Pion magnetic polarisability using the background field method

The magnetic polarisability is a fundamental property of hadrons, which provides insight into their structure in the low-energy regime. The pion magnetic polarisability is calculated using lattice QCD in the presence of background magnetic fields. The results presented are facilitated by the introduction of a new magnetic-field dependent quark-propagator eigenmode projector and the use of the background-field corrected clover fermion action. The magnetic polarisabilities are calculated in a relativistic formalism, and the excellent signal-to-noise property of pion correlation functions facilitates precise values.


Introduction
The electromagnetic polarisabilities of hadrons are of fundamental importance in the low-energy regime of quantum chromodynamics where they provide novel insight into the response of hadron structure to a magnetic field. The pion electric (α π ) and magnetic (β π ) polarisabilities are experimentally measured using Compton scattering experiments, such as γ π → γ π [1][2][3][4] where they enter into the description of the scattering angular distribution [5][6][7][8].
Theoretical approaches to calculating the pion electromagnetic polarisabilities are diverse. Calculations in the framework of chiral perturbation theory have a long history [9,10] while other approaches include dispersion sum rules [11][12][13] and the linear σ model [14]. Here we use the ab initio formalism of lattice QCD with an external background field. This method involves direct calculation of pion energies in an external field where the relativistic energy-field relation [15,16] E 2 (B) = m 2 π + (2 n + 1) |qe B| − 4 π m π β π |B| can be used to extract the magnetic polarisability, β π .
Here the pion has mass m π , charge qe and the term proportional to |qe B| is the Landau-level energy term [17].
In principal there is an infinite tower of energy levels for n = 0, 1, 2, . . . but the lowest lying Landau level is isolated through Euclidean time evolution. This method has previously been used to extract the polarisabilities of baryons [18,19], nuclei [20], and more recently the neutral pion and rho mesons [16,21,22]. The latter meson calculations were made possible through the removal of the spurious Wilson-fermion artefact associated with the background field [16,22].
Email address: ryan.bignell@adelaide.edu.au (Ryan Bignell) Calculations are performed at several non-zero pion masses in order to motivate a chiral extrapolation to the physical regime. These polarisability values are provided with the intent of spurring future chiral effective field theory development to enable extrapolations to the physical regime incorporating finite-volume and sea-quark corrections.

Simulation details & background field method
Four values of the light quark hopping parameter κ ud are used on the 2 + 1 flavour dynamical gauge configurations provided by the PACS-CS [23] collaboration through the IDLG [24]. These provide pion masses of m π = 0.702, 0.572, 0.411 and 0.296 GeV. The lattice spacing varies slightly at each mass due to our use of the Sommer scale [25] with r 0 = 0.49. The lattice volume is L 3 × T = 32 3 × 64. The Background-Field-Corrected clover fermion action of Ref. [16] is used to remove spurious lattice artefacts that are introduced by the Wilson term . This action has a nonperturbatively improved clover coefficient for the QCD portion of the clover term and a tree-level coefficient for the portion deriving from the background field. This combination is effective in removing the additive-mass renormalisation induced by the Wilson term [22].
No background field is present on the gaugefield ensembles and therefore this simulation is electroquenched. This is a departure from the physical world and should be accounted for in future chiral effective field theory work [26].

Background field method
The background field method [27,28] induces a constant magnetic field by adding a minimal electromagnetic coupling to the (continuum) covariant derivative This corresponds to a multiplication of the usual lattice QCD gauge links by an exponential phase factor where a is the lattice spacing. For a uniform field along theẑ axis the spatial periodic boundary conditions induce a quantisation condition, limiting the choice of uniform magnetic field strengths to where k is an integer which governs the field strength for a particle of charge qe and N x = N y = 32 are lattice dimensions. The down quark has the smallest charge magnitude and governs the magnetic field quanta. As the d quark has charge q d e the π + will have charge q π + e = −3 × q d e. That is, the smallest field strength for the π + has k π + = −3 and k d = 1.

Quark Operators
In this work a tuned Gaussian smeared source is used to provide a representation of QCD interactions. The smearing level is varied at zero external field strength (B = 0) and the effective mass examined to determine the smearing which produces the earliest onset of plateau behaviour [29]. The resulting smearing levels are N sm = 150, 175, 300, 250 sweeps for ensembles with masses m π = 0.702, 0.572, 0.411, 0.296 GeV respectively [30].
As charged particles in an external magnetic field, the quarks will experience Landau type effects in addition to the confining force of QCD. To provide greater overlap with the energy eigenstates of the pion we use the SU (3)× U (1) eigenmode quark projection technique introduced in Ref. [30]. In summary, the low-lying eigenmodes |ψ i of the two-dimensional lattice Laplacian with both QCD and background field effects are calculated Here A projection operator can be defined by truncating the completeness relation I = n i=1 |ψ i ψ i |. This truncation filters out the high-frequency modes, an effect similar to (2D) smearing. In the pure U (1) case each quark would have a definitive set of degenerate eigenmodes associated with each Landau level, however the introduction of QCD interactions into the Laplacian causes the U (1) modes associated with the different Landau levels to mix [31]. It is clear that in the case of a charged hadron, it is the hadronic level Landau modes that are respected and as such there is no longer a single definite Landau mode that describes the quark-level physics in the confining phase. We choose n = 96 eigenmodes to construct the quark-level projection operator, in accordance with our previous study Ref. [30] where this number was found to be sufficiently large as to avoid introducing significant noise into the correlation function whilst also small enough to place a focus on the low-energy physics relevant to the isolation of the magnetic polarisability.
As the lattice Laplacian used is two-dimensional, the low-lying eigenspace for each (z, t) slice on the lattice is calculated independently, allowing for the four-dimensional coordinate space representation of an eigenmode to be interpreted as selecting the two dimensional coordinate space representation ψ i, B (x, y) from the eigenspace belonging to the corresponding (z, t) slice of the lattice. Hence the four-dimensional coordinate space representation of the projection operator is where the Kronecker delta functions ensure that the projector acts trivially on the (z, t) coordinates. This projection operator is then applied at the sink to the quark propagator in a coordinate space representation, The use of the SU (3) × U (1) eigenmode quark projection technique has introduced both QCD and magnetic field physics into the quark sink. This, along with a tuned smeared source produces pion correlation functions at nontrivial field strengths that have a strong overlap with the ground state pion, which occupies the lowest lying hadronic Landau level (as detailed in the next section).

U (1) Hadronic Landau Projection
As a charged particle, the pion experiences hadronic level Landau effects, such that the ground state will occupy the lowest Landau level associated with the hadronic charge. In the presence of an external magnetic field along theẑ axis; the energy eigenstates of the π + are no longer eigenstates of the p x and p y momentum components.
In a finite volume lattice the hadronic Landau levels correspond to the eigenmodes of the two-dimensional lattice Laplacian in Eq. 5 where only the U (1) background field is present. As |k π | = |3 k d |, there is a degenerate subspace of |3 k d | eigenmodes to consider at the lowest hadronic Landau level, where k d is the down quark field quanta. We optimise a single U (1) eigenmode, ψ B (x, y) , to project the (x, y) dependence of the two-point correlation function onto the lowest Landau level where r = (x, y, z). The eigenmode ψ B (x, y) in Eq. (9) is chosen to optimise the overlap with the source ρ (x, y) = δ x,0 δ y,0 (assumed to be at the origin) through a rotation of the U (1) eigenmode basis that maximises the value of ρ ψ B . An optional phase can be applied so that ψ B (0, 0) is purely real at the source point. The projection of Eq. (9) is critical to successful isolation of the π + energy-eigenstate in a background magnetic field [32].

Magnetic Polarisability
Defining the following combinations of two-point correlation functions where G (B, t) is the correlation function for p z = 0 in a magnetic field of strength B, then the energy shift is simply Specifically, the effective energies are calculated with δt = 2. This formulation advantageously removes a portion of correlated QCD fluctuations, allowing the magnetic polarisability to be extracted using a simple polynomial fit. In order to constrain the charge of the pion to be q = 1, the fit performed is where c 2 has the units of E 2 (k d ) and is the fit parameter which is related to the magnetic polarisability using Eqs. (4) and (12) where α = 1/137 . . . is the fine structure constant.

Fitting
The two effective-energy shifts (E (B) + m π ) and (E (B) − m π ) generated by the correlator combinations of Eqs. (10) and (11) are required to have plateau behaviour reflecting an isolated energy eigenstate. This isolation is evident in the long constant fits in Figures 1 and 2 for the m π = 0.296 GeV pion. The isolation is a result of our detailed projection treatment of the quark level effects of the background field. This is the first time that plateau behaviour has been observed in these quantities.   14) is considered, with the fits selected through a consideration of the full covariance matrix χ 2 dof . The selected fits are displayed in Figure 3, where the fit for κ ud = 0.13770 of the m π = 0.296 GeV ensemble corresponds to the fit window displayed in Figures 1 and 2. This is the first lattice calculation in which the fully relativistic energy shift of Eq. (1) has been used. This is made possible due to the enhanced precision of pion correlation functions from the the SU (3) × U (1) eigenmode projected quark propagator and Landau-projected hadron sink..
The neutral pion is also amenable using these techniques. Here we consider the neutral, connected pion with quark content dd. The fit of Eq. (14) now needs no explicit subtraction of the Landau energy term as the π 0 d is overall charge-less. The success of the quadratic only and linearly constrained quadratic fits to the highly precise π 0 d and π + energy shifts of Eq. (12) suggests that higher order contributions in B are negligible. These neutral pion results draw from these new techniques, in particular the inclusion of the SU (3) × U (1) quark-propagator Laplacian projection which enables improved energy shift plateaus to be fitted. The magnetic polarisability of the π 0 d may be related to that of the full π 0 by considering the average of the magnetic polarisability of the uu and dd pions where π 0 u is the pion with quark content uu. This pion has relativistic energy As the uu pion is simply the dd pion in a field of twice the magnitude we may write and hence where we have used Eqs. (18) and (19). The magnetic polarisability of the uu pion may then be related to that of the dd pion by Thus the magnetic polarisability of the full neutral pion is The resulting pion magnetic polarisabilities are presented in Table 1.
All quark masses produce similar values for β π + and β π 0 . This is in contrast to the neutron [19] and evident in Figure 4 where our magnetic polarisabilities are plotted as a function of pion mass squared. The full neutral pion results using Eq. (22) are in good agreement with a number of theoretical approaches and experimental measurements [10,33] though we note again that we consider  only the connected portion of the neutral pion correlated here.
The β π 0 d results presented herein utilise the SU (3) × U (1) eigenmode quark-projection technique that we introduced in Ref. [30]. The success of this technique is evident in the improved energy-shift plateaus when compared to the equivalent energy shifts of Ref. [16]. These new results represent an improved understanding of the physics relevant to the extraction of the magnetic polarisability of the pion.

Conclusion
The magnetic polarisability of the charged pion has been calculated using lattice QCD for the first time. This is an important step forward in our understanding of this fundamental property, made possible due to the use of the SU (3) × U (1) eigenmode projection technique, along with a hadronic Landau eigenmode projection. The neutral pion magnetic polarisability is also presented. These results represent the first systematic study of pion magnetic polarisabilities across a range of pion masses with a fermion action which does not suffer from magnetic-field dependent quark-mass renormalisation effects.
To connect these results to experiment, one can draw on chiral effective field theory. By formulating the theory in a finite volume, finite-volume corrections can be determined. Moreover, by separating the contributions of valence and sea quarks, using the techniques of partiallyquenched chiral effective field theory, one can address the electro-quenched aspects of this calculations. Thus our results present an interesting challenge for the effective field theory community.
Future work in lattice QCD for the pion magnetic polarisabilities could focus on calculating the full neutral pion correlator which includes disconnected contributions and thus requires the x-to-x loop propagator [34]. Similarly the electroquenched nature of our calculations could be addressed by extending the background field to the "sea" quarks of the simulation at gaugefield generation time. Such a calculation requires a separate set of gaugefields for each external magnetic field strength which is prohibitively expensive and also removes the advantageous QCD correlations between two-point correlation functions at zero and finite external field strengths.