Can Baryon Chiral Perturbation Theory be used to extrapolate lattice data for the moment ⟨x ⟩ u−d of the nucleon?
CCan Baryon Chiral Perturbation Theory be used to extrapolatelattice data for the moment (cid:104) x (cid:105) u − d of the nucleon? Peter C. Bruns, Ludwig Greil, Rudolf R¨odl, and Andreas Sch¨afer
Institut f¨ur Theoretische Physik, Universit¨at Regensburg, D-93040 Regensburg, Germany (Dated: October 18, 2018)
Abstract
We discuss the question in the title employing manifestly covariant Baryon Chiral PerturbationTheory and recent high-statistics lattice results published in [1].
PACS numbers: 12.38.Gc, 12.39.Fe a r X i v : . [ h e p - l a t ] N ov . INTRODUCTION Lattice Quantum Chromodynamics (LQCD) has now reached the point where fully dy-namical results for nucleon structure observables are available for quark masses almost cor-responding to the physical pion mass M π ≈
135 MeV (for recent reviews, see [2–5]). Forsome, but not all of these quantities, also results extracted from phenomenology exist, andit is crucial to compare the lattice results with the experimental results, in order to judgethe reliablility of LQCD predictions for such quantities which are not easily accessible in ex-periments. For some observables, for example the axial coupling constant g A of the nucleon,and the moment (cid:104) x (cid:105) u − d (isovector quark momentum fraction) of the nucleon, there is sub-stantial tension between the experimental and the lattice outcome. For the latter quantity,which is the topic of the present article, we refer in particular to the very recent results anddiscussion in [1].The quark mass dependence of hadron observables is given by Chiral Perturbation Theory(ChPT) [6–9], the low-energy effective field theory of QCD. It has often been used to ex-trapolate lattice data employing quark masses larger than in the real world down to theexperimental point. Moreover, the dependence on the spatial volume L of the lattice canalso be calculated within the same theoretical framework. This was first demonstrated inthe purely Goldstone-bosonic sector [10–12], and later extended to the one-baryon sector[13–18]. We work in the so-called p -regime where the counting scheme is set up such that M π ∼ L − ∼ O ( p ), where p denotes a small quantity like a (pseudo-)Goldstone boson massor momentum. It has become a common rule of thumb that one should have M π L (cid:38) M π L (cid:38) p -regime). On the other hand, there is a long-standing debate in theliterature about the range of applicability of the chiral extrapolation formulae given by (two-flavor) Baryon Chiral Perturbation Theory (BChPT) in terms of the quark masses or M π ,see [22–35] for relevant references. The opinion most generally shared today seems to be thatearlier applications of BChPT formulae to lattice results for pion masses M π (cid:38)
500 MeVwere not under sufficient control for a reliable extrapolation to the physical point, while(citing the review [31]) “it is fair to say that chiral extrapolations of nucleon properties canbe trusted for pion masses below ∼
350 MeV“. Of course, such statements will also depend2n the particular channel or observable one is considering, on the chiral order (accuracy) ofthe calculation, and on the available information on relevant low-energy constants (LECs)from other sources. Moreover, in a finite spatial volume L , the pion masses should not besmaller than ∼ L − in order to stay in the above-mentioned p -regime.In this work, we will study the chiral extrapolation and the finite volume corrections oflattice results for the moment (cid:104) x (cid:105) u − d of the nucleon (for the pertinent definitions and con-ventions, see e. g. [1, 2, 36–39]). First, in Sec. II, we neglect the finite volume corrections,and use a truncated version of the full one-loop expression for (cid:104) x (cid:105) u − d ( M π ) to test the stabil-ity of the chiral fits with respect to a variation of supplemented terms of leading two-looporder ( O ( p )). Encouraged by this first test, we apply the formulae derived in [38–40] (inthe general setting of manifestly covariant BChPT [41]) to a more complete data set, andexamine the finite volume corrections together with the chiral extrapolation in the pion massin Sec. III, in three different fit scenarios. The plots in Fig. 7 illustrate our main results,corresponding to the fit parameters tabulated in Tab. VII. We discuss our findings in Sec. IV,from two perspectives: • The standard perspective, most commonly found in the literature, where lattice dataat unphysically large quark masses is connected to the region of low quark masses bymeans of a chiral extrapolation, and compared with some known experimental value,and • from a perspective where it is assumed that the experimental value is unknown, andwhere ChPT is applied to test the internal consistency of the lattice data, in particularfor quark masses close to the physical values.With the example of (cid:104) x (cid:105) u − d at hand, we shall demonstrate below that, from the secondperspective, BChPT can play an important role for future lattice simulations, as an indicatorof the possible presence of uncontrolled systematic errors inherent in the data, even if thereare data points at (nearly) physical quark masses and large volumes. Finally, we try to givea well-founded answer to the question posed in the title in our concluding Sec. V.3 I. STABILITY TEST OF CHIRAL FITS FOR (cid:104) x (cid:105) u − d The chiral expansion of (cid:104) x (cid:105) u − d to O ( p ) in BChPT reads [38, 39, 42–47] (cid:104) x (cid:105) u − d = a v , (cid:32) − (cid:18) M π πF (cid:19) (cid:18) ( ◦ g A ) + (3( ◦ g A ) + 1) log (cid:18) M π µ (cid:19)(cid:19)(cid:33) + 4 M π m c r ( µ )+ ◦ g A M π πF m (cid:18)
83 ∆ a v , + 72 ◦ g A a v , − ˜ l (cid:19) + (cid:18) M π m (cid:19) (cid:32) k (cid:18) log (cid:18) M π µ (cid:19)(cid:19) + k log (cid:18) M π µ (cid:19) + k (cid:33) + O ( p ) . (1)For the LECs occuring in the leading-one-loop calculation we have used the nomenclatureof [38], while at O ( p ), ˜ l is a combination of the LECs l ,n , n ∈ { , , , , , } , definedin [39] (see Eq. (A6c) in that reference). m , ◦ g A , F denote the nucleon mass, the axialcoupling constant g A and the pion decay constant, respectively, in the two-flavor chirallimit. For the renormalization scale, we shall use µ = 1 GeV in the following. The terms of O ( p ), parameterized by the three constants k i , are of two-loop order and have not beencomputed so far. Here we have made the reasonable assumption that the expansion of (cid:104) x (cid:105) u − d at the two-loop level is analogous to the ones of the nucleon mass and g A , compare[24, 30, 48].In the following, we want to test the stability of the fits to lattice data employing theone-loop ChPT expressions for (cid:104) x (cid:105) u − d , with respect to variations of subleading (two-loop)order. To do this, we generate a large set of random numbers for { k , k , k } and fit thethree free one-loop parameters a v , , c r ( µ = 1 GeV) and ˜ l to data (see e.g. Sec. 7 of [49] fora similar strategy used to determine the Gasser-Leutwyler-LECs L ri ). Of course, we shouldnot let the coefficients k i become arbitrarily large. We shall assume that BChPT worksreasonably well (along the lines of the usual low-energy power counting) for M π (cid:46)
200 MeV.For Eq. (1), this amounts to the constraint that | k i | (cid:46)
2. To get a robust estimate forpossible higher-order effects, we will allow for a range − < k i < +4 for the random numbersets.We have yet to specify some input: For ∆ a v , , we take the same value as used in [38],which is consistent with information on (cid:104) x (cid:105) ∆ u − ∆ d (compare e.g. [5] for a recent overview).The value of g A in the chiral limit is not well known, and we simply substitute thephenomenological value here [50]. The nucleon mass in the chiral limit is inferred fromTab. B.4 of [33], while the pion decay constant in this limit is taken from Tab. 1 of [51].4 ABLE I: Input values for the chiral fits. m [GeV] ◦ g A F [GeV] ∆ a v , .
893 1 .
270 0 .
086 0 . We collect the input values in Tab. I. In a first step, we fit Eq. (1) to recent lattice datapublished in [1] with M π <
500 MeV, selecting the largest available volumes. In the fits oftype 1, we exclude the point at M π ∼
150 MeV for now, which is measured at a rather smallvalue of M π L ≈ .
5, and is obviously inconsistent with the experimental value withoutapplying finite volume (and possibly other) corrections. For a further discussion of thispoint, see Sec. IV. The following Tab. II gives the results for fits of type 1 with k , , = 0.In the fit marked with a prime, the last point at M π ∼
490 MeV has been dropped.The cyan bands in the plots below consist of 10 fits with prescribed random number values TABLE II: One-loop fit results ( k = k = k = 0) for the fits of type 1.fit a v , c r ˜ l χ / d . o . f . . − .
184 0 .
789 0 . (cid:48) . − .
510 0 .
190 0 . for the k i . The black lines are the fit curves for k i = 0. The fact that the fit parameters h x i u − d M π [ GeV ℄ (a) Fit scenario 1 h x i u − d M π [ GeV ℄ (b) Fit scenario 1’ FIG. 1: The results for fit scenarios 1 and 1’. The first point, which marks the experimental result,and the second point are not included in the fits. M π ≈
500 MeV. Butalso note that there are only five available data points for three free parameters in fit 1’.Moreover, the convergence properties are problematic for this fit class, see below.In the fits of type 2, we also include the phenomenological value taken as (cid:104) x (cid:105) phen. u − d =0 . ± .
005 (following [56], see also [2] for a collection of values and references). However,we still exclude the point at M π ∼
150 MeV. The following Tab. III displays the results forthose fits with k , , = 0. In the fit marked with a prime, the last point at M π ∼
490 MeVhas again been dropped. The inclusion of the experimental point greatly stabilizes the fits.
TABLE III: One-loop fit results ( k = k = k = 0) for the fits of type 2.fit a v , c r ˜ l χ / d . o . f . . − .
116 0 .
896 0 . (cid:48) . − .
095 0 .
947 0 . h x i u − d M π [ GeV ℄ (a) Fit scenario 2 h x i u − d M π [ GeV ℄ (b) Fit scenario 2’ FIG. 2: The results for fit scenarios 2 and 2’. Here, the experimental value (first point) is includedin the fits.
It is striking that the value at M π ∼
150 MeV always lies outside the generated bands. Ofcourse, finite volume effects might play a role here - see Sec. III for a discussion of this issue.Please also note that the fit curves lying close to the borders of the bands correpond toextrapolations with two-loop contributions which are quite large relative to the expectation6rom the chiral power counting, so the present way of quantifying the uncertainty in theextrapolation can be regarded as conservative in this respect.In an attempt to determine the LECs and their uncertainties, it makes sense to includethe physical point. Collecting the resulting fit parameters for 10 fits of type 2 results inthe histograms in Figs. 3(a)-3(c). The black curves are Gauß distributions fitted to thehistograms, with the expectation values (cid:104) LEC (cid:105) and σ (LEC) as free parameters (given inEq. (2) below). N a v , (a) Histogram for a v , N c ( r )8 (b) Histogram for c ( r )8 N ˜ l (c) Histogram for ˜ l FIG. 3: Histograms for N = 10 fits including the fitted Gaussian distribution (black curve). (cid:104) a v , (cid:105) = 0 . , (cid:104) c r (1 GeV) (cid:105) = − . , (cid:104) ˜ l (cid:105) = 0 . ,σ ( a v , ) = 0 . , σ ( c r (1 GeV)) = 0 . , σ (˜ l ) = 1 . . (2)We add some remarks on the convergence properties of the chiral expansion. For M π =200 MeV and the parameters from Tab. II and III, (cid:104) x (cid:105) u − d = a v , (cid:32) (cid:104) x (cid:105) (2) u − d a v , + (cid:104) x (cid:105) (3) u − d a v , + O ( p ) (cid:33) (3) (cid:39) .
125 (1 + 0 .
345 + 0 .
054 + . . . ) for fit 2 , (cid:39) .
124 (1 + 0 .
377 + 0 .
040 + . . . ) for fit 2 (cid:48) , (cid:39) .
134 (1 + 0 .
255 + 0 .
083 + . . . ) for fit 1 , (cid:39) .
170 (1 − .
071 + 0 .
202 + . . . ) for fit 1 (cid:48) . Here the superscripts in round brackets denote the chiral order. It appears that the expansionconverges well in the low-energy region except for the parameters from fit 1’, for which noconvergence is observed even at the physical point. On the one hand, this fit cannot beruled out, since the fit parameters are still of natural size. But on the other hand, it is7ot very meaningful as already remarked above, because the included data points do notsufficiently constrain the three free parameters. In Fig. 4 we show plots for (cid:104) x (cid:105) (2) u − d /a v , (red) and (cid:104) x (cid:105) (3) u − d /a v , (black, dotted) for fit 2 and 2’. It seems that the application of theone-loop approximation becomes problematic somewhere between M π ∼ . . .
450 MeV.This is in accord with our findings in Sec. VII of [34] for the case of baryon masses, andwith the general expectation stated e.g. in [31]. The plotted curves also demonstrate thatit is illegitimate to neglect the M π -term for data with M π (cid:38)
300 MeV, as was often done inapplications. Obviously, some more data points with M π (cid:46)
350 MeV and bigger volumesare necessary to bring the extrapolation under better control.As a further experiment, we include the lattice point at M π ∼
150 MeV as it stands, instead h x i u − d / a v , M π [ GeV ℄ (a) Fit scenario 2 h x i u − d / a v , M π [ GeV ℄ (b) Fit scenario 2’ FIG. 4: Comparison of second order (red) to third order (black) contribution to (cid:104) x (cid:105) u − d /a v , . of the experimental input, neglecting possible finite volume effects for the moment. Letus call this fit scenario 3. As we can see, the χ / d . o . f . becomes worse roughly by a factorof twelve compared to the previous fits, though there are some rare fits of similar quality(necessitating large higher-order effects), as can be read off from the histogram of χ -valuesin Fig. 5(b). Also, the experimental point is always far outside the uncertainty band givenby the estimate of the two-loop effects. No sign of convergence at M π = 200 MeV is visible TABLE IV: One-loop fit results ( k = k = k = 0)fit a v , c r ˜ l χ / d . o . f . . − .
562 0 .
196 2 . h x i u − d M π [ GeV ℄ (a) Fit scenario 3 N χ / d . o . f . (b) Histogram for reduced χ FIG. 5: Results for fit scenario 3, where the lowest lattice data point is included. for this fit class, (cid:104) x (cid:105) u − d (cid:39) .
183 (1 − .
085 + 0 .
196 + . . . ) for fit 3 . (4)This pattern is similar to that for fit 1 (cid:48) . Including some more data points from othercollaborations (the LHPC point of [54] in the large volume (red) and the two RBC pointsof [57] for M π <
500 MeV (orange)) does not change the general picture established above,see Figs. 6(a)-6(c). The χ / d . o . f . values for the corresponding fits lie between 0 . . . . . (cid:48) . The pertaining parameter sets are givenin Tab. V. The main difference to the previous fits is the fact that the ’flat’ behavior offit 1 (cid:48) is now altered to a curve that strongly bends down in the chiral regime, similar tothe other curves, which is mainly due to the high-statistics LHPC point at M π ≈
356 MeV, M π L ≈ .
3. We shall see in the next section that this instability of the first fit scenario(the different behavior of fits 1 and 1 (cid:48) ) is eliminated when the additional information on themodifications in a finite volume is taken into account. Concerning the fits of type 3, we also
TABLE V: One-loop fit results with additional data points ( k = k = k = 0)fit a v , c r ˜ l χ / d . o . f . . − .
060 1 .
011 0 . (cid:48) .
085 +0 .
256 1 .
589 0 .
773 fit a v , c r ˜ l χ / d . o . f . . − .
071 0 .
994 0 . (cid:48) . − .
036 1 .
078 0 . observe that the picture remains qualitatively the same as given in Tab. IV above, when9 h x i u − d M π [ GeV ℄ RegensburgLHPCRBC (a) Fit scenario 1 h x i u − d M π [ GeV ℄ RegensburgLHPCRBC (b) Fit scenario 1’ h x i u − d M π [ GeV ℄ RegensburgLHPCRBC (c) Fit scenario 2
FIG. 6: Results for fit scenario 1, 1’ and 2 with added data points from RBC and LHPC groups.The dashed line corresponds to the previous results without the added data points. including additional data points. The corresponding results are given in Tab. VI below.Here we have again added a fit where the data point with the highest pion mass has beenexcluded from the χ function (labeled as fit 3 (cid:48) ). The pertaining curves look almost exactlythe same as in Fig. 5. Again we find that the χ / d . o . f . values are significantly larger thanin the fit scenarios 1 ,
2, in which the point at M π ∼
150 MeV is not included.
TABLE VI: One-loop fit results with additional data points, including the lowest lattice data point.fit a v , c r ˜ l χ / d . o . f . . − .
462 0 .
392 2 . (cid:48) . − .
614 0 .
071 2 . II. FINITE VOLUME ANALYSIS
As was mentioned in the foregoing section, the data point at M π ≈
150 MeV always liesoutside of the generated stability bands, which could be due to several systematic errorsinherent to every LQCD simulation. In this section we want to analyze how large the finitevolume corrections to (cid:104) x (cid:105) u − d are and whether they can explain the discrepancies that werefound in [1] (see also [58] for a consistent measurement with somewhat larger error bars). Weutilize the full one-loop finite volume corrections calculated in [40] and the input parametersfrom Tab. I. The LECs l ,n used in [39, 40] are set to zero for n = { , , , , , , , , } and the combination ( l , + l , ) is treated as a free fit parameter we call l , . This is anallowed prescription at the order we are working, because there is only one free parameterin the M π -term of (cid:104) x (cid:105) u − d , compare Eq. (1). To the same order p in the chiral counting, wethen find that this parameter can be related to ˜ l from the previous section as follows: l , = 132 m (cid:18) ˜ l −
43 ∆ a v , (cid:19) . (5)For the finite volume analysis, and the comparison with the results of the previous section,we focus on the data set published in [1]. In this work, particular care has been taken todiscriminate the ground state signal from excited state contributions, which is known tobe very relevant for (cid:104) x (cid:105) u − d [56, 59–62]. Just like in the previous section, we explore threedifferent scenarios: in fit scenario 1 fv we fit all data with 200 MeV < M π <
500 MeV andin fit scenario 2 fv we include the experimental point at M π ≈
135 MeV. In fit scenario 3 fvwe fit the lattice data including the point at M π ≈
150 MeV. The fits marked with a primeare variants where the last data point at M π ∼
490 MeV is excluded. We add the threepoints from the RBC and LHPC collaborations [54, 57] also used for the fits of Figs. 6(a)-6(c).11 .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 . m π in [GeV ] . . . . . x ( m π , L = ∞ ) u − d . . . . . . . . m π L (a) Fit scenario 1fv .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 . m π in [GeV ] . . . . . x ( m π , L = ∞ ) u − d . . . . . . . . m π L (b) Fit scenario 1’fv .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 . m π in [GeV ] . . . . . x ( m π , L = ∞ ) u − d . . . . . . . . m π L (c) Fit scenario 2fv .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 . m π in [GeV ] . . . . . x ( m π , L = ∞ ) u − d . . . . . . . . m π L (d) Fit scenario 2’fv .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 . m π in [GeV ] . . . . . x ( m π , L = ∞ ) u − d . . . . . . . . m π L (e) Fit scenario 3fv .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 . m π in [GeV ] . . . . . x ( m π , L = ∞ ) u − d . . . . . . . . m π L (f) Fit scenario 3’fv FIG. 7: The figure shows the results for all fit scenarios. The finite volume corrected data pointsare represented by circles, while the raw data is represented by diamonds, and the experimentalresult is marked with a star. The shaded band represents the one sigma error band. ABLE VII: Fit results including finite volume corrections for all fit scenarios.scenario a v , c r (1 GeV) l , [GeV − ] ˜ l χ / d.o.f.1 fv 0 . − . . . . (cid:48) fv 0 . − . . . .
722 fv 0 . − . . . . (cid:48) fv 0 . − . . . .
773 fv 0 . − . − . . . (cid:48) fv 0 . − . − . − . . We find that, for scenarios 3 fv and 3 (cid:48) fv, the LECs a v , and c r lie outside of the boundsgenerated in Sec. II and that these fit results have significant overlap with the results of fit3 from Sec. II. All in all, our fit results for scenarios 1 fv, 2 fv, 2 (cid:48) fv and noteably also 1 (cid:48) fvare compatible with the bounds obtained from our stability considerations presented in theprevious section, which is a very encouraging result.The results above show that finite volume corrections are too small to account for the dis-crepancy found between the data point at M π ≈
150 MeV and the well-known experimentalvalue at M π ≈
135 MeV.As was mentioned in the introduction, all finite volume formulae obtained from ChPT inthe p -expansion are valid for M π L (cid:29) M π L ≈ . p -expanded finite volume for-mulae. Investigating this question would exceed the limits set for this analysis though.Note that no significant finite volume effects in the data were observed in [1], in agree-ment with the findings of [54, 56, 57, 63] and with the outcome of our fits presented above.It is noteworthy that, at the lowest pion mass, the central value for (cid:104) x (cid:105) u − d is smaller inthe small volume M π L ∼ .
77, compare Tab. I of [1] (within error bars, the shift due tothe different volumes is consistent with zero). In Fig. 8, we show the finite volume shifts δ (cid:104) x (cid:105) u − d := (cid:104) x (cid:105) L →∞ u − d − (cid:104) x (cid:105) u − d ( L ) for three fixed values of M π L , as a function of M π , for thethree fit scenarios 1fv − ∼ M π for sufficiently smallpion masses, as expected from the form of the loop corrections.13 δ h x i u − d M π [ GeV ℄ M π L = 3 . M π L = 4 . M π L = 4 . (a) Fit scenario 1fv -0.05-0.04-0.03-0.02-0.010 0 0.1 0.2 0.3 0.4 0.5 δ h x i u − d M π [ GeV ℄ M π L = 3 . M π L = 4 . M π L = 4 . (b) Fit scenario 2fv -0.01-0.008-0.006-0.004-0.0020 0 0.1 0.2 0.3 0.4 0.5 δ h x i u − d M π [ GeV ℄ M π L = 3 . M π L = 4 . M π L = 4 . (c) Fit scenario 3fv FIG. 8: The finite volume shifts for fit scenarios 1 −
3, for fixed M π L = 3 . , . , . IV. DISCUSSION OF RESULTS
The three fit scenarios studied in Secs. II and III correspond to three different ways ofusing the BChPT formulae. From the point of view of lattice practitioners, our scenarios1(fv) and 3(fv) correspond to the natural way the formulae are applied: Fitting to the latticedata input, it is checked whether the value of the observable at the experimental point canbe predicted from the extrapolation. From our fit 3fv, where the full data set including thepoint at the nearly physical mass ( M π ∼
150 MeV) of [1] is inserted, we have to concludethat this is not possible. The chiral extrapolation together with the extrapolation to L → ∞ is unable to reconcile the lattice data with the phenomenological value: The finite volumecorrections are much too small to allow for a downward shift of the value at the lowest latticepion mass, sufficient to come close to the experimental result. Excluding the low-mass point,we arrive at fit scenario 1fv, where it seems that the opposite conclusion is true - here, theextrapolation returns the phenomenological value to a satisfying degree of accuracy. This is14he situation already encountered some years ago [38, 53, 64–68] and therefore nothing new(note, however, that previous applications of the BChPT framework to (cid:104) x (cid:105) u − d were onlyaccurate at O ( p ), since the M π term was either neglected or taken with a fixed coefficient ∼ ∆ a v , , a v , ). There are pitfalls here, however: The fit result may depend considerably ondata points for relatively high M π , where the one-loop chiral representation is not reliable anymore in a strict sense, as is shown e.g. by the broad band in fit 1 of Fig. 1, the convergencebehavior illustrated in Fig. 4, and the fit results for scenario 1 (cid:48) in Sec. II. Moreover, it isnot clear whether the data points with M π L < p -regime counting. Nonetheless, it is interesting to see thatthe instability observed for fits 1 , (cid:48) in Sec. II is eliminated in fit 1 (cid:48) fv, where the additionalconstraints on the finite volume behavior (as measured on the lattice) are built in. Needlessto say that, for a more reliable extrapolation of lattice data, it is certainly necessary toinclude more data points from large volumes and M π (cid:46)
350 MeV. Concerning the secondperspective mentioned in Sec. I, in view of an application of the BChPT extrapolationformulae for cases in which the experimental value is not (accurately) known, we wouldlike to point out that BChPT can be useful in such a situation, even if the region whereit can be reliably applied is more limited than previously thought. This is demonstratedby the comparison of our fit scenarios 1(fv) and 3(fv): in our opinion, it might be taken asan indication for a problem with some of the lattice data points that the chiral fit curvesgenerically have the tendency to bend down at least below M π (cid:46)
200 MeV, a trend whichcan not be inferred from the lowest lattice data point. This indication is significant, giventhat the finite volume corrections are indeed small, and given that BChPT works at leastclose to the physical point, showing the suppression of higher orders in M π /m ∼ (moduloenhanced chiral logs) imposed by the chiral power counting (compare e.g. Eq. (3)). Underthese conditions, it seems unavoidable that the extrapolation curve bends down appreciably when approaching the region M π (cid:46)
200 MeV. If, on the other hand, the curve is forced todescribe the lowest lattice point (fit 3(fv)), the χ / d.o.f. value increases considerably, givingrise to the reasonable suspicion that the included data points are not compatible with the It is interesting to note that this strong down-bending of (cid:104) x (cid:105) u − d ( M π ) is also seen in the Chiral SolitonQuark Model (CSQM) [69, 70], see Fig. 3 of [71]. Up to a constant shift, the extrapolation curve in thismodel strongly resembles the expected behavior of the BChPT result (like our fits 1,2). The same is truefor the nucleon mass m N ( M π ) in the CSQM [72]. a are not fully under control. Such effects are not grasped by our present extrapolation formulaeither, so this is certainly a reasonable direction of further investigation. Furthermore, ithas been pointed out in [60] that effects due to excited states tend to be larger for smallerpion masses. Considering the apparent problems due to the lowest-mass data point of [1],we think that even though the problem of excited state contamination has been carefullystudied in that work, one should thoroughly continue this route of investigation. Comparingwith results for nearly-physical pion masses of other collaborations, we remark that ETMC[58] finds a value for (cid:104) x (cid:105) u − d consistent with the one of [1], while LHPC [56] obtains a result consistent with the phenomenological value, though with less statistics than achieved in [1].From the point of view of effective field theory, the experimental result is just another datapoint in addition to the lattice data, with M π ∼
135 MeV and L → ∞ , and it is interestingin itself to study the functional form and properties of the chiral extrapolation and thevalues of the LECs using this experimental result as a constraint. The determination ofthe LECs can be useful in the study of other observables, or for a combined fit to severaldifferent nucleon structure properties. It also allows to assess the region of applicability ofthe BChPT formalism, which is also a much-debated topic in the literature on effective fieldtheories [22–35]. For our fit strategy 2 corresponding to this philosophy (related to the firstperspective on chiral extrapolations mentioned in Sec. I), we find a remarkable stability ofthe functional form of the extrapolation, compare e.g. the results of fits 2 and 2 (cid:48) in Tab. III,and also the result of fit 2fv and 2 (cid:48) fv including finite volume corrections (Tab. VII). Inparticular, the comparison shows that we could obtain a rough estimate of the finite volumeeffects already from the fits 2 , (cid:48) of Sec. II, where only the largest volumes were included.The pion mass dependence is also under control for these fits, as can be seen from thecomparison of the full one-loop form (for L → ∞ ), employed in Sec. III, with the truncatedexpansion of Eq. (1). This is illustrated in Fig. 9, which demonstrates that higher-orderterms of O ( p ) contained in the full unexpanded loop functions become important only for M π (cid:38)
400 MeV for these fit results. But even though the results of the fits 2fv, 2 (cid:48) fv seemvery natural and reliable from the BChPT viewpoint, we cannot conclude that they yieldthe correct extrapolation function, unless we have a good argument why only the data pointswith M π (cid:46)
200 MeV are afflicted with some significant systematic error.16 h x i u − d M π [ GeV ℄ O ( p ) unexpanded O ( p ) expanded FIG. 9: The full one-loop expression for (cid:104) x (cid:105) u − d in infinite volume (red), compared with Eq. (1)(black, dashed), for the parameters of Eq. (2) and k i = 0. In our analysis, we have applied the framework outlined in [38–40], where a field corres-ponding to the ∆(1232) resonance is not included as an explicit degree of freedom. Forstudies where this resonance is included explicitly, see [66–68]. It is non-trivial to include the∆(1232) in manifestly covariant BChPT due to problems with the power-counting scheme,and the presence of additional unphysical degrees of freedom in the covariant description ofhigher-spin fields ( s ≥ O ( p ), having three free fit parameters at hand, we expect that, forfixed delta-nucleon mass splitting and sufficiently small pion masses, the extrapolation wouldonly be mildly different when including the ∆ field, as the effects due to the resonance canmostly be absorbed in the coefficients of the local interaction terms. A conclusion pointingin this direction can also be inferred from the results of Ref. [68]. But still, this point clearlydeserves further study. V. CONCLUSION
Baryon Chiral Perturbation Theory can be used to study the chiral extrapolation of (cid:104) x (cid:105) u − d , but it should be applied with great care. In our opinion, one should make sure that:(1) Higher order corrections with a reasonable strength do not alter the resulting extra-polation curve dramatically, (2) the convergence properties of the expanded extrapolationformula are at least roughly in accord with the expectations from chiral power counting,(3) the consideration of the finite volume effects do not lead to unreasonably large shiftsof the fitted LECs, (4) if data with M π (cid:38)
500 MeV is included in the fit, the presence of17hese points has no big influence on the results, and (5) the resulting LECs are of naturalsize. Under these conditions, one can talk of a reliable extrapolation. If the accordingexperimental result is accurately known, this extrapolation can be directly checked, whichmay serve as a further test whether the lattice data is afflicted with a systematic error notincorporated in the effective field theory. As was pointed out in the discussion in the previoussection, this may be also possible if the experimental value is not available: If the chiralfit to the data set only returns results with a high χ / d.o.f. value, with an extrapolationfunction showing a bad convergence of the chiral expansion already at relatively small quarkmasses, and/or LECs of an unexpected size, these features can be seen as an indication(though of course not a proof) that some uncontrolled systematic error is still present inthe lattice calculation. As should have become clear from our previous discussion, thereis a clear indication that such an error source (like e. g. discretization effects) is presentin the case of lattice data for (cid:104) x (cid:105) u − d . This indication can be further sharpened if somesystematics is observed in the examination of this criterion (like in our comparison of fits3(fv) and 1(fv), which provided evidence that the behavior of the data for very low M π isproblematic from the ChPT perspective). In this way, chiral extrapolations can prove usefuleven if simulations are performed at nearly physical pion masses and large volumes. It isthis kind of application of BChPT which we propose to consider in present-day and futurelattice studies of (cid:104) x (cid:105) u − d and other quantities parameterizing the structure of the nucleon.As an extension of the extrapolation framework used here [38–40], one should consider theeffects due to the ∆(1232) resonance along the lines of [73–75]. On a level of higher accuracy,also isospin-breaking corrections should be incorporated. Additionally, a combined fit ofseveral nucleon structure functions should finally be undertaken. Only then, the full strengthof (B)ChPT comes into play, yielding relations between different observables imposed bychiral symmetry (and other symmetries of the strong interaction). Finally, we would liketo mention that the analogues of the moment (cid:104) x (cid:105) u − d for the full baryon ground-state octet(the form factors A ( t ) at t →
0, for all combinations of baryon states and flavor structuresof the operator insertions) have also been calculated in (three-flavor) BChPT at leadingone-loop order [76, 77]. 18 cknowledgments
We thank P. Wein for discussions, and C. Alexandrou for a useful communication. Thiswork was supported by the Deutsche Forschungsgemeinschaft SFB/Transregio 55. [1] G. S. Bali, S. Collins, B. Gl¨aßle, M. G¨ockeler, J. Najjar, R. H. R¨odl, A. Sch¨afer and R. W. Schiel et al. , Phys. Rev. D (2014) 7, 074510 [arXiv:1408.6850 [hep-lat]].[2] D. B. Renner, PoS LAT (2009) 018 [arXiv:1002.0925 [hep-lat]].[3] H. B. Meyer, arXiv:1106.3163 [hep-lat].[4] S. Syritsyn, PoS LATTICE (2014) 009 [arXiv:1403.4686 [hep-lat]].[5] C. Alexandrou, EPJ Web Conf. (2014) 01013 [arXiv:1404.5213 [hep-lat]].[6] S. Weinberg, Physica A (1979) 327.[7] J. Gasser and H. Leutwyler, Annals Phys. (1984) 142.[8] J. Gasser and H. Leutwyler, Nucl. Phys. B (1985) 465.[9] J. Gasser, M. E. Sainio and A. Svarc, Nucl. Phys. B (1988) 779.[10] J. Gasser and H. Leutwyler, Nucl. Phys. B (1988) 763.[11] P. Hasenfratz and H. Leutwyler, Nucl. Phys. B (1990) 241.[12] M. L¨uscher, Lecture given at Cargese Summer Inst., Cargese, France, Sep 1-15, 1983 [13] A. Ali Khan et al. [QCDSF-UKQCD Collaboration], Nucl. Phys. B (2004) 175 [hep-lat/0312030].[14] A. A. Khan, M. G¨ockeler, P. H¨agler, T. R. Hemmert, R. Horsley, D. Pleiter, P. E. L. Rakowand A. Sch¨afer et al. , Phys. Rev. D (2006) 094508 [hep-lat/0603028].[15] S. R. Beane, Phys. Rev. D (2004) 034507 [hep-lat/0403015].[16] S. R. Beane and M. J. Savage, Phys. Rev. D (2004) 074029 [hep-ph/0404131].[17] W. Detmold and M. J. Savage, Phys. Lett. B (2004) 32 [hep-lat/0407008].[18] W. Detmold and C. J. D. Lin, Phys. Rev. D (2005) 054510 [hep-lat/0501007].[19] S. R. Beane, E. Chang, W. Detmold, H. W. Lin, T. C. Luu, K. Orginos, A. Parreno andM. J. Savage et al. , Phys. Rev. D (2011) 014507 [arXiv:1104.4101 [hep-lat]].[20] L. s. Geng, X. l. Ren, J. Martin-Camalich and W. Weise, Phys. Rev. D (2011) 074024[arXiv:1108.2231 [hep-ph]].
21] L. Alvarez-Ruso, T. Ledwig, J. Martin Camalich and M. J. Vicente-Vacas, Phys. Rev. D (2013) 5, 054507 [arXiv:1304.0483 [hep-ph]].[22] A. Walker-Loud, PoS CD (2013) 017 [arXiv:1304.6341 [hep-lat]].[23] G. Colangelo and C. Haefeli, Nucl. Phys. B (2006) 14 [hep-lat/0602017].[24] J. A. McGovern and M. C. Birse, Phys. Lett. B (1999) 300 [hep-ph/9807384].[25] R. D. Young, D. B. Leinweber and A. W. Thomas, Prog. Part. Nucl. Phys. (2003) 399[hep-lat/0212031].[26] V. Bernard, T. R. Hemmert and U.-G. Meißner, Nucl. Phys. A (2004) 149 [hep-ph/0307115].[27] S. R. Beane, Nucl. Phys. B (2004) 192 [hep-lat/0403030].[28] D. Djukanovic, J. Gegelia and S. Scherer, Eur. Phys. J. A (2006) 337 [hep-ph/0604164].[29] J. A. McGovern and M. C. Birse, Phys. Rev. D (2006) 097501 [hep-lat/0608002].[30] M. R. Schindler, D. Djukanovic, J. Gegelia and S. Scherer, Phys. Lett. B (2007) 390[hep-ph/0612164].[31] V. Bernard, Prog. Part. Nucl. Phys. (2008) 82 [arXiv:0706.0312 [hep-ph]].[32] J. M. M. Hall, D. B. Leinweber and R. D. Young, Phys. Rev. D (2010) 034010[arXiv:1002.4924 [hep-lat]].[33] G. S. Bali, P. C. Bruns, S. Collins, M. Deka, B. Gl¨aßle, M. G¨ockeler, L. Greil and T. R. Hem-mert et al. , Nucl. Phys. B (2013) 1 [arXiv:1206.7034 [hep-lat]].[34] P. C. Bruns, L. Greil and A. Sch¨afer, Phys. Rev. D (2013) 5, 054021 [arXiv:1209.0980[hep-ph]].[35] S. R. Beane, W. Detmold, K. Orginos and M. J. Savage, arXiv:1410.2937 [nucl-th].[36] X. D. Ji, J. Phys. G (1998) 1181 [hep-ph/9807358].[37] M. Diehl, Phys. Rept. (2003) 41 [hep-ph/0307382].[38] M. Dorati, T. A. Gail and T. R. Hemmert, Nucl. Phys. A (2008) 96 [nucl-th/0703073].[39] P. Wein, P. C. Bruns and A. Sch¨afer, Phys. Rev. D (2014) 116002 [arXiv:1402.4979 [hep-ph]].[40] L. Greil, P. Wein, P. C. Bruns and A. Sch¨afer, arXiv:1406.6866 [hep-lat].[41] T. Becher and H. Leutwyler, Eur. Phys. J. C (1999) 643 [hep-ph/9901384].[42] J. W. Chen and X. Ji, Phys. Rev. Lett. (2002) 052003 [hep-ph/0111048].[43] A. V. Belitsky and X. Ji, Phys. Lett. B (2002) 289 [hep-ph/0203276].
44] D. Arndt and M. J. Savage, Nucl. Phys. A (2002) 429 [nucl-th/0105045].[45] S. i. Ando, J. W. Chen and C. W. Kao, Phys. Rev. D (2006) 094013 [hep-ph/0602200].[46] M. Diehl, A. Manashov and A. Sch¨afer, Eur. Phys. J. A (2007) 335 [hep-ph/0611101].[47] A. M. Moiseeva and A. A. Vladimirov, Eur. Phys. J. A (2013) 23 [arXiv:1208.1714 [hep-ph]].[48] V. Bernard and U.-G. Meißner, Phys. Lett. B (2006) 278 [hep-lat/0605010].[49] J. Bijnens and I. Jemos, Nucl. Phys. B (2012) 631 [arXiv:1103.5945 [hep-ph]].[50] J. Beringer et al. [Particle Data Group Collaboration], Phys. Rev. D (2012) 010001.[51] R. Baron et al. [ETM Collaboration], JHEP (2010) 097 [arXiv:0911.5061 [hep-lat]].[52] G. Colangelo and S. D¨urr, Eur. Phys. J. C (2004) 543 [hep-lat/0311023].[53] R. G. Edwards, G. Fleming, P. H¨agler, J. W. Negele, K. Orginos, A. V. Pochinsky, D. B. Ren-ner and D. G. Richards et al. , PoS LAT (2006) 121 [hep-lat/0610007].[54] J. D. Bratt et al. [LHPC Collaboration], Phys. Rev. D (2010) 094502 [arXiv:1001.3620[hep-lat]].[55] J. Bl¨umlein and H. B¨ottcher, Nucl. Phys. B (2010) 205 [arXiv:1005.3113 [hep-ph]].[56] J. R. Green, M. Engelhardt, S. Krieg, J. W. Negele, A. V. Pochinsky and S. N. Syritsyn,Phys. Lett. B (2014) 290 [arXiv:1209.1687 [hep-lat]].[57] Y. Aoki, T. Blum, H. W. Lin, S. Ohta, S. Sasaki, R. Tweedie, J. Zanotti and T. Yamazaki,Phys. Rev. D (2010) 014501 [arXiv:1003.3387 [hep-lat]].[58] C. Alexandrou, M. Constantinou, V. Drach, K. Jansen, C. Kallidonis and G. Koutsou, PoSLATTICE (2013) 292 [arXiv:1312.2874 [hep-lat]].[59] S. Dinter, C. Alexandrou, M. Constantinou, V. Drach, K. Jansen and D. B. Renner, Phys.Lett. B (2011) 89 [arXiv:1108.1076 [hep-lat]].[60] J. Green, J. Negele, A. Pochinsky, S. Krieg and S. Syritsyn, PoS LATTICE (2011) 157[arXiv:1111.0255 [hep-lat]].[61] G. S. Bali, S. Collins, M. Deka, B. Gl¨aßle, M. G¨ockeler, J. Najjar, A. Nobile and D. Pleiter etal. , Phys. Rev. D (2012) 054504 [arXiv:1207.1110 [hep-lat]].[62] B. J¨ager, T. D. Rae, S. Capitani, M. Della Morte, D. Djukanovic, G. von Hippel, B. Knipp-schild and H. B. Meyer et al. , arXiv:1311.5804 [hep-lat].[63] C. Alexandrou, J. Carbonell, M. Constantinou, P. A. Harraud, P. Guichon, K. Jansen, C. Kalli-donis and T. Korzec et al. , Phys. Rev. D (2011) 114513 [arXiv:1104.1600 [hep-lat]].
64] M. Dorati, T. A. Gail and T. R. Hemmert, PoS LAT (2007) 071 [arXiv:0710.0541 [hep-lat]].[65] W. Detmold, W. Melnitchouk, J. W. Negele, D. B. Renner and A. W. Thomas, Phys. Rev.Lett. (2001) 172001 [hep-lat/0103006].[66] W. Detmold, W. Melnitchouk and A. W. Thomas, Phys. Rev. D (2002) 054501 [hep-lat/0206001].[67] W. Detmold, W. Melnitchouk and A. W. Thomas, Mod. Phys. Lett. A (2003) 2681 [hep-lat/0310003].[68] P. Wang and A. W. Thomas, Phys. Rev. D (2010) 114015 [arXiv:1003.0957 [hep-ph]].[69] D. Diakonov and V. Y. Petrov, Nucl. Phys. B (1986) 457.[70] D. Diakonov, V. Y. Petrov and P. V. Pobylitsa, Nucl. Phys. B (1988) 809.[71] M. Wakamatsu and Y. Nakakoji, Phys. Rev. D (2006) 054006 [hep-ph/0605279].[72] K. Goeke, J. Ossmann, P. Schweitzer and A. Silva, Eur. Phys. J. A (2006) 77 [hep-lat/0505010].[73] V. Bernard, T. R. Hemmert and U.-G. Meißner, Phys. Lett. B (2003) 137 [hep-ph/0303198].[74] C. Hacker, N. Wies, J. Gegelia and S. Scherer, Phys. Rev. C (2005) 055203 [hep-ph/0505043].[75] V. Pascalutsa and M. Vanderhaeghen, Phys. Lett. B (2006) 31 [hep-ph/0511261].[76] P. C. Bruns, L. Greil and A. Sch¨afer, Eur. Phys. J. A (2012) 16 [arXiv:1105.6000 [hep-ph]].[77] P. E. Shanahan, A. W. Thomas and R. D. Young, Phys. Rev. D (2013) 11, 114515[arXiv:1301.6861 [nucl-th]].(2013) 11, 114515[arXiv:1301.6861 [nucl-th]].