Strange quark contributions to nucleon mass and spin from lattice QCD
aa r X i v : . [ h e p - l a t ] D ec Strange quark contributions to nucleonmass and spin from lattice QCD
M. Engelhardt
Department of Physics, New Mexico State University,Las Cruces, NM 88003, USA
March 30, 2018
Abstract
Contributions of strange quarks to the mass and spin of the nucleon, charac-terized by the observables f T s and ∆ s , respectively, are investigated within latticeQCD. The calculation employs a 2+1-flavor mixed-action lattice scheme, thustreating the strange quark degrees of freedom in dynamical fashion. Numericalresults are obtained at three pion masses, m π = 495 MeV, 356 MeV, and 293 MeV,renormalized, and chirally extrapolated to the physical pion mass. The value ex-tracted for ∆ s at the physical pion mass in the M S scheme at a scale of 2 GeVis ∆ s = − . f T s = 0 . f T s and ∆ s are realized as domain wall fermions, propagators ofwhich are evaluated in MILC 2+1-flavor dynamical asqtad quark ensembles. Theuse of domain wall fermions leads to mild renormalization behavior which provesespecially advantageous in the extraction of f T s . PACS: 12.38.Gc, 14.20.Dh 1
Introduction
Strange quarks represent the lightest quark flavor not present in the valence componentof the nucleon. Their study can thus provide insight into sea quark effects in the nu-cleon in isolated fashion. The two most fundamental properties of the nucleon are itsmass and spin. The investigation presented here focuses on the extent to which thosetwo properties are influenced by the strange quark degrees of freedom. The strangecontributions to nucleon mass and spin can be characterized by the matrix elements f T s = m s m N (cid:20) h N | Z d y ¯ ss | N i − h | Z d y ¯ ss | i (cid:21) (1)and ∆ s = h N, j | Z d y ¯ sγ j γ s | N, j i (2)respectively, where | N, j i denotes a nucleon state with spin polarized in the j -direction.In the case of the scalar matrix element, the vacuum expectation value, i.e., the vacuumstrange scalar condensate, is subtracted; the intention is, of course, to measure thestrangeness content of the nucleon relative to the vacuum. In the case of the axial matrixelement, no subtraction is necessary since the corresponding vacuum expectation valuevanishes. Note that ∆ s measures specifically the contribution of strange quark spin tonucleon spin; strange quark angular momentum constitutes a separate contribution notconsidered here.Aside from representing a fundamental characteristic of the nucleon in its own right,the scalar strange content f T s is also an important parameter in the context of darkmatter searches [1–4]. Assuming that the coupling of dark matter to baryonic matteris mediated by the Higgs field, the dark matter detection rate depends sensitively onthe quark scalar matrix elements in the nucleon, cf., e.g, the neutralino-nucleon scalarcross-section considered in [1]. One a priori reasonable scenario is that the strange quarkfurnishes a particularly favorable channel [1], since, on the one hand, it features a muchlarger Yukawa coupling to the Higgs field than the light quarks, and, on the other hand,is not so heavy as to be only negligibly represented in the nucleon’s sea quark content.As a consequence, an accurate estimate of f T s is instrumental in assessing the discoverypotential for dark matter candidates.The contribution of strange quark spin to nucleon spin ∆ s is, in principle, more di-rectly accessible to experiment than f T s . ∆ s represents the first moment of the strangequark helicity distribution ∆ s ( x ) (including both quarks and antiquarks) as a functionof the momentum fraction x . The helicity distribution can be determined via inclusivedeep inelastic scattering and semi-inclusive deep inelastic scattering [5–7]. However, itsextraction in practice still has to rely to a certain extent on assumptions about thedependence of ∆ s ( x ) on x , even in the semi-inclusive channels (which furnish directinformation on ∆ s ( x )), because of the limitations in accessing small x experimentally.Complementary information about ∆ s is obtained from the strange axial form factor of2he nucleon G sA ( Q ), which can be extracted by combining data from parity-violatingelastic electron-proton scattering and elastic neutrino-proton scattering [8]. Extrapola-tion to zero momentum transfer, Q = 0, again yields an estimate of ∆ s . Dependingon the specific extrapolations and/or model assumptions adopted in determining ∆ s via the various aforementioned avenues, both significantly negative values for ∆ s havebeen put forward [5, 9], as well as values compatible with zero [6]. An independentdetermination of ∆ s via lattice QCD, as undertaken in the present work, thus can beuseful in several ways. Apart from shedding light on the fundamental question of thedecomposition of nucleon spin, it can contribute constraints to phenomenological fits ofpolarized parton distribution functions. Furthermore, it influences spin-dependent darkmatter cross sections [3]; although more accurate determinations of the scalar matrixelements discussed further above constitute the most urgent issue in reducing hadronicuncertainties in dark matter searches, ∆ s also plays a significant role in that context.A number of lattice QCD investigations of strange quark degrees of freedom in thenucleon have recently been undertaken [10–26], the majority of which have focusedspecifically on the scalar content. Studies of the latter have proceeded via two avenues:On the one hand, one can directly determine the matrix element h N | ¯ ss | N i via theappropriate disconnected three-point function; this methodology was adopted in [10,12, 14–17, 19, 22, 24] and also in the present work, as described in detail further below. Astudy of techniques suited to improve the efficiency of this approach has been presentedin [27]. On the other hand, a somewhat less direct inference of the scalar strange quarkcontent of the nucleon is possible via the study of the baryon spectrum, which is relatedvia the Feynman-Hellmann theorem h B | ¯ qq | B i = ∂m B ∂m q (3)to the corresponding sigma terms for the baryon state | B i and quark flavor q . Thisavenue has been pursued in [13, 15, 21, 23], and a related methodology, combining latticehadron spectrum data with chiral perturbation theory, was pursued in [25, 26].The characteristics of these various investigations of the scalar strange quark contentof the nucleon are diverse. They include N f = 2 calculations, in which the strange quarkdegrees of freedom are quenched [12–14,24], but also N f = 2+1 [10,15–17,19,21,23,25,26]and even N f = 2 + 1 + 1 [17,22] calculations. In some cases, lattice data at only one pionmass have been obtained to date and no extrapolation to the physical point has beenattempted. The most stringent results obtained at the physical point including fullydynamical strange quarks were reported in [15, 17, 26]. Ref. [17] quotes m N f T s /m s =0 . N f = 2 + 1 case, and m N f T s /m s = 0 . N f = 2 + 1 + 1case; translated to f T s itself using m s = 95 MeV in the M S scheme at a scale of 2 GeV[28], these correspond to f T s = 0 . N f = 2 + 1 and f T s = 0 . N f = 2+1+1. On the other hand, [15] and [26] report significantly lower values. In [15], Strictly speaking, the method adopted in [16, 17] constitutes a hybrid of the two methods. f T s = 0 . m N f T s = 21(6) MeV, which translates to f T s = 0 . f T s = 0 . s , on the other hand, has also been investigatedin [10,11,24]. Apart from the exploratory study [10], which, however, did not attempt torenormalize the results nor extrapolate them to the physical pion mass, these investiga-tions were based on dynamical quark ensembles containing only the two light flavors inthe quark sea; the present lattice investigation, on the other hand, employs N f = 2 + 1gauge ensembles, thus treating the strange quark degrees of freedom in dynamical fash-ion. The numerical results for ∆ s obtained on this basis are renormalized and chirallyextrapolated, yielding the estimate ∆ s = − . M S scheme at a scale of 2 GeV. Within the uncertainties, this nevertheless remainscompatible with the values obtained in the aforementioned other studies, though it isabout 50% larger in magnitude. This suggests that systematic adjustments to the resultsquoted in those works through unquenching of the strange quark degrees of freedom,renormalization, and chiral extrapolation are not severe.Aside from the two quantities f T s and ∆ s considered in the present work, latticeinvestigations of the strange quark structure within the nucleon have also consideredgeneralizations to non-zero momentum transfer, i.e., form factors [18, 24], includingcalculations of the strange electric and magnetic form factors, which are of interest inthe context of corresponding experimental efforts employing parity-violating electron-proton scattering [29]. Furthermore, also the strange quark momentum fraction andstrange quark angular momenta in the nucleon have been investigated [20].The present lattice investigation, a preliminary account of which was given in [30], isbased on a mixed-action scheme developed and employed extensively by the LHP Collab-oration [31–33]. The nucleon valence quarks as well as the strange quark fields appearingin the operator insertions in eqs. (1) and (2) are realized as domain wall fermions, propa-gators of which are evaluated in the background of (HYP-smeared) 2+1-flavor dynamicalasqtad quark ensembles provided by the MILC Collaboration. Though computationallyexpensive, domain wall fermions lead to benign renormalization properties which proveespecially advantageous in the case of the f T s observable; the substantial systematic un-certainties due to strong additive quark mass renormalizations encountered in analogouscalculations using Wilson fermions [12, 24] are avoided. The computational scheme isdescribed in detail in section 2. Section 3 provides the raw numerical results, the renor-malization of which is discussed in section 4. The renormalized results are extrapolatedto the physical pion mass in section 5, and systematic uncertainties and adjustmentsare considered in section 6, with conclusions presented in section 7.4 Computational scheme
The lattice calculation of the nucleon matrix elements in (1), (2) proceeds in standardfashion via correlator ratios of the type R [ Γ nuc , Γ obs ]( τ, T ) = (cid:10) (cid:2) Γ nucαβ Σ ~x N β ( ~x, T ) ¯ N α (0 , (cid:3) · (cid:2) − Γ obsγρ Σ ~y s ρ ( ~y, τ )¯ s γ ( ~y, τ ) (cid:3) (cid:11)D Γ unpolαβ Σ ~x N β ( ~x, T ) ¯ N α (0 , E (4)with nucleon interpolating fields N of the standard form (quoting here, for definitenessin flavor structure, the proton case) N γ = ǫ abc u cγ u aα ( Cγ ) αβ d bβ (5)where C denotes the charge conjugation matrix. The sums over spatial position ~x projectthe nucleon states onto zero momentum, whereas the sum over spatial position ~y issimply transcribed from (1), (2). Since the nucleon contains no strange valence quarks,the three-point function averaged in the numerator of (4) factorizes, as written, intothe nucleon two-point function and the strange quark loop. I.e., only the disconnecteddiagram, cf. Fig. 1, contributes to the matrix elements under consideration. The strange , T N t = t =t = tG , 0, N Figure 1: Disconnected contribution to nucleon matrix elements. The nucleon propa-gates between a source at t = 0 and a sink at t = T ; the insertion of Γ ≡ Γ obs occurs atan intermediate time t = τ .quark fields have already been reordered in the numerator of (4) such as to make thestandard minus sign associated with quark loops explicit. Finally, the Γ matrices allowone to choose the appropriate nucleon polarization and strange quark operator insertionstructures. The denominator of (4) corresponds to the unpolarized nucleon two-pointfunction, obtained using Γ unpol = 1 + γ . (6)5n the numerator, for the purpose of evaluating f T s , unpolarized nucleon states areappropriate, corresponding to the choice Γ nuc = Γ unpol ; furthermore, the scalar strangequark insertion is obtained by choosing Γ obs = 1, m s m N (cid:0) R [ Γ unpol , τ, T ) − [VEV] (cid:1) ≡ R { f T s } T ≫ τ ≫ −→ f T s (7)with the vacuum expectation value[VEV] = h− Σ ~y s γ ( ~y, τ )¯ s γ ( ~y, τ ) i (8)to be subtracted.On the other hand, ∆ s is obtained by using the projector onto nucleon states polar-ized in the positive/negative j -direction in the numerator of (4),Γ nuc = 1 ∓ iγ j γ unpol , (9)as well as the operator insertion structure Γ obs = γ j γ . Averaging over positive/negative j -direction (with a relative minus sign) as well as the three spatial j to improve statisticsleads one to evaluate − i · · X j =1 R [ ( − iγ j γ /
2) Γ unpol , γ j γ ]( τ, T ) ≡ R { ∆ s } T ≫ τ ≫ −→ ∆ s , (10)where the prefactor 2 compensates for the fact that the ratio (4) is normalized using theunpolarized nucleon two-point function in the denominator, even when the numerator isrestricted to a particular polarization. Lastly, the prefactor ( − i ) cancels the additionalfactor i which the γ -matrix in the operator insertion acquires when the calculation iscast in terms of Euclidean lattice correlators; thus, the ∆ s obtained through (10) isalready Wick-rotated back to Minkowski space-time. Note that (10) does not call forthe subtraction of a vacuum expectation value, since the latter vanishes in the ensembleaverage. In practice, this numerical zero was nevertheless subtracted from (10) in orderto reduce statistical fluctuations. The averages in the correlator ratios (4) were carried out using the three MILC 2 + 1-flavor dynamical asqtad quark ensembles listed in Table 1. HYP-smearing was applied tothe configurations. Both the valence quarks in the nucleon two-point functions and thestrange quark fields appearing in the matrix elements (1) and (2) were implemented usingthe domain wall fermion discretization, with parameters L = 16, M = 1 .
7. The domainwall quark masses, also listed in Table 1, are fixed by the requirement of reproducingthe pion masses corresponding to the MILC ensembles [31]. This mixed action setup has6 m asql am asqs am DW Fl am DW Fs m π m N N f = 2 + 1, 20 ×
64 MILC asqtad ensembles with lattice spacing a = 0 .
124 fmused in the present investigation. Uncertainties in the pion and nucleon masses extractedfrom the corresponding two-point functions are under 1%.been employed extensively for studies of hadron structure by LHPC [31, 33] and furtherdetails concerning its tuning can be found in the mentioned references.The space-time layout of the calculation is shown in Fig. 2. The strange quark looptrace was evaluated using stochastic estimation. Positioning the nucleon source at time t = 0, bulk complex Z (2) stochastic sources with support in all of space within the tem-poral range t = 3 , . . . , τ over the aforementioned timeslices (after having duly divided out the length of the temporal range). The stochasticestimate of the strange quark loop trace was performed employing 1200 of the describedstochastic source vectors per gauge sample. In particular, obtaining a signal for ∆ s depends on accumulating high statistics in the stochastic estimator. The scalar matrixelement, on the other hand, requires less statistics in the strange quark loop trace, but ismore susceptible to gauge fluctuations. The sink time T at which the nucleon two-pointfunctions are contracted and projected onto zero momentum remains variable in thislayout.While this scheme provides a high amount of averaging in relation to the numberof strange quark propagator inversions, it precludes testing for a plateau in the three-point function by varying the operator insertion time τ . The positioning of the bulkstochastic source was motivated by previous work [31, 33] in the same mixed-actionscheme, which investigated (connected contributions to) a wide variety of observables,and the results of which indicate that, having evolved in time from t = 0 to t = 3,excited state contaminations in the system are already small compared to the statisticaluncertainty of the calculation performed here. To achieve this suppression of excitedstate contributions, the quark fields in the nucleon sources (5) are Wuppertal-smearedsuch as to optimize the overlap with the nucleon ground state [31–33]. Despite therelative positioning of the nucleon source and the strange quark insertion time rangebeing fixed in this manner, it does nevertheless remain possible to exhibit the extentto which results depend on the separation between operator insertion and nucleon sinktime T , which is still variable in the present scheme. Correlator ratios will be shownfurther below as a function of T , with asymptotic behavior being seen to emerge threelattice time steps beyond the end of the operator insertion range, i.e., for sink times7 Figure 2: Setup of the lattice calculation. The nucleon source is located at lattice time t = 0. An average over operator insertion times τ is performed in the range τ = 3 , . . . , T of the nucleon sink is variable. T ≥
10. This corroborates the suitability of the choice for the bulk stochastic sourcetime range adopted in the present calculation.The correlator ratio (4) exhibits statistical fluctuations not only due to the strangequark loop factor discussed above, but also due to the nucleon two-point function factors.To reduce these fluctuations, it is equally necessary to sample the latter to a sufficientextent. To this end, multiple samples of the nucleon two-point function were obtainedby employing eight different spatial positions for the nucleon source on the source timeslice. In addition to this eight-fold sampling of the nucleon two-point function, the de-scribed scheme can be accommodated several times in temporally well-separated regionson the lattice; in practice, three replicas of the entire setup specified above were placedon the lattice, separated by 16 lattice spacings in the time direction, thus further en-hancing statistics. Each gauge configuration therefore yielded altogether 24 samples ofthe numerator and denominator in the correlator ratio (4). Note again that averagingwas further improved in the case of ∆ s by taking into account nucleon polarization axesaligned with all three coordinate axes, as already made explicit in (10). The numerical results for the correlator ratios R { f T s } and R { ∆ s } , cf. (7) and (10),averaged over the insertion time τ as described in section 2.2, are shown in Figs. 3 and4 as a function of sink time T . The correlator ratios start out near vanishing values atsmall sink times T , and then accumulate strength as T traverses the region of supportof the stochastic strange quark source. Beyond this region, R { f T s } and R { ∆ s } begin tolevel off and approach their asymptotic values f T s and ∆ s , respectively. There is only8 limited time window in which the latter behavior can be observed due to the increasein statistical fluctuations for large T . m π R { ∆ s }| T =10 R { ∆ s }| T =10 ,...,
293 MeV -0.023(17) -0.023(25)356 MeV -0.030(9) -0.030(12)495 MeV -0.021(4) -0.022(5)Table 2: Correlator ratio R { ∆ s } at sink time T = 10, and averaged over T = 10 , . . . , R { ∆ s } at neighboring sink times are taken into account.The correlator ratio R { ∆ s } consistently behaves in line with the expectation ad-vanced in section 2.2, namely, there is no significant trend in the correlator ratio beyond T = 10. This is quantified in Table 2, which compares the value of R { ∆ s } at T = 10with its T = 10 , . . . ,
12 average.The behavior of the lattice data for the correlator ratio R { f T s } is not as smooth as inthe case of R { ∆ s } . In particular, in the m π = 356 MeV correlator ratio, one notices anenhancement in the T = 12 , . . . ,
14 region compared with the value at T = 10. On theother hand, both in the m π = 495 MeV and the m π = 293 MeV data, plateaux appearto be reached at T = 10. Table 3 again provides a comparison of the T = 10 value of R { f T s } with its T = 10 , . . . ,
12 average; in the case of m π = 356 MeV, the results arebarely compatible within the statistical uncertainties. Nevertheless, an interpretationof this behavior as a statistical fluctuation, as opposed to a systematic effect, seemsmost plausible: The direction of the deviations from the expected plateau behaviorin the correlator ratio beyond T = 10 is not consistent across the data sets, with R { f T s } decreasing slightly for m π = 495 MeV, while it rises in the other two cases.Furthermore, there is no clear trend as a function of pion mass, with the largest upwarddeviation occurring for the middle pion mass, m π = 356 MeV, while at m π = 293 MeVthe deviation is again quite small.In summary, thus, at the level of statistical uncertainty achieved in the presentcalculation, no systematic excited state effects stemming from a too restricted source-sink separation can be reliably extracted; or, in other words, such effects are smallerthan the aforementioned statistical uncertainties. In view of this, in the following, the T = 10 values of the correlator ratios R { ∆ s } and R { f T s } , as reported in Tables 2 and 3,will be regarded as the most reliable estimates for their asymptotic limits ∆ s and f T s ,respectively. Systematic uncertainties, including the ones due to excited states, will berevisited and discussed in detail in section 6.9 f T T s {} R m 495 MeV= p -0.02 0 0.02 0.04 0.06 0.08 0.1 0 2 4 6 8 10 12 14 f T T s {} R m 356 MeV= p -0.02 0 0.02 0.04 0.06 0.08 0.1 0 2 4 6 8 10 12 14 f T T s {} R m 293 MeV= p Figure 3: Correlator ratio R { f T s } , cf. (7), averaged over insertion time τ as describedin section 2.2, as a function of sink time T , for the three pion masses considered.10 D T s {} R m 495 MeV= p -0.1-0.08-0.06-0.04-0.02 0 0.02 0.04 0 2 4 6 8 10 12 14 D T s {} R m 356 MeV= p -0.1-0.08-0.06-0.04-0.02 0 0.02 0.04 0 2 4 6 8 10 12 14 D T s {} R m 293 MeV= p Figure 4: Correlator ratio R { ∆ s } , cf. (10), averaged over insertion time τ as describedin section 2.2, as a function of sink time T , for the three pion masses considered.11 π R { f T s }| T =10 R { f T s }| T =10 ,...,
293 MeV 0.060(10) 0.064(14)356 MeV 0.043(6) 0.051(8)495 MeV 0.046(3) 0.045(4)Table 3: Correlator ratio R { f T s } at sink time T = 10, and averaged over T = 10 , . . . , R { f T s } at neighboring sink times are taken into account. To establish a connection with phenomenology, quantities measured on the lattice needto be related to their counterparts in a standard renormalization scheme such as the
M S scheme at a scale of 2 GeV. An advantage of the domain wall fermion discretization, usedin the present treatment to represent the strange quark fields entering the quark bilinearoperator insertions in the matrix elements (1) and (2), lies in its good chiral symmetryproperties, which lead to a mild renormalization behavior. As a result, even thoughnot all elements necessary for a complete renormalization of the quantities consideredare available (e.g., matrix elements of gluonic operators in the nucleon, with which theflavor singlet parts of the strange quark bilinears may mix), it is possible to estimate theassociated uncertainties in the renormalization, indicating that these do not dominateover the statistical uncertainties of the calculation.
Consider first the renormalization of f T s , specifically, the proposition that the combi-nation m s h N | ¯ ss | N i can be treated as independent of the renormalization scheme andscale, m s h N | ¯ ss | N i| renorm ? = m s h N | ¯ ss | N i| bare . (11)As has been emphasized previously in [14], this behavior is contingent upon chiral sym-metry being maintained in the lattice evaluation of the bare quantities. In general,the strange scalar operator ¯ ss can mix both with light quark and gluonic operators;decomposing it into flavor singlet and octet components,¯ ss = 13 h ¯ qq − √ qλ q i (12)(where q ∈ { u, d, s } ), the flavor singlet part can mix with the relevant gluonic operator,(¯ qq ) renorm = Z S (¯ qq ) bare + Z GS ( F ) bare (13)12nd also the renormalization constant Z S of the octet part,(¯ qλ q ) renorm = Z S (¯ qλ q ) bare (14)in general is not identical to Z S . However, these mixing effects would have to proceed[14] via diagrams in which the ¯ qq insertion occurs within a disconnected quark loop;furthermore, since ¯ qq = ¯ q L q R + ¯ q R q L , a chirality flip would have to take place withinthat loop. Gluon vertices coupling to the loop cannot effect such a chirality flip. Thus,if all explicit chiral symmetry breaking effects are excluded, both in the dynamics aswell as by adopting a mass-independent renormalization scheme, the gluonic admixturein (13) is avoided. By extension [14], also no mixing of ¯ ss with ¯ uu + ¯ dd can take place,i.e., the renormalization factors of the flavor singlet and octet parts in (12) are equalunder the assumption of strict chiral symmetry. Then, ¯ ss renormalizes in a purelymultiplicative manner.Complementarily, also the strange quark mass m s only renormalizes multiplicatively, m renorm s = Z m m bare s , when chiral symmetry is maintained. For generic non-chiral latticefermion discretizations, the renormalized strange quark mass m renorm s acquires strong ad-ditive contributions supplementing the aforementioned multiplicative renormalization,invalidating the behavior (11). This is again avoided when chiral symmetry is respected.Indeed, when renormalization is purely multiplicative, in view of the Feynman-Hellmanntheorem, m s h N | ¯ ss | N i = m s ∂m N ∂m s , (15)factors Z m acquired by the strange quark mass under renormalization cancel on theright-hand side of (15), implying that also the left-hand side is invariant.Chiral symmetry of the lattice fermion discretization is therefore instrumental inestablishing a simple renormalization behavior of f T s , and, indeed, is realized to a goodapproximation in the present treatment due to the use of domain wall fermions in thecoupling to the operator ¯ ss . Whereas exact chiral symmetry would only be achievedusing an infinite fifth dimension in the domain wall construction, in practice, the extentof this dimension has been chosen sufficiently large as to render the residual mass m res quantifying the violation of chiral symmetry an order of magnitude smaller than thelight quark mass in all ensembles considered [31]. One must, however, be careful inassessing the significance of this smallness, since, on the other hand, the light quark andgluonic operators with which ¯ ss mixes may have nucleon matrix elements much largerthan h N | ¯ ss | N i ; for h N | ¯ uu + ¯ dd | N i , this is certainly the case. This can potentially offsetthe suppression of m res . One may speculate that mixing with the operator ¯ uu + ¯ dd constitutes the strongest effect, since the light quark fields are special in that theyform the valence component of the nucleon, which has no counterpart in the vacuumexpectation value that is subtracted off throughout, cf. (1). On the other hand, thepresence of the valence quarks also strongly distorts the gluon field in the nucleon. Noestimate of the gluonic admixture to ¯ ss is available, but the light quark admixture under13enormalization will be argued below to constitute an effect of the order of 1%. In viewof the statistical uncertainties associated with the present determination of f T s , whichamount to about 20%, a putative gluonic mixing effect would have to be an order ofmagnitude larger than the light quark mixing effect in order to appreciably influencethe final result for f T s . This seems a rather implausible scenario. For this reason, thestrong constraint on mixing with light quarks derived below will be taken as indicationthat violations of (11) are negligible at the present level of statistical accuracy of f T s .Concentrating thus on the effect stemming from the mixing with the operator ¯ uu + ¯ dd ,an estimate of the possible residual violation of (11) can be obtained from the Feynman-Hellmann theorem as follows [14]. The residual breaking of chiral symmetry can beparametrized to leading order via the additive mass renormalization m res,q , which ingeneral depends on the flavor q for which one is considering the domain wall Diracoperator, m renorm q = Z m ( m bare q + m res,q ) . (16)Using (15), one has m s h N | ¯ ss | N i| bare = m bare s X q ∂m N ∂m renorm q ∂m renorm q ∂m bare s (17) ≈ m s h N | ¯ ss | N i| renorm + m renorm s ∂m res,l ∂m bare s h N | ¯ uu + ¯ dd | N i renorm where m res,l denotes the light quark flavor residual mass, and where only the dominantcorrection has been kept in the second line by using m bare s ≫ m res,s and h N | ¯ uu + ¯ dd | N i ≫h N | ¯ ss | N i . No direct data for the factor ∂m res,l /∂m bare s are available, but a rough order ofmagnitude estimate can be constructed from related data on the residual mass obtainedwithin the LHPC program and in the present work. This estimate is discussed in theAppendix, with the result ∂m res,l ∂m bare s (cid:12)(cid:12)(cid:12)(cid:12) am s =0 . ,am l =0 . ≈ − . h N | ¯ uu +¯ dd | N i renorm is given by its magnitude at the physical point; combining a typical phe-nomenological value for the nucleon sigma term, m l h N | ¯ uu + ¯ dd | N i ≈
60 MeV, cf.,e.g., [34] for a recent discussion, with the physical light quark mass [28], m l ≈ . h N | ¯ uu + ¯ dd | N i ≈
17. Supplementing this with the value of the strange quarkmass [28], m s = 95 MeV (all aforementioned quark masses being quoted in the M S scheme at 2 GeV), yields δ ( m s h N | ¯ ss | N i ) ≈ − . f T s as one translates the quantity to the M S scheme at 2 GeV.14ompared to the statistical uncertainties of the present calculation, a correction of thismagnitude is negligible, and in the following, (11) will therefore be taken to hold for thepresent calculation within its uncertainties .It should again be noted that the mixing with gluonic operators has not been quan-tified in the above considerations. Whereas the weakness of the mixing with light quarkoperators (which, after all, is mediated by the coupling to the gluonic fields) makes itseem improbable that mixing with gluonic operators themselves is significant comparedto other uncertainties of the calculation, explicit corroboration of this expectation wouldbe desirable. Turning to the axial vector matrix element, chiral symmetry again provides importantconstraints, although it cannot completely exclude mixing effects, due to its anoma-lous U A (1) breaking. The domain wall fermion discretization admits a five-dimensionalpartially conserved axial vector current A aµ obeying a Ward-Takahashi identity of theform [35, 36] ∆ µ A aµ = 2 mJ a + 2 J a q (20)The first term on the right-hand side represents the explicit breaking of chiral symmetryby the quark masses, whereas the second term in the flavor-octet case encodes theresidual chiral symmetry breaking present for a fifth dimension of finite extent [36], J a q ≈ m res J a . Thus, up to these chiral symmetry breaking effects, which will be revisitedfurther below, the flavor octet part of the current A aµ is conserved and undergoes norenormalization. On the other hand, in the flavor singlet case, the second term onthe right-hand side of (20) in addition encodes the coupling to the gluonic topologicaldensity which leads to the anomalous U A (1) breaking of chiral symmetry. This opensthe possibility of operator mixing under renormalization in the flavor singlet component.With respect to direct gluonic admixtures, the axial case differs somewhat from thescalar case discussed further above. The continuum axial vector current A (cont) µ receivesno direct gluonic admixtures [37], since the relevant gluonic operator would be theoperator K µ which, upon taking the divergence, yields the gluonic topological density, ∂ µ K µ = g e F F . However, K µ itself is not gauge invariant, and therefore the gaugeinvariant operator A (cont) µ receives no admixtures from K µ . While it is not clear to whatextent this argument is modified at finite lattice spacing, the fact that direct gluonicadmixtures to the flavor-singlet axial vector current must vanish in the continuum limitsuggests that any such modifications would be small compared to the mixing of thestrange quark axial current with the light quark axial currents. Concentrating thus on Note that the consideration of the factor ∂m res,l /∂m bare s given in the Appendix relies on a numberof assumptions and should be viewed as no more than a rough estimate of what is most likely an upperbound on the magnitude of this correction factor. In view of this, applying the correction (19) to thenumerical data does not seem warranted. sγ µ γ s = 13 h ¯ qγ µ γ q − √ qγ µ γ λ q i (21)(where q ∈ { u, d, s } ), and allowing for different renormalization constants for the singletand octet parts, (¯ sγ µ γ s ) renorm = 13 Z A ¯ qγ µ γ q − √ Z A ¯ qγ µ γ λ q (22)= Z A ¯ sγ µ γ s + 13 Z A − Z A Z A Z A ¯ qγ µ γ q (23)The relative strength of mixing ( Z A − Z A ) /Z A is not available for the specific lat-tice scheme used in the present work, but has been estimated for the case of cloverfermions [11, 38]. To the extent that this encodes the effect of the axial anomaly asopposed to specific lattice discretization effects, it can be taken as indicative of thestrength of mixing also in other lattice schemes such as the one used in the present in-vestigation. Inasfar as it is influenced by the lattice scheme, it presumably can be takento represent an upper bound for the strength of mixing in schemes which better respectchiral symmetry such as the one used here (assuming there are no accidental cancella-tions). Quantitatively, for the conversion into the M S scheme at the scale µ = 2 GeV,one obtains in the two-flavor clover fermion case [11] Z A − Z A Z A = 0 . .
765 = 0 .
011 (24)Correcting by a factor 3 / Z A ≈ .
1, as obtained below, as well as ∆( u + d ) ≈ .
42 (at thephysical pion mass, cf. [33]), the uncertainty from mixing with light quarks, i.e., thesecond term in (23) is estimated to amount to δ (∆ s ) ≈ . s obtained inthis work, but will be treated as a systematic uncertainty.It should be remarked that the considerations for the clover case [11] referred toabove pertain to lattices with a substantially finer spacing than used here (0 .
073 fmvs. 0 .
124 fm). However, adjusting for this is expected to modify (25) merely by a fewpercent and thus is negligible in the present context. This can be inferred from themagnitude of the O ( a ) improvement corrections to the renormalization constants quotedin [11]; note also that the present HYP-smeared mixed action scheme, which is fully O ( a )-improved, suffers only from very benign finite lattice spacing effects, as evidenced,16.g., by the congruence between nucleon mass measurements in this same scheme andcorresponding MILC determinations on much finer, a = 0 .
06 fm lattices [32].Turning to the renormalization constant Z A , the axial vector quark bilinear usedin practice in evaluating ∆ s is the local ¯ s ( x ) γ µ γ s ( x ) as opposed to the correspondingflavor component of the partially conserved A µ of eq. (20). To renormalize this localoperator, the standard scheme [36] can be applied, with the modification that, in thepresent case, it is not the pion current but the η current which is relevant: Calculatingthe (connected parts of the) two-point functions of both the conserved and the localcurrent, C ( t + 1 /
2) = X x hA ( x, t ) η (0 , i| conn L ( t ) = X x h ¯ qγ γ ( λ / q η (0 , i| conn , (26) Z A /Z A can be extracted from an appropriate ratio which takes into account the tem-poral offset between the two currents,12 (cid:18) C ( t + 1 /
2) + C ( t − / L ( t ) + 2 C ( t + 1 / L ( t ) + L ( t + 1) (cid:19) t/a ≫ −→ Z A Z A (27)Note that the full correlators h ¯ qγ γ ( λ / q η (0 , i and hA ( x, t ) η (0 , i acquire dis-connected contributions for unequal strange and light quark masses; however, for thespecific purpose of extracting Z A /Z A , any ratio of quantities which only differ by thisoverall renormalization factor is suitable, including using only the connected parts ofthe aforementioned correlators, as indicated in (26). Table 4 lists the values obtainedfor Z A /Z A , which are applied to the bare lattice measurements of ∆ s . m bare l Z A /Z A Z A /Z A at varying m bare l .A systematic uncertainty is associated with this scheme of renormalizing ∆ s , due toresidual sources of chiral symmetry breaking. One of these sources is the finite extentof the fifth dimension in the domain wall fermion construction. At finite m res , therenormalization constant of the partially conserved current, Z A , can deviate from theunit value it would take if chiral symmetry were strictly observed [39]. Secondly, notethat, at finite lattice spacing a , there is a certain tension between adopting a mass-independent lattice renormalization scheme and maintaining O ( a ) improvement [40].The renormalization constant Z A in general contains dependences of order O ( m q a ),evident in the slight variation displayed in Table 4. Since the lattice data necessary to17erform the continuum limit a → Z A /Z A to the chiral limit, thus obtaining a mass-independentrenormalization scheme in more direct correspondence to the M S scheme, but spoiling O ( a ) improvement. On the other hand, by retaining the leading quark mass dependence,i.e., applying the finite- m q renormalization constants in Table 4 ensemble by ensemble, O ( a ) improvement is maintained, at the expense of introducing a slight mass dependenceinto the renormalization scheme at finite a . The mass dependence, implying a breakingof chiral symmetry in addition to the one encoded in the residual mass m res , is thenexpected to be of order O ( m q a ). Since the present investigation yielded lattice data atonly a single lattice spacing a , precluding a direct estimate of the effects of finite latticespacing, maintaining O ( a ) improvement seems sufficiently desirable to elect the latteralternative, i.e., applying the renormalization constants in Table 4 ensemble by ensembleand thus introducing a slight mass dependence into the renormalization scheme.A way of estimating the systematic uncertainty in the renormalization of ∆ s re-sulting from the residual breaking of chiral symmetry due to the above sources lies inthe mismatch between the axial vector and the vector renormalization factors, Z A /Z A vs. Z V /Z V , which would remain equal if chiral symmetry were strictly maintained [41].In the lattice scheme used here, these factors typically differ by 3% [33] . An additionalsystematic uncertainty of this magnitude will therefore be attached to the renormalizedvalue of ∆ s . Chiral extrapolation formulae for strange quark matrix elements in the nucleon havebeen given in [42]. At leading order (LO), both h N | ¯ ss | N i and ∆ s are constant in m π .On the other hand, when one evaluates one-loop effects, including ∆-resonance degreesof freedom, one obtains next-to-next-to-leading-order (NNLO) formulae which containtoo many parameters to be effectively constrained by the restricted set of lattice dataat three pion masses accessed in this work. However, practicable fits can be constructedby reducing the NNLO formulae given in [42] to the chiral effective theory without ∆-resonance degrees of freedom, which is achieved by setting g ∆ N = 0 and correspondinglyalso eliminating the counterterms associated with the ∆-resonance degrees of freedom.In this case, the behavior of h N | ¯ ss | N i reduces to a linear function in the light quarkmass, i.e., in m π , h N | ¯ ss | N i = S + S m π (28)with the two fit parameters S and S , whereas ∆ s retains a chiral logarithm,∆ s = D (cid:20) − g A π f m π log( m π /µ ) (cid:21) + D m π (29) To be precise, [33] considered the isovector currents, i.e., determined Z A /Z A and Z V /Z V . D and D ; the dependence on the scale µ is of courseabsorbed by the D counterterm. The pion decay constant f is normalized such that f ∼
132 MeV, and the physical axial coupling constant is g A ∼ .
26. Correspondingchiral fits to the renormalized lattice data for m s h N | ¯ ss | N i = m N f T s and ∆ s are shownin Fig. 5. The LO constant fits and the (reduced) NNLO fits are consistent with oneanother. Positing the LO constant behavior in m π leads to artificially low estimates ofthe uncertainties; in this case, the statistical error bars are dominated by the most ac-curately determined m π = 495 MeV data points, at which, on the other hand, the chiralextrapolation formulae are the least trustworthy. Plausible estimates of the statisticaluncertainties of the extrapolated values are given by the (reduced) NNLO fits, whichallow for variation of the observables with m π . The estimates of f T s and ∆ s resultingfrom the (reduced) NNLO fits at the physical pion mass in the M S scheme at a scaleof 2 GeV are (before taking into account systematic effects, which are discussed in thenext section), f T s = 0 . , (30)where the physical nucleon mass has been used to convert the fit result from Fig. 5 backto f T s , and ∆ s = − . . (31)Note again that the quoted uncertainties at this point contain only the statistical errorfrom the lattice measurement, propagated through the chiral extrapolation; systematicuncertainties and adjustments of the results (30) and (31) are elaborated upon in thenext section. Several sources of systematic uncertainty should be taken into account with respect tothe two results (30) and (31):
Renormalization uncertainties:
Uncertainties associated with renormalization werealready discussed in section 4. In the case of f T s , uncertainties due to mixing with lightquark operators generated by residual breaking of chiral symmetry were estimated tobe of the order of 1% and will thus not be considered further here. In the case of ∆ s ,mixing effects were less well constrained because of the anomalous breaking of chiralsymmetry. The potential correction to ∆ s was estimated to amount to δ (∆ s ) ≈ . Z A due to residual sources of chiral symmetry breaking. Theseincluded finite- m res effects as well as effects of adopting a not fully mass-independentlattice renormalization scheme, in order to preserve O ( a ) improvement. Adding theuncertainty from operator mixing and the one associated with Z A in quadrature impliesan uncertainty in ∆ s of +0 . / − . a m f (m /GeV) T s p N -0.06-0.04-0.02 0 0 0.05 0.1 0.15 0.2 0.25 0.3 D (m /GeV) s p Figure 5: Pion mass dependence of the results for m s h N | ¯ ss | N i = m N f T s and ∆ s .Filled circles represent renormalized lattice data; in the case of ∆ s , these are obtainedby multiplying the T = 10 values from Table 2 by the corresponding renormalizationconstants from Table 4, whereas m N f T s is obtained by multiplying the T = 10 valuesfrom Table 3 by the corresponding nucleon masses from Table 1. Open symbols showchiral extrapolations of the lattice data to the physical pion mass, cf. main text. Opencircles represent the LO (constant) chiral extrapolations, whereas open squares representthe reduced NNLO extrapolations obtained by dropping the ∆-resonance degrees offreedom, with the dashed lines showing the pion mass dependences of the central valuesin the latter case. 20 inite lattice spacing effects: Since the present investigation only employed ensem-bles at a single lattice spacing, a = 0 .
124 fm, no direct assessment of the a -dependenceof the results was possible. This motivated the insistence on a fully O ( a )-improved cal-culational scheme, in order to minimize the influence of the finite lattice spacing fromthe outset (and, in the process, part of the lattice spacing dependence was already sub-sumed under the uncertainty in renormalization, as noted above). In the case of f T s ,which is related via the Feynman-Hellmann theorem to the nucleon mass, an estimateof the uncertainty due to discretization effects can be inferred from the a -dependenceof the nucleon mass. As shown in [32], already at a = 0 .
124 fm, the nucleon mass inthe present HYP-smeared mixed action scheme coincides with the MILC a = 0 .
06 fmresults. Given that the MILC results themselves still change by about 10% going from a = 0 .
124 fm to a = 0 .
06 fm, the residual O ( a ) effect in the latter case is expected tobe around 3%. This will therefore be taken as the generic estimate of the magnitude offinite lattice spacing effects in the present calculation, both for f T s and ∆ s . Uncertainty due to truncation of the chiral perturbation series:
Taking thedeviation between the LO and the (reduced) NNLO fits at the physical pion mass,cf. Fig. 5, as a measure of the uncertainty due to truncation of the chiral perturbationseries, a 6% uncertainty is attached to the value of f T s and a 14% uncertainty to ∆ s . Effects of inadequate lattice dimensions:
Both the spatial extent of the latticeand the temporal separations between nucleon source, sink and operator insertion arelimited. Consequently, results are influenced both by interactions with periodic copies ofthe lattice as well as excited state admixtures. Neither of the corresponding uncertaintieswere directly quantifiable within the present calculation. On the one hand, ensembleswith only a single lattice extent were employed; on the other hand, no systematic excitedstate effects could be gleaned from the sink position dependence of the lattice data atthe present level of statistical accuracy, as discussed in section 3. An indication ofthe possible magnitude of such effects can be inferred from lattice calculations of thenucleon axial charge g A , the isovector light quark analogue of ∆ s , which represents a well-studied benchmark quantity. Calculations of g A within the present lattice scheme [33]and others [43, 44] exhibit a deviation from the phenomenological expectation of up to10%, with the cause of this deviation attributed to either excited state contaminationsor finite lattice size effects. This will therefore be taken as an estimate of the systematicuncertainty due to such effects also in the present calculation. Adjustment of the strange quark mass:
The strange quark mass am asqs = 0 .
05 inthe gauge ensembles used in the present calculation lies appreciably above the physicalstrange quark mass, which a posteriori was determined to be am asq,physs = 0 .
036 [16].For the case of the strange scalar matrix element, a corresponding correction factor wasestimated in [16], namely, ∂∂m asqs ∂m N ∂m asqs = − . · .
31 fm = − .
68 fm (32)21ultiplying this by the shift in m asqs , δm asqs = (0 . − . /a , yields, in view of theFeynman-Hellmann theorem, an enhancement of the strange scalar matrix element by δ ( h N | ¯ ss | N i ) asq = δm asqs ∂∂m asqs ∂m N ∂m asqs = 0 .
077 (33)To translate this to the present scheme, one needs to rescale the strange quark mass, m asqs = ( m asq /m DW F ) m DW Fs . The rescaling factor m asq /m DW F varies only weakly be-tween am asqs = 0 .
05 and am asqs = 0 .
02, namely, from 0 .
617 in the former case to 0 . am asqs = 0 .
05 and am asqs = 0 . m asq /m DW F = 0 . m asqs in the denominator and one inthe numerator, altogether, translated to the present scheme, δ ( h N | ¯ ss | N i ) DW F = m asq m DW F δ ( h N | ¯ ss | N i ) asq = 0 .
048 (34)To obtain a measure of the relative change in h N | ¯ ss | N i implied by this, note that theresult (30), multiplied by m N /m DW Fs = 7 .
3, yields h N | ¯ ss | N i DW Fam
DWFs =0 . = 0 .
42, andtherefore the adjustment (34) corresponds to an enhancement of h N | ¯ ss | N i by a factor1 .
115 as one shifts the strange quark mass to its physical value.On the other hand, f T s itself acquires an additional factor m DW F,physs /m DW Fs ≈ m asq,physs /m asqs = 0 .
72 as one shifts the strange quark mass to the physical point, im-plying that f T s is reduced by a factor 0 . · .
115 = 0 .
80. The reduction of m s in factovercompensates the enhancement of h N | ¯ ss | N i . Altogether, thus, (30) will be adjustedby a factor 0 .
80 to arrive at the physical value; in addition, a systematic uncertainty of3% will be associated with that adjustment in view of a 15% uncertainty in (32), cf. [16],as well as the variability in m asq /m DW F .For the case of ∆ s , no similarly detailed consideration is available. However, it seemsplausible that the leading effect of a lowering of the strange quark mass is an overallenhancement of the strange quark density, with the detailed dynamics governing anygiven strange quark unaffected to a first approximation. In this case, one would expectthe enhancement factor 1.115 to equally apply to the axial matrix element ∆ s . In viewof the rough nature of this argument, a systematic uncertainty of 12% will be associatedwith this enhancement, thus covering the range of no enhancement of ∆ s up to twicethe enhancement seen in the case of the scalar matrix element.Summarizing the diverse uncertainties and adjustments discussed above, the finalestimates for the physical values of f T s and ∆ s are as follows: f T s = 0 . s = − . +3 − )(1)(4)(3)(4) (36)where the uncertainties are, in the order written, statistical, due to renormalization,due to finite lattice spacing, due to truncation of the chiral perturbation series, due toinadequate lattice dimensions, and due to the adjustment of the strange quark massto the physical value. To quote a succinct final result, if one combines all the system-atic uncertainties discussed in this section together with the statistical uncertainty inquadrature, f T s = 0 . s = − . This investigation focused on two of the most basic signatures of strange quark degreesof freedom in the nucleon, namely, the strange quark contribution to the nucleon mass,characterized by f T s , and the portion of the nucleon spin contained in strange quarkspin, ∆ s . A high amount of averaging not only of the disconnected strange quark loop,but also of the nucleon two-point function led to clear signals for both f T s and ∆ s especially at the heaviest pion mass, m π = 495 MeV, with the signals deteriorating, butnot disappearing, as the pion mass is lowered to m π = 356 MeV and m π = 293 MeV.Combining all the lattice data, the signals survive chiral extrapolation to the physicalpion mass. Systematic uncertainties remain under adequate control; the only source ofsystematic uncertainty which was not quantified is gluonic operator admixtures to f T s under renormalization. However, as discussed in section 4.1, a scenario in which theseadmixtures rise to the level at which they begin to appreciably influence the conclusionsreached regarding f T s seems highly implausible. The gluonic admixtures would haveto be an order of magnitude larger than the related light quark admixtures, whichwere constrained to the 1% level. Nevertheless, a more quantitative corroboration ofthis argument would be desirable. All other systematic uncertainties were quantified,cf. section 6, and, while some of them are still sizeable, none rise to the level of thestatistical uncertainties. With respect to controlling systematic uncertainties, the use ofa (to a very good approximation) chirally symmetric discretization of the strange quarkfields in the matrix elements (1),(2) proved very advantageous, since it provides forbenign renormalization properties, including the almost complete suppression of lightquark admixtures to f T s alluded to above. This stands in contrast to, e.g., the caseof Wilson fermions, in which the evaluation of f T s is considerably complicated by thepresence of strong additive mass renormalizations [12, 24].The magnitudes obtained for both f T s and ∆ s appear natural. Neither quantity isabnormally enhanced; strange quarks contribute about 4.5% of the nucleon mass, and23he magnitude of the strange quark spin, which is polarized opposite to the nucleon spin,amounts to about 3% of the latter. The conditions provided by nucleon structure fordark matter detection via coupling of the Higgs field specifically to the strange quarkcomponent thus do not appear to be as favorable as assumed in the most optimisticscenaria. There is also no indication from the result for ∆ s that an unnaturally largecontribution to the spin of the nucleon is hidden in the small- x strange quark sector thathas hitherto eluded experimental study. The strange quark spin does indeed appear tobe polarized slightly in the direction opposite to the nucleon spin, as also indicated bythe preponderance of phenomenological studies. However, the magnitude of ∆ s foundin the present calculation is smaller than the magnitudes extracted in the analyses [5, 9]which assume a substantial enhancement of the strange quark helicity distribution atsmall momentum fraction x . Acknowledgments
Fruitful exchanges with W. Freeman, H. Grießhammer, P. H¨agler, K. Orginos, S. Pateand D. Toussaint are gratefully acknowledged. The computations required for this inves-tigation were carried out at the Encanto computing facility operated by NMCAC, usingthe Chroma software suite [45] and gauge ensembles provided by the MILC Collabora-tion. This work was supported by the U.S. DOE under grant DE-FG02-96ER40965.
Appendix: Estimate of ∂m res,l /∂m bare s The quantity ∂m res,l /∂m bare s enters the estimate of operator mixing effects in f T s , cf. sec-tion 4.1. No direct data for this quantity are available, but an order of magnitude es-timate can be constructed from related data on the residual mass obtained within theLHPC program and in the present work, summarized in Table 5. It should be noted am bare l · am res,l · am res,s m res,l and m res,s at varying m bare l .that all these results were obtained at a constant lattice spacing a . Fitting parabolae tothe data in Table 5 yields the following derivatives at the SU (3)-flavor symmetric point am s = am l = 0 .
081 and at the lightest m l considered in this work, am l = 0 . ∂m res,l ∂m bare l (cid:12)(cid:12)(cid:12)(cid:12) am s = am l =0 . = − . ∂m res,l ∂m bare l (cid:12)(cid:12)(cid:12)(cid:12) am s =0 . ,am l =0 . = − .
020 (39)24 m res,s ∂m bare l (cid:12)(cid:12)(cid:12)(cid:12) am s = am l =0 . = − . ∂m res,s ∂m bare l (cid:12)(cid:12)(cid:12)(cid:12) am s =0 . ,am l =0 . = − . ∂m res,l /∂m bare s from this can be obtained by noting that, atthe SU (3)-flavor symmetric point, ∂m res,l ∂m bare s (cid:12)(cid:12)(cid:12)(cid:12) am s = am l =0 . = 12 ∂m res,s ∂m bare l (cid:12)(cid:12)(cid:12)(cid:12) am s = am l =0 . = − . ∂/∂m l = ∂/∂m u + ∂/∂m d ). Assumingthat the derivative of m res,l in the m s -direction varies only weakly as one changes m l ,one arrives at the estimate that also ∂m res,l ∂m bare s (cid:12)(cid:12)(cid:12)(cid:12) am s =0 . ,am l =0 . ≈ − . m l becomes smaller.A check on this estimate can be constructed by the following alternative chain of rea-soning. First, note that the quantity ∂m res,l /∂m bare l contains two contributions, namely,one from the explicit variation of m l in the Dirac operator which m res,l characterizes, andthe other from the implicit dependence of the gauge field ensemble on m l . To estimate ∂m res,l /∂m bare s , one therefore needs to apply two correction factors to ∂m res,l /∂m bare l :A factor characterizing the proportion of the variation of m res,l due specifically to theimplicit variation of the gauge fields, and a factor characterizing the strength of thatvariation with m s as opposed to m l . The order of magnitude of these correction factorscan be inferred as follows. The former factor is available at the SU (3)-flavor symmetricpoint as the ratio ∂m res,s /∂m bare l ∂m res,l /∂m bare l (cid:12)(cid:12)(cid:12)(cid:12) am s = am l =0 . = 0 . . (43)For the purposes of the present argument, it will be assumed that this factor only variesmildly as m bare l is lowered to am bare l = 0 . ∂m res,s ∂m bare s (cid:12)(cid:12)(cid:12)(cid:12) implicit ,am l =0 . ≈ ∂m res,s ∂m bare s (cid:12)(cid:12)(cid:12)(cid:12) implicit ,am l =0 . (44)i.e., the implicit variation of m res,s via the dependence of the gauge fields on m s changesonly mildly as a function of m l . Note that, while this assumption is analogous to theone leading to (42), it is better founded since m res,s itself varies less with m l than m res,l .25gain, one would expect the left-hand side to represent an upper bound for the right-hand side. Noting that the left-hand side is identical to one-half the quantities in theleft-hand identity in (40), one thus has0 .
08 = − . / − . ∂m res,s /∂m bare s | implicit ,am l =0 . ∂m res,s /∂m bare l | am l =0 . (45) ≈ ∂m res,s /∂m bare s | implicit ∂m res,s /∂m bare l (cid:12)(cid:12)(cid:12)(cid:12) am l =0 . (46) ≈ ∂m res,l /∂m bare s ∂m res,l /∂m bare l | implicit (cid:12)(cid:12)(cid:12)(cid:12) am l =0 . (47)where in the final step it is assumed that changes in the numerator and denominatordue to changing the quark mass in the Dirac operator from m s to m l approximatelycancel . This is the desired conversion factor characterizing the strength of the implicitvariation of m res,l with m s relative to the one with m l . Correcting, as proposed above, ∂m res,l /∂m bare l by the two factors (43) and (47), one finally arrives at the alternativeestimate ∂m res,l ∂m bare s (cid:12)(cid:12)(cid:12)(cid:12) am s =0 . ,am l =0 . ≈ . · . · ( − . − . , (48)consistent in order of magnitude with (42), especially in view of the latter being expectedto represent an overestimate.The estimate (42) for ∂m res,l /∂m bare s at the lowest light quark mass considered inthe numerical calculations in this work is used in section 4.1 to constrain the influenceof operator mixing effects in the renormalization of f T s . References [1] A. Bottino, F. Donato, N. Fornengo and S. Scopel, Astropart. Phys. (2002) 205.[2] J. Ellis, K. A. Olive, Y. Santoso and V. C. Spanos, Phys. Rev. D 71 (2005) 095007.[3] J. Ellis, K. A. Olive and C. Savage, Phys. Rev.
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