone, they do not undergo any additive renormalization and do not mix with an term.. Therefore the scalar lattice matrix elements will renormalize with the inverse matrix above: the different quark contributions will mix in the mass non-degenerate case.
We now turn to the situation of interest of light sea quarks, with a quenched strange quark. This means that the singlet mass will not depend on the strange quark mass anymore. (The superscript “lat” has been added for clarity.) Eq. (30) can now be written as
where the presence of the strange quark is not felt by the sea quarks. However, the definition of involves . Inverting the above matrix yields333Here and occasionally below we omit specifying the external state (in our case ) in cases where identities between hadronic matrix elements are independent of this state.
For the light quark matrix element (i.e. the -term) this means that
while for the strangeness matrix element we obtain
It is evident from Eq. (33) that the so-called -ratio renormalizes as follows,
As discussed in Sec. II above we employ three hopping parameter values, , and , that correspond to the pseudoscalar masses MeV, 450 MeV and 720 MeV, respectively, see Eqs. (II) – (10). We use all these -values for the valence quarks () as well as for the current insertions (). This amounts to nine combinations for the disconnected ratios while for the connected part .
We will explore the dependence of the lattice matrix elements on the current quark mass and investigate finite size effects, using a partial summation method. Subsequently, unrenormalized and renormalized valence and sea quark contributions are studied. Finally, we compute the light -term, the mass contributions and and the -ratio.
iv.1 Dependence of the lattice matrix elements on the current quark mass
From the heavy quark expansion it is evident that for . This can also be seen on the lattice where the leading non-vanishing contribution in the hopping parameter expansion is proportional to the plaquette. To confirm this expected saturation we compute the scalar matrix element for three additional current quark masses up to and above the charm quark mass on a subset of 576 lattices, using our smallest valence quark mass. We remark that for an improvement of the operator , mixing with needs to be considered. These improvement terms will become large if is big. In fact the renormalized becomes negative at about twice the strange quark mass if we neglect such effects. We have not investigated gluonic contributions as yet and therefore the behaviour at large masses is beyond the scope of the present study. For very small quark masses we would expect . Indeed, these expectations are confirmed for the unrenormalized , as can be seen in Fig. 4. Note that the third data point from the left corresponds to the strange quark mass while the charm quark can be found at the value . The arctan-curve is drawn to guide the eye.
iv.2 Finite size effects: partial sums
In order to investigate finite size effects we find it worthwhile to replace the numerator of Eq. (12) by a partial sum:
The subscript (connected part) indicates that we subtract the product of the two individual vacuum expectation values from the numerator. Since no zero momentum projection is performed at the source of the two-point function, that resides at the spatial position , the result will depend on the cut-off . We expect the summand at large to fall off exponentially in so that these values will eventually not contribute to the signal anymore but just increase the statistical noise. At very large spatial volumes one may therefore consider to perform such a partial sum only, thereby reducing the statistical error, and to estimate the induced bias by parameterizing the asymptotic fall-off.
We display the partial sums for the and lattices for the strangeness current at in Fig. 5. We do not detect any statistically significant dependence of the curves on the value of and show the results obtained at . At small we see the naively expected volume scaling. This becomes flatter around but only saturates to a constant once the boundary of the box is hit at . Increasing beyond this value means that in the case of this smaller lattice only the lattice “corners” are summed up. However, the data saturate at the same distance, rather than at , indicating that indeed the nucleon is well accommodated within this box size and that finite size effects are small.
iv.3 Sea and valence quark contributions
The results of the bare disconnected scalar matrix elements for the two volumes (and three quark masses ) are displayed in Table 1. For disconnected terms can differ not only from but also from the of the nucleon’s valence quarks. In Fig. 6 we display the dependence of the unrenormalized -values on the valence quark mass of the proton for both volumes, together with linear chiral extrapolations. The right-most data points correspond to the strange quark mass and the left-most points to the present sea quark mass. The volume dependence is not significant and neither are the differences between the values obtained at the smallest mass point and the chirally extrapolated numbers. The results need to be renormalized and this is possible at .
To enable the calculation of the light -term and the renormalization of the strangeness matrix element, we also compute the connected contribution for , using the traditional sequential propagator method. We obtain and 8.35(43) for the and lattices, respectively. This means that at the pseudoscalar mass MeV the relative contribution of the disconnected matrix element reads
The unrenormalized values of Table 1 seem to be fairly independent of the current quark mass. The ratio of the strangeness matrix element over a light sea quark contribution undergoes the renormalization
The renormalization of the numerator is obtained from Eq. (33) while the denominator can be split into two parts that renormalize with and with , respectively: . The flavour symmetry of the unrenormalized sea obviously cannot disappear when subtracting the same (large) terms from the numerator and the denominator. However, this subtraction results in large statistical uncertainties. For instance on the large volume the above ratio reads with an additional systematic uncertainty of from the value of the renormalization parameter , Eq. (29). We conclude that the renormalized sea is symmetric within a factor of about five.
The disconnected fraction of Eq. (38) will undergo the renormalization
The value quoted is obtained on the volume using the VWI prescription, with a systematic error that incorporates the difference between the two determinations of the renormalization constant ratios Eq. (29) and their respective uncertainties. For the renormalized -parameter that is defined in Eq. (III) this implies that
where the approximation holds within a factor of five, see also Eq. (47) below.
iv.4 The light and strange -terms
Combining all information results in the renormalized values for the pion-nucleon -term at the simulated sea quark mass
at the two different volumes, where the second error is due to the uncertainty of the chirally extrapolated -value. The above -term can also be obtained from the derivative of the nucleon mass with respect to the logarithm of the light quark mass. Using the fact that at small , , we can write
To leading order in chiral perturbation theory . This linear assumption suggests to multiply the result Eq. (42) with the ratio to obtain the physical -term MeV. In general, however, higher order corrections will lead to some curvature.
Fortunately, we do not only know the -term at but also the nucleon mass sternbeck () at other values of , at () and at (), see Table 2. A combined covariant BPT Becher:2001hv () fit to these data within the window , imposing our directly obtained value of as an additional constraint, results in the preliminary number sternbeck ()
at the physical point. The error includes both the statistical uncertainty of the fit and the systematics from varying the low energy parameters , and within their phenomenologically allowed ranges Becher:2001hv (); Frink:2004ic (); flag (). A detailed analysis will be presented in Ref. sternbeck (). We display the result of this extrapolation in Fig. 7 for the ratio in units of , together with our direct determinations. The broad error band indicates the result of the same fit, without using our constraint at MeV.
We now use Eq. (35) with given in Eq. (29) to obtain the renormalized strangeness matrix element from the values given above. This amounts to subtracting numbers of similar sizes from each other. There is no noticeable finite size effect between the and volumes. For our simulation point at a low pion mass MeV we obtain the values, and 0.008(6) for the two determinations of the renormalization parameter from the VWI and AWI, respectively.
Of particular phenomenological interest is the dimensionless strange quark contribution to the nucleon mass
Again, we quote the value obtained from the VWI prescription, with a systematic error that incorporates the difference between the two determinations of the renormalization constant ratios and their respective uncertainties. This may be indicative of effects. The problem of large cancellations cannot be overcome easily. One needs to get closer to the continuum limit so that approaches zero. For instance, at , Gockeler:2004rp (), significantly reducing the subtraction of the connected diagram (and probably the value of that will contain a smaller light quark contribution).
The result obtained is interesting insofar as it suggests a scalar strangeness of less than of the nucleon mass, MeV. In spite of the relative enhancement by the ratio this is not bigger than the pion-nucleon -term above. This is quite consistent with the finding of Eq. (40) of a tiny renormalized light sea quark participation in . We remark that taking the combination without the proper subtraction would have resulted in , even bigger than the light quark mass contribution of about 0.09, at our light quark mass value that exceeds the physical one by a factor of about four. Neglecting the mixing with light quarks in the renormalization is probably the main reason why this contribution was overestimated in the pioneering lattice studies, see e.g. Ref. Gusken:1999te () and references therein. Early results are also summarized in Ref. Young:2009ps ()
We can constrain the scale-independent -ratio of Eq. (III),
where the errors are statistical and the difference between the two determinations of , respectively. Again, as the central value, we have taken the result from the VWI renormalization factor. From our determination of the pion-nucleon -term we know that the denominator of Eq. (III) will increase by a factor 1.4–1.5 when extrapolated to the chiral limit. Based on the weak observed dependence of on the valence quark mass, see Fig. 6, we would expect the numerator to exhibit a less pronounced quark mass dependence. Thus a confidence-level upper limit on the -parameter should also apply at physically light sea quark masses.
Finally, we also predict the gluonic and heavy sea quark contribution of Eq. (4),
This means that the light and strange quark flavours contribute a fraction between 3 % and 8 % to the nucleon mass.
We directly calculate the light quark and strangeness -terms on lattices with spatial extents up to , a lattice spacing GeV and a pseudoscalar mass MeV. At this mass point and lattice spacing the quark line disconnected contribution amounts to a fraction of of the full result. After renormalization however we find this number to drop below the 5 % level, see Eq. (40).
At our fixed mass point we obtain the renormalized values, MeV and MeV, for the strangeness and light quark -terms. Assuming the latter value to depend linearly on , as predicted by leading order chiral perturbation theory, this corresponds to at the physical point. However, from nucleon mass data obtained at pseudoscalar masses, , it is clear that there exists a non-vanishing curvature. Our direct determination can be used to strongly constrain an covariant baryon chiral perturbation theory extrapolation of the nucleon mass sternbeck (). The combined fit yields the preliminary value at the physical point. Without the direct information on the slope at one point the statistical and systematic uncertainties would have been much larger, see Fig. 7. It would be difficult to significantly reduce this large error, without nucleon mass data at physical and, possibly, smaller than physical quark masses.
We are also able to exclude values of the -parameter, with a confidence level of 95 %. This means that the strangeness contribution to the scalar coupling of the nucleon is much smaller than that due to the light quark -term. To determine the strangeness -term using the indirect method, i.e. the Feynman-Hellmann theorem, requires sea quark flavours. Even then the dependence of the nucleon mass on the strange quark mass will be very weak and the tiny slope (and its error) will be amplified by the mass ratio . Therefore, for an accurate prediction of , and in particular for a non-vanishing lower bound on its value, an additional direct determination at one or a few mass points will be crucial.
We remark that the results presented here have not been extrapolated to the continuum limit. Neither has the effect of quenching the strange quark been addressed, except within the renormalization of the strangeness -term.
Acknowledgements.We thank Peter Bruns and Ludwig Greil for discussion. This work was supported by the European Union under Grant Agreement number 238353 (ITN STRONGnet) and by the Deutsche Forschungsgemeinschaft SFB/Transregio 55. Sara Collins acknowledges support from the Claussen-Simon-Foundation (Stifterverband für die Deutsche Wissenschaft). André Sternbeck was supported by the EU International Reintegration Grant (IRG) 256594. James Zanotti was supported by the Australian Research Council under grant FT100100005 and previously by the Science & Technology Facilities Council under grant ST/F009658/1. 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. We thank the support staffs of these institutions. The Chroma software suite Edwards:2004sx () was used extensively in this work and gauge configurations were generated using the BQCD code Nakamura:2010qh () on QPACE and BlueGenes.
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