aa r X i v : . [ h e p - l a t ] J un Baryon structure from Lattice QCD
C. Alexandrou , Department of Physics, University of Cyprus,P.O. Box 20357, Nicosia, CY-1678 Cyprus Computation-based Science and Technology Research Center,Cyprus Institute, 20 Kavafi Street Nicosia 2121, Cyprus
We present recent lattice results on the baryon spectrum, nucleon electromagnetic and axial formfactors, nucleon to ∆ transition form factors as well as the ∆ electromagnetic form factors. Themasses of the low lying baryons and the nucleon form factors are calculated using two degenerateflavors of twisted mass fermions down to pion mass of about 270 MeV. We compare to the resultsof other collaborations. The nucleon to ∆ transition and ∆ form factors are calculated in a hybridscheme, which uses staggered sea quarks and domain wall valence quarks. The dominant dipolenucleon to ∆ transition form factor is also evaluated using dynamical domain wall fermions. Thetransverse density distributions of the ∆ in the infinite momentum frame are extracted using theform factors determined from lattice QCD.
PACS numbers: 11.15.Ha, 12.38.Gc, 12.38.Aw, 12.38.-t, 14.70.Dj
I. INTRODUCTION
During the last five years we have seen tremendousprogress in dynamical lattice simulations using a num-ber of different fermion discretization schemes with quarkmasses reaching closer to the physical pion mass. Manycollaborations are contributing to this progress. The Eu-ropean Twisted Mass Collaboration (ETMC) is usingtwisted mass fermions (TMF), which provide an attrac-tive formulation of lattice QCD that allows for automatic O ( a ) improvement, infrared regularization of small eigen-values and fast dynamical simulations [1, 2]. Automatic O ( a ) improvement is obtained by tuning only one param-eter requiring no further improvements on the operatorlevel. A drawback of twisted mass fermions is the O ( a )breaking of isospin symmetry, which is only restored inthe continuum limit. In the baryon sector it has beenshown that this isospin breaking is consistent with zerowithin our statistical accuracy by evaluating the massdifference between ∆ ++ (∆ − ) and ∆ + (∆ ) [3, 4]. This isin agreement with a theoretical analysis [5, 6] that showspotentially large O ( a ) flavor breaking effects to appearin the π -mass but to be suppressed in other quantities.A number of collaborations, as for example QCDSF [7],PACS-CS [8], BMW [9] and CERN [10] are using im-proved Clover fermions for their simulations. It is worthmentioning that PACS-CS has simulations very close tothe physical pion mass albeit in a small volume, whereasthe Wuppertal group recently calculated meson massesand the decay constants using N F = 2 + 1 configurationssimulated at the physical pion mass [11]. A number ofgroups adopted a hybrid approach to compute hadronicmatrix elements taking advantage of the efficient simula-tion and availability of staggered sea fermions producedby the MILC collaboration [12] and the chiral symmetryof domain wall fermions. The Lattice Hadron PhysicsCollaboration (LHPC) has been particularly active inproducing results on a number of key observables [13, 14],some of which will be discussed in Sections IV and V. A very promising recent development is the simula-tion of dynamical chiral fermions using large volumesand at small enough pions masses. The RBC-UKQCDcollaboration is generating gauge configurations using N F = 2 + 1 domain wall fermions (DWF) [15], whereasthe JLQCD Collaboration is producing dynamical config-urations with two flavors of overlap fermions [16]. Mostof the current simulations are done using volumes of spa-tial length L such that m π L > . II. HADRON SPECTRUM
The masses of the lowest lying hadrons of a given setof quantum numbers are readily calculated by comput-ing the two-point function at zero momentum: C h ( t ) = P x h | J h ( x , t ) J † h (0) | i . Choosing good interpolatingfields and applying smearing techniques ensure groundstate dominance at short time separation t so that gaugenoise is kept small [18]. In Figs. 1 and 2 we compare re-cent results on the low lying baryon spectrum using dy-namical twisted mass [3, 19] and clover fermions [8] andwithin the hybrid approach [20] (staggered sea and do-main wall valence quarks). The level of agreement of lat-tice QCD results using a variety of fermion discretizationschemes seen in Figs. 1 and 2 before taking the contin-uum limit or other lattice artifacts into account is quiteimpressive. Small discrepancies seen mainly in the decu-plet masses can be attributed to lattice artifacts and asystematic analysis of these effects is performed by eachcollaboration before extracting the final continuum val-ues. In particular results using staggered fermions may FIG. 1: Comparison of masses for the low lying octet baryons.Results using N F = 2 TMF are shown by the filled (black)triangles for L = 2 . L = 2 . a = 0 .
089 fm and with the open (red) triangles for L = 2 . a = 0 .
070 fm. Results with the hybrid action are shownwith the (green) asterisks for a = 0 .
124 fm and results using N F = 2+1 Clover fermions with the open (orange) circles and a = 0 . N F = 2+1 staggered fermions (filled (light blue) circles). Thephysical masses are shown by the (purple) star. suffer the most from cut-offs effects since the lattice usedis rather coarse as compared to those using twisted massand Clover fermions which have lattice spacings smallerthan 0.1 fm.Having reliable methods to extract the masses of thelow lying hadrons one can investigate techniques for theextraction of the masses of excited states. A number ofapproaches exist. A commonly used method is based onthe variational approach [21]: For a given N × N corre-lator matrix C kn ( t ) = h | J k ( t ) J † n (0) | i one defines the N principal correlators with λ k ( t, t ) as the eigenvalues of FIG. 2: Comparison of masses for the low lying decupletbaryons.The notation is the same as that of Fig. 1. C ( t ) − / C ( t ) C ( t ) − / , where t is small. Sincelim t →∞ λ k ( t, t ) = e − ( t − t ) E k (cid:0) e − t ∆ E k (cid:1) , k = 1 , . . . , N (1)the N principal effective masses tend (plateau) to the Nlowest-lying stationary-state energies of the hadrons withthe same quantum numbers. It is crucial to use very goodoperators so noise does not swamp signal and constructspatially extended operators using smearing of the quarkfields as well as applying link variable smearing. The useof a large set of appropriately constructed operators isalso very important. Despite recent calculations usingthis method [22] the issue of the ordering of the Roperresonance as compared to the negative parity partnerof the nucleon still remains unresolved. Maximum en-tropy methods have also be developed for the analysisof hadron two-point correlators and recent results can befound in Ref. [23]. A new method that relies solely on χ -minimization with an unbiased evaluation of errors canbe applied to extract the masses of the states on whichthe two-point correlator is sensitive on [24]. This methodwas applied to extract the excited states of the nucleonusing local correlators that are easily produced in latticesimulations. For this study two interpolating fields are FIG. 3: Probability distributions for the amplitudes andmasses in lattice units extracted from local correlators using N F = 2 Wilson fermions at pion mass 500 MeV on a latticeof spatial length 1.8 fm at β = 6 . J N .FIG. 4: As in Fig. 3 but using the interpolating field J ′ N . considered: J N ( x ) = ǫ abc ( u a Cγ d Tb ) u c J ′ N ( x ) = ǫ abc ( u Ta Cd b ) γ u c . (2)As can be seen from the histograms shown in Figs. 3and 4, one clearly identifies the first excited state in thepositive parity channel of the nucleon using rather low quality data. In addition, we observe that the state oflowest mass that is present in the mass spectrum of thecorrelator computed with J N is absent when using J ′ N .Instead the correlator with J ′ N has a lowest state thatdoes not show up when using J N . The conjecture is thatthis state is the Roper. III. FORM FACTORS
To extract information on hadron structure one needsto calculate coupling constants, such as the nucleon axialcharge g A , the πN and πN ∆ coupling constants, formfactors, moments of parton distribution functions andgeneralized form functions. In order to compute thesequantities we need to calculate the relevant three-pointfunctions, which, in addition to the forward propaga-tor needed for the calculation of the masses, require theevaluation of the sequential propagator. The three-pointfunction, related to the matrix element of the operator O between hadron states | h ′ > and | h > , is given by h G h ′ O h ( t , t ; p ′ , p ; Γ) i = X x , x exp( − i p ′ · x ) exp(+ i q · x ) h Ω | Γ βα T h J αh ′ ( x , t ) O ( x , t ) ¯ J βh ( , i | Ω i , (3)where for O we consider the electromagnetic and axialcurrents. We use sequential inversions through the sink,which allows us to obtain the three-point function for anymomentum transfer q and operator insertion but fixes thequantum numbers of the initial and final baryons. A. Nucleon Electromagnetic form factors
The elastic nucleon electromagnetic form factors arefundamental quantities characterizing important featuresof neutron and proton structure that include their size,charge distribution and magnetization. An accurate de-termination of these quantities in lattice QCD is timelyand important because of a new generation of preciseexperiments. The matrix element of interest is h N ( p ′ , s ′ ) | A µ | N ( p, s ) i = m N E N ( p ′ ) E N ( p ) / ¯ u ( p ′ , s ′ ) γ µ F ( q ) + iσ µν q ν m N F ( q ) u ( p, s ) , (4) where p ( s ) and p ′ ( s ′ ) denote initial and final momenta(spins) and m N is the nucleon mass, F (0) = 1 for theproton and F (0) measures the anomalous magnetic mo-ment. These form factors are connected to the electric, G E ( q ), and magnetic, G M ( q ), Sachs form factors bythe relations G E ( q ) = F ( q ) + q (2 m N ) F ( q ) G M ( q ) = F ( q ) + F ( q ) . (5)To extract the nucleon matrix element from lattice mea-surements, we calculate, besides the three point function G Nj µ N ( t , t ; p ′ , p ; Γ), the nucleon two-point function, G NN ( t, p ), and look for a plateau in the large Euclideantime behavior of the ratio R ( t , t ; p ′ , p ; Γ; µ ) = h G Nj µ N ( t , t ; p ′ , p ; Γ) ih G NN ( t , p ′ ; Γ ) i " h G NN ( t − t , p ; Γ ) i h G NN ( t , p ′ ; Γ ) i h G NN ( t , p ′ ; Γ ) ih G NN ( t − t , p ′ ; Γ ) i h G NN ( t , p ; Γ ) i h G NN ( t , p ; Γ ) i / t − t ≫ ,t ≫ ⇒ Π( p ′ , p ; Γ; µ ) . (6) where h G NN ( t, p ; Γ) i = X x e − i p · x Γ βα h Ω | T J α ( x , t ) ¯ J β ( , | Ω i . (7) We use the lattice conserved electromagnetic current, j µ ( x ), symmetrized on site x and projection matrices forthe Dirac indicesΓ i = 12 (cid:18) σ i
00 0 (cid:19) , Γ = 12 (cid:18) I
00 0 (cid:19) . (8)Throughout this work we use kinematics where the fi-nal nucleon state is produced at rest and therefore q = p ′ − p = − p . For the polarized matrix element one canconstruct an optimal linear combination for the nucleonsink, which in Euclidean time is given by S m ( q ; i ) = X k =1 Π( − q ; Γ k ; µ = i ) = C m N ( ( p − p ) δ ,i +( p − p ) δ ,i + ( p − p ) δ ,i ) G M ( Q ) (9) with Q = − q . This construction provides the maxi-mal set of lattice measurements from which G M ( Q ) canbe extracted requiring one sequential inversion. No suchimprovement is necessary for the unpolarized matrix el-ements given byΠ( , − q ; Γ ; µ = i ) = C q i m N G E ( Q ) (10)andΠ( , − q ; Γ ; µ = 4) = C E N + m N m N G E ( Q ) , (11)which yield G E ( Q ) with an additional sequential inver-sion. C = q m N E N ( E N + m N ) is a factor due to the normal-ization of the lattice states. FIG. 5: Nucleon isovector electric form factor using N F = 2TMF.FIG. 6: Nucleon isovector magnetic form factor using N F = 2TMF. Besides using an optimal nucleon source, the other im-portant ingredient in the extraction of the form factors isto take into account simultaneously in our analysis all thelattice momentum vectors that contribute to a given Q .This is done by solving the overcomplete set of equations P ( q ; µ ) = D ( q ; µ ) · F ( Q ) (12)where P ( q ; µ ) are the lattice measurements of the ratiogiven in Eq. (6) having statistical errors w k and usingthe different sink types, F = (cid:18) G E G M (cid:19) and D is an M × M being the number of current directions and momentumvectors contributing to a given Q . We extract the formfactors by minimizing χ = N X k =1 P j =1 D kj F j − P k w k ! (13)using the singular value decomposition of D . The anal-ysis described in this Section to extract G E ( Q ) and G M ( Q @ ) is also applied to the analysis of all form factorspresented in this work.The γ N → N transition contains isoscalar photoncontributions. This means that disconnected loop di- FIG. 7: Nucleon magnetic moment using N F = 2 TMF (filledcircles), N F = 2 + 1 DWF [31] (crosses) and N F = 2 Wilsonfermions [32](open triangles). The physical value is shown bythe star. agrams also contribute. These are generally difficultto evaluate accurately since the all-to-all quark propa-gator is required. In order to avoid disconnected dia-grams, we calculate the isovector form factors. Assuming SU (2) isospin symmetry, it follows that h p | ( ¯ uγ µ u − ¯ dγ µ d ) | p i − h n | ( ¯ uγ µ u − ¯ dγ µ d ) | n i = h p | (¯ uγ µ u − ¯ dγ µ d ) | p i and therefore by calculating the proton three-point function related to the matrix element of theright hand side of the above relation we obtain the isovector nucleon form factors G pE ( q ) − G nE ( q ) and G pM ( q ) − G nM ( q ).The results for the isovector electric and magnetic formfactors using N F = 2 twisted mass fermions are shownin Figs. 5 and 6 [25]. The lattice results on the electricform factor fall off slower as compared to a parametriza-tion of the experimental data [26] shown by the solidline, whereas the magnetic form factor is closer to ex-periment. This is consistent with recent high accuracyresults obtained by the LHP Collaboration [27] and theRBC-UKQCD Collaboration [28]. Fitting the magneticform factor to a dipole form we extract G M (0), whichdetermines the anomalous magnetic moment. We showits dependence on the pion mass in Fig. 7. Using chiraleffective theory with explicit nucleon and ∆ degrees offreedom to one-loop order the isovector anomalous mag-netic moment [29], the Dirac and Pauli radii can be ex-trapolated to the physical point [18, 29, 30]. There arethree fit parameters for the magnetic moment and thebest fit to the twisted mass data with the associated er-ror band is shown in Fig. 7. Multiplying the Pauli radiussquared with the magnetic moment yields an expressionwith only one-parameter like the Dirac radius that canshift the curves but does not affect their slopes. As canbe seen the physical magnetic moment is within the errorband whereas for the radii results closer to the physicalpoint are needed to check the predicted slope. FIG. 8: The Dirac radius squared (top) and Pauli radiussquared multiplied by the magnetic moment (bottom). Thenotation is the same as that of Fig. 7.FIG. 9: The nucleon axial charge using N F = 2 TMF and N F = 2 + 1 DWF. B. Nucleon axial form factors
The matrix element of the weak axial vector currentbetween nucleon states can be written as h N ( p ′ , s ′ ) | A µ | N ( p, s ) i = i m N E N ( p ′ ) E N ( p ) ! / ¯ u ( p ′ , s ′ ) " G A ( q ) γ µ γ + q µ γ m N G p ( q ) τ u ( p, s ) (14)where the axial isovector current A µ = ¯ ψ ( x ) γ µ γ τ ψ ( x ). FIG. 10: Top: The nucleon axial form factors G A ( Q ) for N F = 2 TMF and N F = 2 + 1 DWF. The dashed line is adipole fit to the lattice data whereas the solid line to experi-ment. Bottom: G p ( Q ) for N F = 2 TMF. The dashed line ispredicted from G A ( Q ) and Eq. (15). The dotted lines is thebest fit with the associated error band. Having computed the nucleon electromagnetic form fac-tors we can obtain the axial ones with no additionalinversions[25, 32]. The advantage here is that only theconnected diagram contributes. In addition at zero mo-mentum transfer we obtain the nucleon axial charge g A ,a quantity that is very accurately measured experimen-tally. We show in Fig. 9 results obtained using N F = 2twisted mass [25] and domain wall fermions [33]. Theleading one-loop chiral perturbation theory result for g A in the small scale expansion [34] can be used to extrapo-late lattice results to the physical point. Making a three-parameter fit to the twisted mass results we obtain thesolid curve shown in Fig. 9 together with the error banddetermined by allowing the fit parameters to vary withina χ increase by one unit from the minimum. Note thatthis error band does not include uncertainties in the fixedparameters. We obtain, at the physical point, a valuewith a large error that is lower than the experimentalvalue by an amount slightly larger than one standarddeviation. Results closer to the physical pion mass areneeded to reduce the error due to the chiral extrapola-tion.The Q -dependence of the nucleon axial form factors G A ( Q ) and G p ( Q ) using N F = 2 twisted mass fermionsis shown in Fig. 10. Our results for G A ( Q ) are in agree-ment with those obtained using N F = 2 + 1 domain wall fermions at a comparable value of the pion mass. The Q -dependence of G A ( Q ) can be well described by adipole Ansatz g / ( Q /m A + 1) as shown by the dashedline. This is what is usually used to describe experimen-tal data for G A ( Q ) where a value of m A ∼ . m A extracted from the lattice data is larger resulting ina slower fall off as compared to experiment shown by thesolid line. Assuming pion pole dominance G p ( Q ) can beobtained in terms of G A ( Q ) as G p ( Q ) = 4 m N /m π Q /m π G A ( Q ) . (15)In Fig. 10 we show with the dashed line what pion poledominance predicts if we use the fit determined from G A ( Q ). The error band shows the best fit to G p ( Q ) ifwe instead fit the strength and mass of the monopole inEq. 15. IV. N TO ∆ TRANSITION FORM FACTORS
The determination of the N to ∆ electromagnetic andaxial transition form factors requires the evaluation of thethree-point function h G ∆ O Nσ ( t , t ; p ′ , p ; Γ) i with a newset of inversions, where for O we consider the electro-magnetic and axial currents. The γ ∗ N ∆ matrix elementis given by h ∆( p ′ , s ′ ) | j µ | N ( p , s ) i = i s m ∆ m N E ∆ ( p ′ ) E N ( p ) ¯ u σ ( p ′ , s ′ ) G M ( Q ) K M σµ + G E ( Q ) K E σµ + G C ( Q ) K C σµ u ( p , s ) . The evaluation of the two subdominant electromagneticform factors G E ( Q ) and G C ( Q ), which are of pri-mary interest as far as the question of deformation isconcerned, require high accuracy. A lattice QCD calcu-lation accurate enough to exclude a zero value to onestandard deviation would point to deformation in thenucleon/∆ system. This is particularly relevant giventhe fact that extraction of these form factors from exper-iment involves modeling and therefore a non-zero valuefrom a first principles calculation even to one standarddeviation is an important result. Optimized sinks areconstructed to isolate the subdominant form factors [35]along the same lines as discussed for the polarized nu-cleon matrix element. In experimental searches for de-formation, it is customary to quote the ratios of the elec-tric and Coulomb quadrupole amplitudes to the magneticdipole amplitude, EMR or R EM = − G E ( Q ) G M ( Q ) R EM andCMR or R SM = − | ~q | m ∆ G C ( Q ) G M ( Q ) , in the rest frame of the∆. FIG. 11: The EMR calculated using quenched (stars) anddynamical (filled circles) Wilson fermions and in the hybridapproach (squares).FIG. 12: The CMR calculated using Wilson fermions and inthe hybrid approach.
Results on these ratios obtained using Wilson fermionsand the hybrid action are shown in Figs. 11 and 12. Lat-tice results at low Q are non-zero. The lattice valuesof CMR at small Q are less negative than experiment.As the pion mas decreases lattice results tend to becomemore negative approaching experiment. Therefore oneanticipates that for even smaller pion masses the dis-crepancy between lattice and experiment will be reducedsince pion cloud effects are expected to make CMR morenegative as we approach the physical regime [36].In Fig. 13 we compare results for the dipole form factor G M ( Q ) obtained within the hybrid approach and usingdynamical domain wall fermions at about the same mass.As can be seen there is very good agreement showing thatresults obtained within the non-unitary hybrid action arereliable.The invariant proton to ∆ + weak matrix element isexpressed in terms of four transition form factors as < ∆( p ′ , s ′ ) | A µ | N ( p, s ) > = i r „ m ∆ m N E ∆ ( p ′ ) E N ( p ) « / FIG. 13: G M ( Q ) in the hybrid approach and using N F =2 + 1 domain wall fermions. Quenched Wilson results are alsoincluded for comparison.FIG. 14: The ratio of N to ∆ axial transition form factors C A ( Q ) /C A ( Q ). The dotted line shows the pion pole dom-inance prediction for the hybrid case. The dashed and solidlines are fits to a monopole form for the hybrid and quenchedresults respectively.¯ u λ ∆ + ( p ′ , s ′ ) "„ C A ( q ) m N γ ν + C A ( q ) m N p ′ ν « ( g λµ g ρν − g λρ g µν ) q ρ + C A ( q ) g λµ + C A ( q ) m N q λ q µ u P ( p, s ) (16) where C A ( Q ) and C A ( Q ), the dominant form fac-tors, can be related assuming pion pole dominance like G A ( Q ) and G p ( Q ) are related in the nucleon case.In Fig. 14 we plot the ratio C A ( Q ) /C A ( Q ). Thedoted line shows the prediction assuming pion pole dom-inance after a dipole fit to the hybrid results on C A ( Q )is performed. As in the nucleon case the ratio does notfall off as rapidly and a fit to a monopole form to thesame hybrid results gives the dashed line. in GeV G E quenched Wilson, m p = 410 MeVhybrid, m p = 353 MeVdynamical Wilson, m p = 384 MeV FIG. 15: The Q -dependence of G E ( Q ). The green (red)line and error band show a dipole fit to the mixed action(quenched ) results. The value of G E , in units of e/ (2 m ),at Q = 0 are − . ±
291 for the quenched calculation, − . ±
67 for N F = 2 Wilson case and − . +1 . − . for thehybrid calculation. V. ∆ ELECTROMAGNETIC FORM FACTORSAND DENSITY DISTRIBUTION
Since the ∆(1232) decays strongly, experiments tomeasure its form factors are harder than for the N to∆ transition and yield less precise results. The ∆ formfactors can be computed using lattice QCD more accu-rately than can be currently obtained from experiment.The decomposition for the on shell γ ∗ ∆∆ matrix elementis given by h ∆( p f , s f ) | j µ EM | ∆( p i , s i ) i = A ¯ u σ ( p f , s f ) O σµτ u τ ( p i , s i ) O σµτ = − g στ (cid:20) a ( q ) γ µ + a ( q )2 m ∆ (cid:16) p µf + p µi (cid:17)(cid:21) − q σ q τ m (cid:20) c ( q ) γ µ + c ( q )2 m ∆ (cid:16) p µf + p µi (cid:17)(cid:21) , (17)where a ( q ), a ( q ), c ( q ), and c ( q ) are knownlinear combinations of the electric charge form factor G E ( q ), the magnetic dipole form factor G M ( q ), theelectric quadrupole form factor G E ( q ), and the mag-netic octupole form factor G M ( q ). An optimized sinkis constructed that isolated the subdominant electricquadrupole form factor [14, 37]. The results are shownin Fig. 15.The electric quadrupole form factor is particularly inter-esting because it can be related to the shape of a hadron.Just as the electric form factor for a spin 1/2 nucleon canbe expressed precisely as the transverse Fourier trans-form of the transverse quark charge density in the infinitemomentum frame [38], a proper field-theoretic interpre-tation of the shape of the ∆(1232) can be obtained byconsidering the quark transverse charge densities in this FIG. 16: Quark transverse charge density in a ∆ + polarizedalong the x -axis, with s ⊥ = +3 /
2. The light (dark) regionscorrespond with the largest (smallest) values of the density. frame. Fig. 16 shows the transverse density ρ ∆ T s ⊥ for a∆ + with transverse spin s ⊥ = +3 / + quark chargedensity is elongated along the axis of the spin (prolate). VI. CONCLUSIONS
Lattice QCD simulations are now being carried out inthe chiral regime by a number of collaborations. We haveshown that there is agreement among recent lattice re-sults using different fermion discretization schemes on thelow lying baryon spectrum and the nucleon form factors.Recent results on the low lying hadron spectrum wherelattice artifacts have been carefully examined are in per-fect agreement with experiment providing validation ofthe lattice approach and QCD itself [9]. Furthermore wehave shown that lattice QCD provides a framework forthe computation of quantities that can not be accuratelymeasured in experiment such as the ∆ form factors pro-viding valuable insight into the structure of such hadrons.We anticipate that other key hadronic quantities will becomputed to sufficient accuracy and with lattice artifactstaken into account thereby providing direct comparisonto experiment.
Acknowledgments
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