# Finite volume study of electric polarizabilities from lattice QCD.

###### Abstract

Knowledge of the electric polarizability is crucial to understanding the interactions of hadrons with electromagnetic fields. The neutron polarizability is very sensitive to the quark mass and is expected to diverge in the chiral limit. Here we present results for the electric polarizability of the neutron, neutral pion, and neutral kaon on eight ensembles with nHYP-smeared clover dynamical fermions with two different pion masses (227 and 306 MeV). These are currently the lightest pion masses used in polarizability studies. For each pion mass we compute the polarizability at four different volumes and perform an infinite volume extrapolation for the three hadrons. Along with the infinite volume extrapolation we conduct a chiral extrapolation for the kaon polarizability to the physical point. We compare our results for the neutron polarizability to predictions from chiral perturbation theory.

Finite volume study of electric polarizabilities from lattice QCD.

Michael Lujan^{†}^{†}thanks: Speaker. , Andrei Alexandru, Walter Freeman, and Frank Lee

The George Washington University, Washington DC, USA

E-mail: mlujan@gwmail.gwu.edu, aalexan@gwu.edu, wfreeman@gwu.edu, fxlee@gwu.edu

\abstract@cs

## 1 Introduction

To lowest order, the response of a composite particle to an electromagnetic field can be parameterized by the effective Hamiltonian:

(1.0) |

where and are the electric and magnetic dipole moments, respectively, and and are the electric and magnetic polarizabilities. Due to time-reversal symmetry of the strong interaction the electric dipole moment vanishes. Furthermore, by considering the simplified case of a weak constant electric field, the leading order interaction comes from the polarizability term at . The polarizability is a first-order structure constant which measures the rigidity of the hadron in the presence of the external field.

In this work, we use lattice QCD to compute the electric polarizability (). We employ the background field method to extract the polarizability. Previous lattice calculations [1, 2, 3, 4, 5, 6] were done at relatively heavy pion masses leaving the chiral region largely unexplored. Here, we use 2-flavor n-HYP clover fermions with two different pion masses (227 MeV and 306 MeV) to study the chiral behavior of the polarizability. Moreover, for each mass we compute on four different lattice volumes to study the volume dependence.

We analyze three neutral hadrons: neutral pion, neutral kaon, and the neutron. For each hadron we performed an infinite volume extrapolation. For the kaon, we also performed a chiral extrapolation to the physical point. The results of our neutron polarizability, will be compared to predictions from chiral perturbation theory. We note that our work, though done on dynamical configurations, uses electrically-neutral sea quarks throughout.

## 2 Methodology

### 2.1 Background field method

We use the background field method to place the electric field on the lattice. The method uses minimal coupling which augments the static electromagnetic vector potential () to the covariant derivative, i.e.

(2.0) |

where are the gluon field degrees of freedom. In practice, this amounts to an overall multiplicative phase factor to the original gauge links which appear in :

(2.0) |

The polarizability is extracted by computing the variation of the hadron’s ground state energy with and without the presence of an electric field.

In order to extract the polarizability we need to use a weak enough electric field so that higher order terms in the field
expansion can be safely neglected.
In this work we instead use Dirichlet boundary conditions (DBC).
The advantage of this is that we can use arbitrarily small values of the field. DBC also protects against the
vacuum instability due to the Schwinger
mechanism [7].^{1}^{1}1This instability only
occurs for real electric fields and not for imaginary electric fields as used in most lattice studies. However,
to use imaginary fields we need to rely on the analyticity of the theory around the point where the electric field is
zero. Schwinger mechanism signals that this is not an analytical point generically. DBC offers one way to restore this
analyticity.
However, using DBC creates boundary effects. One of them is the introduction of a non-zero momentum for
the hadron of magnitude
. The lowest energy state of the system is then , where is the mass of the hadron.
To account for this motion
we compute the mass shift () motion due to the polarizability
using the relation

(2.0) |

The hadron’s mass () is computed using periodic boundary conditions.

### 2.2 Fitting Procedure

The form of the correlators, for neutral hadrons in an constant electric field, retain their exponential fall off, allowing us to use some of the standard spectroscopy techniques to measure the shift in hadrons’ energies.

The main difference in the fitting analysis is the fact that we need to extract the energy shift from the three correlation functions: , and , which are the correlation functions for the zero-field, and non-zero fields in the positive and negative -direction, respectively. Since all three correlators are computed from the set of gauge configurations they are highly correlated; we therefore need to properly account for the correlations among them. To do this we construct the following difference vector as

(2.0) | |||||

where is the fit window, and . We minimize the function,

in the usual fashion, where is the correlation matrix which encodes the correlations among the three different correlators.

## 3 Ensemble Details and Results

Ensemble | Lattice | (fm) | ||
---|---|---|---|---|

EN1 | 0.1245 | 0.12820 | 230 | |

EN2 | 0.1245 | 0.12820 | 300 | |

EN3 | 0.1245 | 0.12820 | 300 | |

EN4 | 0.1245 | 0.12820 | 270 | |

EN5 | 0.1215 | 0.12838 | 230 | |

EN6 | 0.1215 | 0.12838 | 450 | |

EN7 | 0.1215 | 0.12838 | 670 | |

EN8 | 0.1215 | 0.12838 | 500 |

In this work we use 2-flavor nHYP-clover fermions [8] with two different pion masses (227 MeV and 306 MeV) in order to study the chiral behavior of the polarizability. For each pion mass we compute on four different volumes in order to study the volume effects of the polarizability. A description of the ensembles are tabulated in Table 1. In our simulations we use a field size of , where is the charge of the down quark. For a detailed discussion for the choice of we refer the reader to our previous work [9].

### 3.1 Finite Volume Effects

The polarizabilities for the neutral pion, neutral kaon, and neutron were extracted for each ensemble using the fitting procedure described in Sec. 2.2. For a given pion mass we studied the infinite volume behavior of the polarizability by extrapolating to the infinite volume limit using various polynomial degrees as fit models. We found, for the pion, that a constant extrapolation describes the data well. For the neutron and kaon we used a linear approximation. Figures 1 and 2 illustrate the results of the infinite volume extrapolation for both pion masses.

The volume dependence analysis assumes that the finite size corrections are mainly driven by the extent of the lattice in the direction of the electric field (which is in -direction for this work). To verify this, we take our EN4 lattice which has the spatial dimension and place the electric field along the -direction which has only 24 lattice units. We choose this ensemble because the difference in the and directions are the largest which gives us the best comparison. If the finite volume corrections associated with the transverse directions are small, we expect our results to be comparable to the results of the EN2 ensemble which has the spatial dimension . We display our results on the bottom panel of Fig. 2 for the EN2 lattice and the EN4 lattice for the electric field in the direction and the electric field in the direction. Our expectations are in very good agreement with our findings. That is, we found that dominant source of the finite size effects are connected to the extent of the lattice in the direction of the field.

### 3.2 Chiral Behavior

In this section we analyze the pion mass behavior for each of the particles we analyzed. On the left panel of Fig. 3 we plot our infinite volume extrapolation results for the pion polarizability as a function of . We also include the values calculated by Detmold et al. [3] at MeV along with quenched results calculated by Alexandru and Lee [10]. The negative trend, which has been seen in our previous work [11], is still present; determining its origin is an ongoing study.

Our results for the kaon polarizability is illustrated on the right panel of Fig. 3. The plot also includes the value determined by Detmold et al. [3]. The kaon polarizability becomes larger as we lower the pion mass. In our previous work [9] we performed an extrapolation to the physical point using only the EN2 and EN6 lattices. We had found fm. Here we do the same analysis but now using our infinite volume results together with the value determined in [3]. We find . In this fit we assumed that the finite volume corrections at MeV are small since they seem to decrease as we increase the pion mass.

On the left panel of Fig. 4 we plot our infinite volume results for the neutron polarizability as a function of along with quenched data that were computed in [10]. We also compare our data to two different PT predictions: PT [12, 13] and PT [14] to gauge systematic errors of our calculation. For a more detailed comparison of the two PT predictions we refer the reader to [9].

Similar to our findings in [9], which only analyzed the EN2 and EN6 ensembles, we find that our infinite volume results are compatible with the quenched ones. Moreover, our results are now in excellent agreement with the PT curve. This was not the case for our previous analysis which did not take into account the volume effects. In Fig. 4 we add the experimental point along with two other lattice calculations [2, 1] for the neutron polarizability. Our results have the smallest pion masses used in polarizability studies and the smallest statistical errors.

## 4 Conclusion

We have presented a calculation of the electric polarizabilities for the neutron, neutral pion, and neutral kaon in the framework of lattice QCD. We used two different pion masses (227 MeV and 306 MeV) to study the chiral behavior of the polarizability. Currently, these are the smallest masses used in polarizability studies. We employed the background field method along with Dirichlet boundary conditions to place a constant electric field onto the lattice. A finite volume study was performed for each pion mass by computing the polarizability on four different lattice volumes. This was one of the most important results in this work. For the neutron we find that the finite volume corrections are significant. Without the infinite volume extrapolation our results did not agree with PT calculations at our lowest pion mass where PT is expected to be more accurate.

## 5 Acknowledgements

This work was done on the following GPU clusters: GWU IMPACT clusters, GWU CCAS Colonial One cluster, JLab clusters, Fermilab clusters, and UK clusters. This work is supported in part by the NSF CAREER grant PHY-1151648 and the U.S. Department of Energy grant DE-FG02-95ER-40907.

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