The nucleon to Delta electromagnetic transition form factors in lattice QCD
The electromagnetic nucleon to transition form factors are evaluated using two degenerate flavors of dynamical Wilson fermions and using dynamical sea staggered fermions with domain wall valence quarks. The two subdominant quadrupole form factors are evaluated for the first time in full QCD to sufficient accuracy to exclude a zero value, which is taken as a signal for deformation in the nucleon- system. For the Coulomb quadrupole form factor the unquenched results show deviations from the quenched results at low bringing dynamical lattice results closer to experiment, thereby confirming the importance of pion cloud contributions on this quantity.
pacs:11.15.Ha, 12.38.Gc, 12.38.Aw, 12.38.-t, 14.70.Dj
Despite several decades of scrutiny, the intrinsic shape of the nucleon, a fundamental building block of our world, is still not fully resolved Isgur et al. (1982); Capstick and Karl (1990). Although the nucleon is experimentally easily accessible in exclusive and inclusive scattering its spectroscopic quadrupole moment is zero since it has spin . However, this does not mean that the nucleon is spherically symmetric, since it can have an intrinsic deformation. Deformation is a common phenomenon in nuclear and atomic physics. Quantum mechanically, a multiphoton coincidence experiment could determine that a ground state of a diatomic molecule or nucleus has a deformed shape. However, usually in electromagnetic probes of microscopic systems, we are constrained to make measurements associated with one-photon exchange, corresponding to a matrix element of a one-body operator. In the case of a diatomic molecule, the one-body charge density of the state is spherically symmetric, and cannot reveal the deformation that is present in the system. For , however, when a nuclear or atomic system is well approximated by a deformed intrinsic state, it is still possible to observe its deformation using a one-body electromagnetic operator. In this case, the quadrupole moment is non-zero in the laboratory frame if the state is deformed. For collective rotation of the deformed intrinsic state Bohr and Mottelson (1975), the relation between the spectroscopic quadrupole moment, , measured in the laboratory frame and the intrinsic quadrupole moment, , in the body-fixed intrinsic frame is given by
where is the total angular momentum of the system in the laboratory frame, is the projection of onto the z-axis of the body-fixed intrinsic frame, and we have considered the sub-state with azimuthal quantum number . In the previous example of the diatomic molecule, although , Eq. (1) yields so that the deformation is unobservable. Similarly, in the case of a nucleon with , is zero although may not be Alexandrou et al. (2002); Alexandrou (2004). Since the has a non-zero quadrupole moment can be measured Alexandrou et al. (2007a). The electric and Coulomb transition amplitudes E2 and C2 between the nucleon and its resonance have the same property of revealing the presence of deformation in the nucleon, the , or both. Therefore, as in experiment, in this work we detect deformation by measuring E2 and C2.
In recent years, we have seen tremendous progress in experimental measurements of the subdominant quadrupole amplitudes Blanpied et al. (1996); Beck et al. (2000); Mertz et al. (2001); Pospischil et al. (2001); Joo et al. (2002); Sparveris et al. (2005); Stave et al. (2006); Sparveris et al. (2007), yielding very accurate results particularly at low . For a recent review of the experimental situation see Ref. Bernstein and Papanicolas (2007). These accurate measurements have motivated several recent theoretical studies both in lattice QCD Alexandrou et al. (2005a, b, 2006a) and in chiral effective theories Vanderhaeghen (2007); Gail and Hemmert (2007). On the lattice, hadron deformation can also be studied by investigating directly the charge distribution using density-density correlators Alexandrou et al. (2002, 2003); Alexandrou et al. (2004a), and it was shown that the rho meson is deformed Alexandrou and Koutsou (2007). The shows deviations from a spherical shape albeit with large statistical errors. The issue of deformation of the using density-density correlators is under study with improved lattice techniques Alexandrou et al. (2006b, c). For concise reviews see Alexandrou (2007a, 2004).
The focus of this work is a calculation of the nucleon - electromagnetic transition form factors within QCD. We use two different types of simulations: The first uses two degenerate flavors of dynamical Wilson fermions and the second a hybrid action. The hybrid action uses dynamical staggered sea quarks, with two degenerate light quarks and one fixed to the mass of the strange quark. These dynamical quark configurations are produced by the MILC collaboration Bernard et al. (2001) and represent a state-of-the-art simulation of the QCD vacuum with the three lightest flavors of quarks taken into account. For the light valence quarks we use two degenerate domain wall fermions. This approach has been used successfully to evaluate nucleon structure functions Negele et al. (2004); Renner et al. (2005); Hägler et al. (2005); Edwards et al. (2006a, b, c); Hägler et al. (2007) and the N to axial-vector form factors Alexandrou et al. (2007b, c, 2006d, d). In this work, we compare results calculated using the hybrid action to the results obtained using two degenerate flavors of dynamical Wilson fermions. Given the different lattice artifacts involved in the two approaches, agreement between them provides a consistency check of our lattice methodology. Comparison between the dynamical results and our quenched results Alexandrou et al. (2005a); Alexandrou (2007b) probes pion cloud contributions.
The first lattice study of the N to electromagnetic transition form factors Leinweber et al. (1993) was carried out in the quenched approximation with limited statistics. Although the mean value was negative, the statistical errors on the suppressed quadrupole amplitudes were large and a zero value could not be excluded. Nonetheless, this pioneering work set up the methodology for a more elaborate study that would soon become feasible once sufficient computing resources became available. Using the approach of Ref. Leinweber et al. (1993) we evaluated the transition form factors using quenched and dynamical Wilson quarks going to smaller quark masses than those of Ref. Leinweber et al. (1993) but only at the lowest -value allowed on our lattices Alexandrou et al. (2004b). However, although we increased statistics, the quadrupole form factors still had large errors. Constructing optimized sources that led to a large sample of statistically independent measurements for a given -value and carrying out sequential inversions through the source instead of through the current, we were finally able to obtain both quadrupole form factors to sufficient accuracy for a range of -values. This calculation, carried out in the quenched approximation, confirmed a non-zero value with the correct sign for both the quadrupole amplitudes Alexandrou et al. (2005a). Using this new methodology we extend in this work the calculation to the unquenched case. Initial unquenched results have been reported in Refs. Alexandrou et al. (2004c, 2005b, 2006a).
Lattice calculations at the physical quark mass are currently prohibitively expensive. Recently, progress in both hardware performance and algorithms Schroers (2007) has extended the range of accessible quark masses to lower masses, bringing lattice calculations into the region where they can be extrapolated using chiral perturbation theory. In this work, we use pions with masses as low as about 350 MeV. In a previous work Alexandrou et al. (2005a), we found a discrepancy for the CMR ratio between the quenched lattice results and experiment at low values of . On the other hand, for larger -values, the quenched results and experiment were in quantitative agreement. Our current results for CMR show deviations between unquenched and quenched results at the lowest momentum transfer. In fact, the previously observed discrepancy is reduced by our new unquenched results. This could indicate that pion cloud effects are quite important at low and at light pion masses, as also discussed in the framework of chiral perturbation theory Pascalutsa and Vanderhaeghen (2005). With pion masses in the range of 300 MeV as planned in the future one hopes to make progress in reliable chiral extrapolations to the physical regime. Such progress has been demonstrated recently in the extrapolation of the nucleon axial coupling within chiral effective theory Edwards et al. (2006a); Khan et al. (2006).
This paper is organized as follows: Section II gives the decomposition of the N to matrix element on the hadronic level, Section III and Section IV detail the lattice analysis and outline our strategy for extracting observables. Section V contains our results for the transition form factors. Finally, Section VI contains our conclusions and an outlook for further studies we intend to perform in the field.
Ii N to Delta matrix element
The electromagnetic transition matrix element can be expressed in terms of the three Sachs form factors Jones and Scadron (1973),
with the Lorentz-structure
The kinematic prefactors in Euclidean space are given by
where and denote initial and final momenta (spins), , and is a spin-vector in the Rarita-Schwinger formalism. We define and , is the lattice momentum transfer giving . The Rarita-Schwinger spin sum for the in Euclidean space is given by
and the Dirac spin sum
The magnetic dipole form factor, , is the dominant form factor. The electric and Coulomb quadrupole form factors, , and are subdominant. They are scalar functions depending on the momentum transfer squared. On the lattice we can only access space-like and therefore . The reason for using the Sachs parametrization in a lattice computation, as pointed out in Ref. Leinweber et al. (1993), is that the Sachs form factors do not depend strongly on the difference between the nucleon and the mass.
It is customary to quote the ratios of the electric and Coulomb quadrupole amplitudes to the magnetic dipole amplitude, (EMR) and (CMR). In the rest frame of the , they are given by Jones and Scadron (1973); Gellas et al. (1999)
Deformation of the nucleon- system will produce non-zero values of and . Using accurate measurements of the longitudinal-transverse response function at GeV it has been shown that a spherical nucleon - system is inconsistent with the experimental results Papanicolas (2003); Sparveris et al. (2005).
Iii Lattice matrix element
The most computationally demanding part in this evaluation is the calculation of the three-point correlation function given by
where an initial state with the quantum numbers of the nucleon is created at time zero and the final state with the quantum numbers of the is annihilated at a later time . The photon couples to one of the quarks in the nucleon at an intermediate time producing a . For the spin- source, , and the spin- source, , we use the interpolating fields
and for the projection matrices for the Dirac indices
For large Euclidean time separations and , the time dependence and field normalization constants cancel in the following ratio
where and are the nucleon and two-point functions given respectively by
For the Wilson fermionic action we use the lattice conserved electromagnetic current, , given by
symmetrized on site by taking where is the charge of a quark of flavor and is its hopping parameter. For domain wall fermions, we use the local four-dimensional electromagnetic current . This is not conserved and therefore, to relate lattice and continuum results, we need the renormalization constants , which are known. Throughout this work we choose a frame where the is at rest. For these kinematics we have , . At sufficiently large time separations, and , becomes independent of time yielding the desired ratio .
At the hadronic level, with the inclusion of complete sets of baryonic states and the use of Dirac and Rarita-Schwinger spinors the ratio of Eq. (12) leads to the relations
with the kinematical coefficients
A convenient method for the evaluation of three-point functions is the sequential inversion through the sink. This requires fixing the hadronic state at to be the with fixed vector index . The projection matrices at the sink are also fixed, but the operator inserted at any time can be left arbitrary. Therefore, with one sequential inversion one can evaluate the three-point function for a large set of lattice momentum transfer values , any current direction and any operator insertions at any intermediate time . One then looks for a plateau as a function of that determines . Eqs. (15-17) constitute a system of equations for the form factors and at each value of . It must be noted that while three independent measurements of suffice for the determination of the form factors, increasing the combinations of and photon momentum which are measured improves the statistical accuracy of the form factors. With that goal in mind, we observe that Eqs. (15-17) are identically zero for several values of , e.g. Eq. (15) is zero when or . Furthermore, for a given selection of , not all lattice rotations of are giving non-zero contributions. We therefore search for the linear combinations of Eqs. (15-17) which maximize the number of non-zero contributions in a lattice rotationally invariant fashion and construct the following optimal combinations
for . For we have
The three-point functions involved in and require one sequential inversion if one uses the appropriate linear combination directly in the construction of the sink. Thus, with two inversions we obtain the maximal number of lattice determinations of the form factors for all the allowed lattice photon momenta . The -type matrix element determines while the quadrupole form factors are extracted from the -type matrix element. The method is clearly superior to the method used in Refs. Leinweber et al. (1993); Alexandrou et al. (2004b) since the same CPU cost allows the evaluation of the form factors at all . It should be noted that the evaluation using the -type sink does not determine at the lowest allowed photon momentum (and at the equivalent momentum in the other two directions). For this reason we use, in addition, the combination
to get for the values of for which the -type sink vanishes. This linear combination gives
The full set of lattice measurements for the type , and matrix elements for all contributing values of and at a given are analyzed simultaneously. We denote by the lattice measurements for the ratios using the three-point function constructed using the sink-types described in Eqs. (18-21) and with their statistical error. If we denote the solution vector by we are led to the overcomplete set of equations
The kinematical prefactors, , are known analytically and we have inserted the continuum expressions as detailed in Eqs. (3) and (II). As already pointed out, we have lattice measurements for all possible vector current components and photon momentum vectors which contribute to a given value of . For such measurements, the matrix is an matrix of kinematical coefficients. The solution vector is determined from the minimization of the total
Defining and , the solution is obtained using the singular value decomposition of the matrix
where is an matrix, is a matrix and the are the non-negative, singular values of . The form factors are therefore given by
This strategy has been developed in Hägler et al. (2003) and also applied in Göckeler et al. (2004) and subsequent publications. The errors in the lattice measurements and the final error on the form factors are determined from the jackknife procedure. Having decided to use sequential inversions through the sink, the sink-source separation must be kept fixed. One would like to use the smallest sink-source separation that guarantees that excited state contributions are negligible. In our previous work we found that a sink-source separation,, of about 5 GeV is sufficient. We check in the next Section that this is also sufficient for this calculation by comparing to the results obtained when we increase by about 25%.
|V||# of confs||(GeV)||(GeV)||(GeV)|
|185 Orth et al. (2005)||0.1575||0.691(8)||0.701(9)||1.485(18)||1.687(15)|
|157 Orth et al. (2005)||0.1580||0.509(8)||0.566(12)||1.280(26)||1.559(19)|
|200 Urbach et al. (2006)||0.15825||0.384(8)||0.453(27)||1.083(18)||1.395(18)|
|# of confs||(fm)||(GeV)||(GeV)||(GeV)|
In the hybrid action approach we have applied hypercubic (HYP)-smearing Hasenfratz and Knechtli (2001) to the gauge fields. We have performed the computation using two different boundary conditions (b.c.) in the temporal direction. In the first case we impose Dirichlet b.c. on time slices 0 and 31 using only the first half of the lattice and in the second we use the full lattice with antiperiodic b.c. consistent with the b.c. used in the production of the gauge fields. The lattice spacing fm has been determined from heavy-quark spectroscopy Aubin et al. (2004) with a statistical uncertainty of . For Wilson fermions we use antiperiodic b.c.
To improve convergence to the nucleon and ground states we use Gaussian or Wuppertal smearing to create smeared quark fields as described in Refs. Alexandrou et al. (2007c, 2006e) with smearing parameters and .
Domain wall fermions (DWF) Kaplan (1992); Shamir (1993); Narayanan and Neuberger (1993) introduce an additional fifth dimension of length . They preserve the Ward-Takahashi identity Furman and Shamir (1995) even at finite lattice spacing in the limit . At sufficiently small values of the lattice spacing , the effect of a finite value of can be parameterized by an additional residual mass term in the Ward-Takahashi identity Blum et al. (2004); Blum (1999). This behavior describes a residual explicit breaking of chiral symmetry which can be minimized by choosing a sufficiently large extra dimension, . We have found that a value of in lattice units is sufficient to keep the residual mass, , at most one order of magnitude smaller than the domain wall quark mass, . The height parameter of the domain wall action has been chosen to be .
The DWF quark masses displayed in Tables 2 have been tuned by adjusting the lightest pseudoscalar meson in the Asqtad calculation Bernard et al. (2001) to have the same mass as the pseudoscalar meson using domain-wall fermions. For technical details of this tuning procedure, see Renner et al. (2005); Hägler et al. (2007).
Iv Extraction of form factors from lattice results
In this section we discuss several technical issues, beyond the general methodology described in the previous section, that must be addressed before the N to form factors can be extracted reliably from lattice measurements.
As mentioned already, given that sequential inversions are the most time consuming part of the calculation, the method of choice to calculate form factors that are functions of the momentum transfer squared is to perform sequential inversions through the sink. However, this approach requires that we fix the initial and final hadron states as well as the sink-source time separation, . Changing requires a new sequential inversion.
Therefore we must first determine the optimal value of . The criterion is to choose as small as possible so that statistical errors due to the exponential decrease of the signal are minimized but large enough so that we ensure that excited states with the nucleon and quantum numbers are suppressed. In order to decide on the value of we compare results at two different sink-source separations in the quenched theory and in the hybrid approach. The quenched theory is used to provide a guideline because the statistical errors are the smallest and therefore deviations due to contamination from excited states can be seen more easily. We use two time separations, namely and and perform the calculation at the smallest pion mass. In Fig. 2 we show results for the ratio defined as in Eq. (12) but using the optimal source of Eq. (18) for the three-point function and normalized with the appropriate kimematical factors such that the plateau value yields . Note that this is not what is actually done in the overconstrained analysis where the plateau value for each value of the momentum vector is extracted. Nevertheless shown in Fig. 2 gives an idea of the quality of the plateaus that are obtained. We show results for the four lowest -values when the sink-source separation is taken . We compare the results obtained for sink-source separation at the two lowest values and at the largest value shown in the figure. The third lowest value corresponding to for is not included since it will make the figure difficult to read. As can be seen the two time separations yield consistent results over a time range . Although in this comparison we use the same number of configurations the statistical errors are much smaller for the shorter time separation. Fitting to a constant over the plateau range but now for individual momentum vectors within the overconstrained analysis we obtained . The results are shown in Fig. 2, for the two different time separations using 100 configurations in each case. As can be seen there is very good agreement showing that a time separation of about 5 GeV is sufficient.
It is important to ensure that what we find in the quenched case carries over to the hybrid action. We expect pion cloud effects to be important for dynamical quarks and therefore we must check that the time evolution is large enough to allow the pion cloud to fully develop. Guided by our findings in the quenched theory we choose or about 5 GeV and or about 6.3 GeV. In Fig. 2 we show the ratio for the two heaviest quark masses whereas in Fig. 4 we show the results for at the smallest quark mass for the two different time separations. As can be seen, at the two lowest values yields the same plateau value for both time separations. For the statistical errors are larger associated with the larger time separation. The values extracted for at the smallest pion mass, where pion cloud effects are expected to be the largest, are also consistent for the two time separations, as can be seen in Fig. 4. The deviations observed at are due to the large statistical noise associated with the larger time evolution. We therefore conclude from this analysis that a time separation of about 5 GeV is also sufficient for our unquenched study. As a result we fix GeV or in lattice units to in the quenched case, to for dynamical Wilson fermions and to in the hybrid approach.
Another potential source of a systematic error is the spatial size of our lattices. Given that for the quenched case we use a lattice of spatial size of about 3 fm we expect finite volume effects to be negligible. Since we do not have dynamical Wilson configurations on a larger volume we test for finite size effects in the hybrid scheme for which, at the smallest quark mass, there are MILC configurations for spatial lattice size fm and fm giving and , respectively. In Fig. 4 we show results for for these two spatial sizes. Results on the smaller lattice are consistent with results on the larger lattice. This indeed shows that finite volume effects are small for . Since for all our quark masses, except the lightest mass dynamical Wilson fermions , we expect finite volume effects to be small.
V Results for the nucleon to transition form factors
The lattice results for the dominant dipole form factor are shown in Fig. 6 as we change the quark mass in the three types of simulations considered in this work, namely in the quenched theory denoted by , for two degenerate flavors of dynamical Wilson fermions, denoted by and in the hybrid scheme. They are also given in Tables III, IV and V of the Appendix. All the results discussed in this section in the hybrid approach are obtained on the lattices of temporal extent and using antiperiodic boundary conditions in the temporal direction.
In all three cases the magnetic dipole decreases with the quark mass. The dashed lines are fits to an exponential Ansatz whereas the dotted lines to a dipole Ansatz at the lightest quark mass. As can be seen, both provide a good description to the lattice results. The dipole mass that we find in the hybrid approach at the smallest pion mass is GeV. For comparison, a dipole fit to the experimental results yields GeV reflecting the faster fall off of the experimental results. In Fig. 6 we compare quenched and unquenched results for at the three quark masses. For similar pion mass the results are in agreement even for pion mass as low as 350 MeV. As can be seen, lattice results fall approximately on the same curve having a weaker dependence than the experimental results. In the momentum range considered here experimental results can be well described by a dipole form. In fact, whereas the experimental results fall off faster than the dipole form factor of the nucleon with , the lattice results display a weaker dependence yielding larger values for . It remains an open question whether decreasing the quark mass towards the physical limit will modify this dependence.
In Figs. 8 and 8 we show the results for the EMR and CMR ratios at the lightest quark mass, whereas we give the complete set of numbers in Tables III, IV and V. In the case of Wilson fermions we use both sink types and to extract and . For domain wall fermions, for which inversions are very expensive, we only use , which means that is not determined at the lowest momentum transfer as well as at some intermediate values. However, we obtain results at the second lowest momentum transfer squared which for the lowest pion mass on the lattice is about the same as the lowest momentum on our quenched lattice, enabling a nice comparison. The main conclusion of this study is that these ratios are non-zero and negative. As can be seen in Fig. 8, there is agreement between the quenched results for EMR and those obtained within the hybrid approach at the smallest quark mass. For the CMR ratio there is a notable difference: At the lowest value of the quenched results yield a value that it is clearly negative but smaller in magnitude than experiment. Results in the hybrid approach at the smallest quark mass are negative and larger in magnitude decreasing the gap between the lattice value and experiment. This confirms that pion cloud contributions are significant at small modifying the quenched results. This is consistent with the expectation from chiral perturbation theory Pascalutsa and Vanderhaeghen (2005) that pion cloud contributions become significant at low . Large pion contributions are also needed in phenomenological approaches Tiator et al. (2001, 2003); Sato and Lee (2001).
We have presented results for the N to electromagnetic transition form factors calculated within full QCD, using two degenerate flavors of dynamical Wilson fermions, and using a hybrid action with staggered sea quarks and domain wall valence quarks. Comparison of these results with previous quenched calculations Alexandrou et al. (2005a) do not reveal large unquenching effects in the case of the dominant dipole form factor, , for pion masses down to about 350 MeV. All lattice results for at these quark masses show a weaker dependence than experiment. Comparison of results with the hybrid action at two different volumes indicates that volume effects are not larger than our statistical errors. Therefore we can not attribute this weaker dependence to finite volume effects. Agreement between results using dynamical Wilson fermions and Domain Wall fermions that have different finite lattice spacing systematics suggests that it is unlikely to be due to discretization errors. However, we cannot presently extrapolate the lattice results to the continuum limit. Such an extrapolation would require three different lattice spacings at similar volumes and quark masses, which are currently not available.For the EMR and CMR ratios, on the other hand, unquenched lattice results begin to show agreement with experiment. One of the main results of this study is the quark mass dependence of the CMR ratio at low . Whereas quenched lattice results underestimate the magnitude of CMR at low , results in full QCD become more negative, bringing lattice results closer to experiment. This demonstrates the importance of full QCD dynamics in the quadrupole form factors as one enters the chiral regime.
The largest conceptual question as we enter the chiral regime in full QCD, is how to fully incorporate the physical effect of the decay of the into a pion and nucleon on the transition form factors. Even when the pion is sufficiently light that the could decay in an infinite box, the -wave pion in a small box can still be above the decay threshold. Naive extrapolation to the chiral limit without including the decay threshold could produce qualitative errors in form factors, such as the discrepancy presently observed between the lattice and experimental magnetic form factors. In the event that chiral perturbation theory is sufficiently accurate at and above the decay threshold, finite volume chiral perturbation theory would be an attractive framework for addressing decay channel physics. Otherwise, extension of finite volume techniques presently used for phase shifts will be required to address form factors. This presents an exciting and important research challenge.
Acknowledgements.We would like to thank B. Orth, Th. Lippert and K. Schilling Orth et al. (2005) as well as C. Urbach, K. Jansen, A. Shindler and U. Wenger Urbach et al. (2006) and the MILC collaboration for providing the unquenched configurations used in this work, as well as the LHP collaboration for providing forward propagators Renner et al. (2005). A.T. would like to acknowledge support by the University of Cyprus and the program “Pythagoras” of the Greek Ministry of Education and G. K. by the Cyprus Research Promotion Foundation. This work is supported in part by the EU Integrated Infrastructure Initiative Hadron Physics (I3HP) under contract RII3-CT-2004-506078, by the DFG (Forschergruppe Gitter-Hadronen-Phänomenologie), by the U.S. Department of Energy (D.O.E.) Office of Nuclear Physics under contract DE-FG02-94ER40818 and by the National Science Council of Taiwan under the grant numbers NSC96-2112-M002-020-MY3 and NSC96-2811-M002-026. This research used computational resources provided by the IBM machine at NIC, Jülich, Germany, by the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098, and by the MIT Blue Gene computer, supported by the DOE under grant DE-FG02-05ER25681.
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Appendix A Appendix
|(GeV)||EMR %||CMR %|
|(GeV)||EMR %||CMR %|
|(GeV)||EMR %||CMR %|
|Hybrid action, GeV|