Isospin splittings in the light baryon octet from lattice QCD and QED
Sz. Borsanyi, S. Dürr, Z. Fodor, J. Frison, C. Hoelbling, S.D. Katz, S. Krieg, Th. Kurth, L. Lellouch, Th. Lippert, A. Portelli, A. Ramos, A. Sastre, K. Szabo
IIsospin Splittings in the Light-Baryon Octet from Lattice QCD and QED
Sz. Borsanyi, S. D¨urr,
1, 2
Z. Fodor,
1, 2, 3
J. Frison,
4, 5, ∗ C. Hoelbling, S.D. Katz,
3, 6
S. Krieg,
1, 2
Th. Kurth, L. Lellouch,
4, 5
Th. Lippert, A. Portelli,
4, 5, 7
A. Ramos,
4, 5, † A. Sastre,
4, 5 and K. Szabo (Budapest-Marseille-Wuppertal Collaboration) Department of Physics, Wuppertal University, Gaussstrasse 20, D-42119 Wuppertal, Germany IAS/JSC, Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany Institute for Theoretical Physics, E¨otv¨os University, P´azm´any Peter s´etany 1/A, H-1117 Budapest, Hungary Aix-Marseille Universit´e, CNRS, CPT, UMR 7332, 13288 Marseille, France Universit´e de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France MTA-ELTE Lend¨ulet Lattice Gauge Theory Research Group, Budapest, Hungary School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom (Dated: June 19, 2018)While electromagnetic and up-down quark mass difference effects on octet baryon masses are verysmall, they have important consequences. The stability of the hydrogen atom against beta decay isa prominent example. Here we include these effects by adding them to valence quarks in a latticeQCD calculation based on N f =2+1 simulations with 5 lattice spacings down to 0.054 fm, latticesizes up to 6 fm and average up-down quark masses all the way down to their physical value. Thisallows us to gain control over all systematic errors, except for the one associated with neglectingelectromagnetism in the sea. We compute the octet baryon isomultiplet mass splittings, as well asthe individual contributions from electromagnetism and the up-down quark mass difference. Ourresults for the total splittings are in good agreement with experiment. The existence and stability of atoms and ordinary mat-ter rely heavily on the fact that neutrons are slightlymore massive than protons. The difference in the massof these two particles has been measured very preciselyand is only 0.14% of their average mass [1]. Althoughit has yet to be shown from first principles, we believethat this tiny difference results from the competition be-tween electromagnetic (EM) effects proportional to thefine structure constant α ≡ e / (4 π ) and mass isospinbreaking effects proportional to the mass difference of upand down quarks δm ≡ m u − m d . Here, we study thisissue in the light-baryon octet. In particular, we computemass splittings in the nucleon ( N ), Σ, and Ξ isospin mul-tiplets using lattice QCD, to which we add QED in thevalence quark sector. Although one would also have toaccount for QED contributions from sea quarks to have acomplete calculation, these effects are suppressed, as dis-cussed below. Moreover, the approach taken here allowsus to use a very rich set of QCD gauge configurationsthat we have already generated [2–4]. Eliminating theuncertainty associated with neglecting QED sea-quarkcontributions would require performing completely newsimulations, implementing reweighting techniques [5, 6],or using EM current insertion methods [7] and includingquark-disconnected contributions. Such a computationis beyond the scope of the present work.Because mass and EM isospin symmetry breaking cor-rections are small and of comparable size, it is legitimateto expand the standard model in powers of δm and α , as-suming O ( δm ) ∼ O ( α ) [8]. This expansion is expected toconverge very rapidly, with each subsequent order con-tributing ∼
1% of the previous one. Given the size ofother uncertainties in our calculation, we can safely work at LO in this expansion, i.e. at O ( δm, α ). The physical point.–
In the absence of weak interac-tions and for energies smaller than the charm-anticharmthreshold, the standard model of quarks has five param-eters that must be fixed by comparison to experiment.Here we trade these parameters for observables whichare particularly sensitive to them: 1) the lattice spacing, a , for the mass of the decuplet baryon Ω − –alternativelythe isospin averaged Ξ mass–as in [9], 2) the average u - d mass, m ud , for M π + , 3) the strange mass, m s , for M K χ ≡ ( M K + + M K − M π + ) /
2, 4) δm for the mass-squared difference ∆ M K ≡ M K + − M K , and 5) bare α for its renormalized value because it does not renormalizein our quenched QED calculation. The physical point isthen reached by tuning these observables to their physi-cal values given in [1], while taking the continuum a → L → ∞ limits. Separating EM and δm contributions.– In addition tocomputing the total splittings, it is interesting to sep-arate them into a contribution coming from δm and onecoming from QED. We define the EM contribution bysetting δm = 0 via ∆ M = M uu − M dd = 0, with allother parameters tuned to their physical values. Here, M ¯ qq is the mass of a neutral meson ¯ qq , q = u, d , whosepropagator includes only quark-connected diagrams. Us-ing the χ PT results of [10], it is straightforward to showthat the difference of these mesons’ squared masses is∆ M = 2 B δm + O ( αm ud , δmm ud , αδm, α ), where B is the N f =2 quark condensate parameter. Close to thephysical point, O ( m ud ) can be counted like O ( δm ). Thus,our definition of the EM contribution differs from anyother valid one by corrections of the size of NLO isospin a r X i v : . [ h e p - l a t ] J a n breaking terms, which are one order higher than the oneto which we work here. To obtain the contribution from δm , we set α = 0 and all other parameters to their phys-ical values. In particular, the physical value of ∆ M isobtained from the analysis of ∆ M K briefly described be-low and by computing the value of ∆ M correspondingto the physical ∆ M K . This analysis and its implicationsfor Dashen’s theorem [11] and m u,d , which are very in-teresting in their own right, will be discussed elsewhere[12]. Simulation details and parameters.–
We start from our47, isospin symmetric, N f =2+1, QCD ensembles, ob-tained from simulations with pion masses down to120 MeV, lattice sizes up to 6 fm, five lattice spacingsdown to 0.054 fm, and more than 1000 trajectories each[4]. To these ensembles, we add QED gauge degrees offreedom in a way which has now become standard [13–15]. For each QCD gauge configuration, we generate anEM field A µ ( x ) defined on the links, using the noncom-pact EM action (in Coulomb gauge) and the methodsdetailed in [14]. The gauge potential is then exponen-tiated as U QED µ ( x ) = exp [ iqeA µ ( x )]. Unlike the QCDlinks, our QED links undergo no smearing before beingcoupled to quarks. Similarly, we have not added a cloverimprovement term for the U(1) field. The U(1) fields arethen multiplied with the SU(3) gauge variable on eachlink and inserted into the Wilson-Dirac operator associ-ated with the quark of charge q before inversion. The re-sulting quark propagators are combined into meson andbaryon two-point functions. The extended sources andsinks used are the same as in [3, 4].For most of our SU(3) ensembles, we have generatedtwo valence data sets, which include QED with the phys-ical value of α . In the first set (set 1), the bare, valence u , d , and s quark masses are individually tuned so thattheir PCAC values approximately reproduce the corre-sponding ensemble’s light and strange sea-quark PCACmasses. Thus, we subtract the α/a divergences in thevalence bare quark masses, which come from the EMself-energy, as described in [15]. In the second set (set2), we choose m d to be slightly more massive than in thefirst set so that ∆ M scatters around its physical valuefrom ensemble to ensemble. We have one additional va-lence data set (set 3) in which α is varied. The latterincludes a point with α ∼ α ph and ∆ M (cid:39) ∆ M , ph , asecond with α ∼ α ph / M , and a thirdwith α ∼ M (cid:39)
0. The superscript ph indicatesthat we are referring to the physical value of a quantity.We have 74 valence points in total, which are shown inthe M uu - M dd plane in Fig. 1. This rich collection of dataallows us to gain full control over the dependence of thesplittings on all of the relevant parameters. Analysis of meson and baryon correlators.–
The time de-pendence of the π + , K + , K , and the Ω − or Ξ two-pointfunctions is fitted, in the asymptotic regime, to a hy- M dd ( M e V ) M uu (MeV )mass isospin m u = m d linephysical point physical m u m d region physical δm region FIG. 1.
Valence data sets plotted in the M dd vs M uu plane. The red squares (set 1) lie along the mass isospin line M uu = M dd , and the blue circles (set 2) are scattered aroundan estimate of the ∆ M , ph region, obtained from the resultsof [1, 16]. The green triangles (set 3) are points in which α isvaried away from its physical value. For clarity, points with M π >
450 MeV are not shown. perbolic cosine and an exponential, respectively. For theisospin multiplets whose splittings we wish to determine,we perform a simultaneous, correlated fit to the two-pointfunctions of the two members of the multiplet in which wereplace the individual hadron mass parameters by theiraverage and their difference. The time ranges for corre-lator fits are determined after a systematic study of thegoodness of fit as a function of initial and final fittingtimes. The choices made here are very similar to thoseof [3, 4].
Interpolating to the physical point and determining theindividual EM and δm contributions.– Having deter-mined the isospin splittings and relevant hadron massesin lattice units for each of our QCD plus QED data sets,we have to convert them to physical units and extrap-olate them to the continuum and infinite-volume limits.We also must interpolate the splittings to the physicalmass point, as well as to the mass and EM isospin limits.We determine the five lattice spacings simultaneouslyfrom a combined fit of the data with ∆ M (cid:39)
0, forthe isospin symmetric observable aM Ω − or, alternatively, aM Ξ , using the techniques of [3, 4, 9]. The isospin masssplitting ∆ M X of a hadron X is naturally described bythe LO isospin expansion∆ M X = A X α + B X ∆ M , (1)where ∆ M substitutes for δm . The coefficients A X and B X still depend on the isospin symmetric parameters ofthe theory, e.g. m ud or m s . We find that their depen-dences on these parameters are well described by a linearexpansion in M π + and in M K χ for the range of massesretained below. − . − − . . .
52 0 20 40 60 80 100 120 ∆ Q E D M Ξ ( M e V ) /L (MeV) a = 0 .
11 fm a = 0 .
09 fm a = 0 .
07 fm a = 0 .
06 fm a = 0 .
05 fm
FIG. 2.
Example of FV corrections to ∆ QED M Ξ , plotted asa function of /L . The dependence of the lattice results onall other variables has been subtracted using a fit of the typedescribed in the text. Results with a same /L and a are av-eraged because they show no systematic residual dependenceon the other simulation parameters, in particular on quarkmass. The linear fit in /L , which is performed for pointswith M π + ≤
500 MeV , has a χ / DOF = 59 ./ . It is plottedas a solid curve, with its 1 σ band. We must also account for discretization and finite-volume (FV) effects. The latter are particularly im-portant because of the presence of the massless pho-ton. Using techniques from [17], and performing ap-propriate asymptotic expansions, it is straightforward toshow that the leading finite-volume term in scalar andspinor QED is proportional to 1 /L . We find these cor-rections to be generically large. For instance, in boxeswith L = 1 . . QED M Ξ , the QEDcontribution to ∆ M Ξ ≡ M Ξ − M Ξ − , ranges from 123%to 76%. This is illustrated in Fig. 2. In our calcula-tion, L extends up to 6 fm, where the figure indicatesa 36% FV correction. While still large, our correctionsare sufficiently small that they may be described witha low-order polynomial in 1 /L . This is confirmed bythe data in Fig. 2, which show no sensitivity to termsbeyond linear order in 1 /L . The same is true of our re-sults for ∆ M N ≡ M p − M n , which have a slope in 1 /L which is compatible with that of ∆ QED M Ξ , but withlarger statistical errors. Not surprisingly, the slope in∆ M Σ ≡ ∆ [∆ I =2] M Σ = M Σ + − M Σ − is consistent withzero: the absolute values of the two particles’ charges areequal.Concerning discretization effects, the improvement ofthe QCD action implies O ( α s a, a ) corrections to A X and B X . However, due to the lack of improvement in theQED sector, discretization effects on A X are O ( a ). Inour analysis, we include O ( a ) QED discretization effects to A X as well as O ( α s a, a ) QCD ones to B X .Combining all of this information yields a nine param-eter description of each of the mass splittings. In thenotation of Eq. (1), this corresponds to A X = a X + a X [ M π − ( M ph π ) ] + a X [ M K χ − ( M ph K χ ) ]+ a X a + a X L , (2) B X = b X + b X [ M π − ( M ph π ) ] + b X [ M K χ − ( M ph K χ ) ]+ b X f ( a ) (3)where the a Xi and b Xi are the parameters and f ( a ) = α s a or a , alternatively. For each splitting, among the ninepossible parameters, we have retained all combinationswhich are such that adding one more dependence to thefit causes the associated parameter to be consistent withzero within one standard deviation. Error estimation–
Our analysis methodology makes noassumptions beyond those of the fundamental theory, ex-cept for the isospin symmetry which is maintained in thesea and whose consequences we discuss below. Howeverthe analysis does depend on several choices that can besources of systematic uncertainties.To deal with these uncertainties, we proceed with themethod put forward in [9]. More specifically, we considerthe following variations in our analysis procedure. Forthe time ranges of the correlator fits, we consider twoinitial red fit times, one for which we expect negligibleexcited state contributions and a red second, more ag-gressive one. This estimates the uncertainty due to con-tributions from excited states. Regarding the choice ofscale setting quantities, we consider two possibilities: themass of the Ω − and that of the isospin averaged Ξ. To es-timate the uncertainty associated with the truncation ofthe Taylor expansion used to interpolate these two massesto physical M π + , we vary the fit ranges by excluding alldata with pion mass above 400 and 450 MeV. To esti-mate part of this same uncertainty for the isospin split-tings, we consider cuts at M π + = 450 and 500 MeV, sincetheir M π + dependence is very mild. These cuts also pro-vide an estimate of the uncertainty associated with FVcorrections, as our simulations keep LM π ∼
4, implyingcuts on 1 /L as low as 1 /L <
100 MeV. Part of the un-certainty associated with the continuum extrapolation isdetermined by considering either α s a or a discretizationerrors. Finally, to estimate any additional uncertaintyarising from the truncation of these expansions, we con-sider the result of replacing either A X or B X by Pad´eexpressions. These are obtained by considering that theexpansions of A X and B X in Eqs. (2) and (3) are the firsttwo terms of a geometric series which we resum. This re-summation is not applied to the FV corrections. Instead,we try adding a 1 /L term to either the Taylor or Pad´eforms. In all cases, we find the coefficient of this term tobe consistent with zero. X ∆ M X ∆ QED M X ∆ QCD M X N − . . − . − . . − . − . − . − . Isospin breaking mass differences in MeV for mem-bers of the baryon octet. The first error is statistical and thesecond is systematic. As discussed in the text, we guesstimatethe QED quenching uncertainties on the EM contributions tobe O (10%) . Propagating the uncertainty in ∆ QED M K yieldsan O (4%) error on the δm contributions. The quenching un-certainties on the total splittings can then be obtained byadding those of the EM and δm contributions in quadrature.These guesstimates are not included in the results. These variations lead to 2 = 128 different fits foreach of the isospin splittings and parameter combina-tions. Correlating these with the 128 fits used to deter-mine ∆ M , ph and allowing various parameter combina-tions but discarding fits with irrelevant parameters, weobtain between 64 and 256 results for each observable.The central value of a splitting is then the mean of theseresults, weighted by the p -value. The systematic error isthe standard deviation. Because we account for all cor-relations, these fit qualities are meaningful. The wholeprocedure is repeated for 2000 bootstrap samples and thestatistical error is the standard deviation of the weightedmean over these samples. We have also checked that theresults are changed only negligibly (far less than the cal-culated errors) if they are weighted by 1 instead of by the p -value.The δm corrections that we do not include in the seaare NLO in isospin breaking and can safely be neglected.The neglected O ( α ) sea-quark contributions break fla-vor SU(3). Moreover, large- N c counting indicates thatthey are O (1 /N c ). Combining the two suppression fac-tors yields an estimate ( M Σ − M N ) / ( N c M N ) (cid:39) .
09. Asmaller estimate is obtained by supposing that these cor-rections are typical quenching effects [19] that are SU(3)suppressed, or by using [20] the NLO χ PT results of [10].However, in the absence of direct quantitative evidence,it is safer to assume that the EM contributions to thesplittings carry an O (10%) QED quenching uncertainty. Final results and discussion.–
Combining the methodsdescribed above, we obtain our final results for the totaloctet baryon isospin splittings ∆ M N , ∆ M Σ , and ∆ M Ξ defined above. These results, together with those ob-tained for the EM and δm contributions, are summa-rized in Table I. We also plot them in Fig. 3, with theexperimental values for the full splittings. Our resultsare compatible with experiment.Concerning the separation into δm and EM contribu-tions, there exist very few determinations of these quan-tities up to now. In the review [21], hadron EM split-tings were estimated using a variety of models and Cot- − − − − − − − − − M N ∆ M Σ ∆ M Ξ ( M e V ) totalQCDQEDexp. FIG. 3.
Results for the isospin mass splittings of the octetbaryons (total), the individual contributions to these split-tings from the mass difference m u − m d (QCD) and from EM(QED). The bands denote the size of these results. The errorbars are the statistical and total uncertainties (statistical andsystematic combined in quadrature). For comparison, the ex-perimental values for the total splittings are also displayed. tingham’s formula for the nucleon. These estimates arecompatible with our results within ∼ . σ . The EM nu-cleon splitting has recently been reevaluated with Cot-tingham’s formula in [22], yielding a result which is inbetter agreement with ours. ∆ M N has further been stud-ied with sum rules in [23].Besides the entirely quenched, pioneering work of [24],ours is the only one in which the baryon octet isosplit-tings are fully computed. The only other lattice calcula-tion of the full nucleon splitting is presented in [25]. Likeours, it implements QED only for valence quarks. Whiletheir ∆ QCD M N agrees very well with ours, agreementis less good for the EM contribution and total splitting,which they find to be 0 . − . δm contributionsto the baryon splittings, in N f =2 [7, 26] and N f =2+1[27, 28] simulations. The results of [26–28] rely on impre-cise phenomenological input to fix m u /m d or ( m u − m d ).The estimate for ∆ QED M K of [16] is used directly in[26, 28] and that of [29], indirectly in [27]. The most re-cent N f =2 calculation [7] actually determines ∆ QED M K in quenched QED, as we do here for N f =2+1. ∆ QCD M N is computed in [7, 26, 27] while all three QCD splittingsare obtained in [28]. The latter is also true in [30], where N f =2+1 lattice results are combined with SU(3) χ PTand phenomenology. Agreement with our results is typ-ically good. In all of these calculations, the range ofparameters explored is smaller than in ours, making itmore difficult to control the physical limit.L.L. thanks Heiri Leutwyler for enlightening discus-sions. We also thank J´erˆome Charles and Marc Knechtfor helpful conversations. Computations were performedusing the PRACE Research Infrastructure Resource JU-GENE at FZ J¨ulich, with further HPC resources pro-vided by GENCI-[IDRIS/CCRT] (Grant No. 52275) andFZ J¨ulich, as well as using clusters at Wuppertal andCPT. This work was supported in part by the OCEVULabex (ANR-11-LABX-0060) and the A*MIDEX Project(ANR-11-IDEX-0001-02) funded by the “Investissementsd’Avenir” French government program managed by theANR, by CNRS Grants GDR No. 2921 and PICS No.4707, by EU Grants FP7/2007-2013/ERC No. 208740and No. MRTN-CT-2006-035482 (FLAVIAnet), and byDFG Grants No. FO 502/2 and No. SFB-TR 55. ∗ Present address: School of Physics and Astronomy, Uni-versity of Edinburgh, Edinburgh EH9 3JZ, UK † Present address: NIC, DESY Platanenallee 6, D-15738Zeuthen, Germany[1] J. Beringer et al. (Particle Data Group), Phys.Rev.
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