Pion in a uniform background magnetic field with clover fermions

Background field methods provide an important nonperturbative formalism for the determination of hadronic properties which are complementary to matrix-element calculations. However, new challenges are encountered when utilising a fermion action exposed to additive mass renormalisations. In this case, the background field can induce an undesired field-dependent additive mass renormalisation that acts to change the quark mass as the background field is changed. For example, in a calculation utilising Wilson fermions in a uniform background magnetic field, the Wilson term introduced a field-dependent renormalisation to the quark mass which manifests itself in an unphysical increase of the neutral-pion mass for large magnetic fields. Herein, the clover fermion action is studied to determine the extent to which the removal of $\mathcal{O}(a)$ discretisation errors suppresses the field-dependent changes to the quark mass. We illustrate how a careful treatment of nonperturbative improvement is necessary to resolve this artefact of the Wilson term. Using the $32^3 \times 64$ dynamical-fermion lattices provided by the PACS-CS Collaboration we demonstrate how our technique suppresses the unphysical mass renormalisation over a broad range of magnetic field strengths.

Background field methods provide an important nonperturbative formalism for the determination of hadronic properties which are complementary to matrix-element calculations. However, new challenges are encountered when utilising a fermion action exposed to additive mass renormalisations. In this case, the background field can induce an undesired field-dependent additive mass renormalisation that acts to change the quark mass as the background field is changed. For example, in a calculation utilising Wilson fermions in a uniform background magnetic field, the Wilson term introduced a field-dependent renormalisation to the quark mass which manifests itself in an unphysical increase of the neutral-pion mass for large magnetic fields. Herein, the clover fermion action is studied to determine the extent to which the removal of O(a) discretisation errors suppresses the field-dependent changes to the quark mass. We illustrate how a careful treatment of nonperturbative improvement is necessary to resolve this artefact of the Wilson term. Using the 32 3 × 64 dynamical-fermion lattices provided by the PACS-CS Collaboration we demonstrate how our technique suppresses the unphysical mass renormalisation over a broad range of magnetic field strengths.
Bali et al. highlighted this important problem [11,12] arising from the Wilson fermion action with the background field method. In particular they determined that in the free-field limit, that the mass of a Wilson quark is shifted by an amount a 2 |qe B| to where a is the lattice spacing, B the magnetic field strength, qe is the quark charge, and m w is the Wilson quark mass. Here we have introduced the notation that a subscript label in square brackets represents a quantity that is affected by lattice background field artefacts due to the fermion action, in this case the Wilson quark mass m [w] (B). This notation will be used throughout this work in order to distinguish between energy values with and without the additive background field mass renormalisation respectively (e.g. to distinguish the pion mass m [π] (B) and m π ). It was shown in Refs. [8,11] that the overlap quark formalism [13,14] does not suffer from this problem of * ryan.bignell@adelaide.edu.au field-dependent mass renormalisation. As the overlap action is many times more computationally expensive than the Wilson action, we turn our focus to the Wilson clover fermion action to determine its suitability with respect to the additive mass renormalisation arising from the background field.
The clover fermion action [15] is designed to remove O(a) lattice artefacts arising from the Wilson term. Thus, it is interesting to examine the extent to which the Wilson fermion artefacts survive in the clover fermion formulation. Herein, the clover fermion is studied in both the free-field limit and full QCD to determine its efficacy in removing the unphysical energy change caused by the uniform background magnetic-field.
Pion correlation functions are the natural choice with which to investigate this issue; they are free of complications associated with the magnetic moment, provide precision at low computational cost, and offer insight into interesting physics through the difference between neutral and charged pions. The neutral pion is of particular interest as it also has no hadronic-level Landau contribution.
The free-field limit allows the unphysical additive mass renormalisation to be studied, without the complications of QCD. This aids in formulating a solution for this problem using clover fermions. When full QCD interactions are present, the competing effects of QCD and the background magnetic-field make isolating and understanding the additive mass renormalisation more challenging. Moreover, the use of a nonperturbatively improved clover coefficient in a full QCD calculations further complicates the transition from free-field to full QCD interactions.
Here, the clover fermion action is studied for both the free-field and full-QCD cases to determine the conditions for the removal of the field-dependent additive quark mass renormalisation. In Section II a brief overview of the background field method is presented while Section III details the calculations performed in the free-field limit. These confirm the presence of the additive mass renormalisation and the utility of the clover fermion action in this limit. Full QCD is considered in Section IV and Section V details how the clover fermion action can be tuned to remove the field-dependent additive quark mass renormalisation. The magnetic polarisability of the neutral pion is presented in Section V B and conclusions are summarised in Section VI.

II. BACKGROUND FIELD METHOD
In this approach, a constant background magnetic field is introduced and the eigenstates are examined in the basis of the full Hamiltonian [3]. We commence in the continuum where a minimal electromagnetic coupling is added to form the covariant derivative Here qe is the electric charge of the fermion field and A µ is the electromagnetic four potential. On the lattice, the equivalent modification is to multiply the QCD gauge links by an exponential phase factor To generate a uniform magnetic field along theẑ axis in the continuum, one considers This does not uniquely specify the electromagnetic potential. The choice commonly selected over the interior of the lattice is A x = −B y which gives a constant magnetic field of magnitude B in the +ẑ direction. Due to the periodic boundary conditions on the lattice, we set A y = +B N y x along the boundary in theŷ dimension to maintain the constant magnetic field across the boundary. Now, considering the entirety of the lattice, a quantisation condition [1] for the magnetic field strength is required where a is the lattice spacing, N x , N y are the spatial dimensions of the lattice and k B is an integer specifying the field strength in terms of the minimum field strength.
In this work the field quanta k B is in terms of the charge of the down quark, i.e., q = −1/3 such that Hence a field with k B = 1 will be in the −ẑ direction.

III. FREE-FIELD CASE
The free-field simulation is investigated first; here the quarks couple to the external magnetic field through their electric charges but do not experience any QCD effects. The energy of such a charged particle contains a Landau energy term proportional to the charge of the particle. In the non-relativistic approximation with B = Bẑ, this energy spectrum is equivalent to that of a harmonic oscillator The relativistic generalisation of the Landau energy applies to each fermion responding to the field. For a free quark, the energy is Here p z is the quark momentum in theẑ direction, | s| = 1/2 and the quark has charge qe.

A. Wilson Fermions
Wilson fermions will have an additional energy term according to Eq. (1), and thus a free-field energy as the Wilson term is the discretised lattice Laplacian which also describes the lowest lying Landau level [9]. As a is the lattice spacing, this additional term is absent in the continuum limit. The additional term in Eq. (9) is a lattice artefact identified with field-strength dependent additive quark mass renormalisation.
To demonstrate the presence of this additive mass term, the free-field pion mass is calculated. Here we consider both the charged pion energy, E π ± and the neutral connected pion energy E π 0 u/d (B) and focus on the lowest lying states. Standard pseudoscalar interpolating fields χ =q γ 5 q are considered, where the quark flavours are either uu or dd corresponding to π 0 u and π 0 d . For a neutral pion with quark content uu or dd it is possible to have the the spin-dependent term of Eq. (9), 2 s · qe B cancel the Landau energy term (2 n + 1) |qe B| for n = 0. This cancellation of terms occurs for both the quark and the anti-quark. Consider for example, the uu pseudoscalar. As a spin-zero state, the two quarks have opposite spin orientations. Similarly, the quark and antiquark have opposite electric charges. If the u quark is spin down and the u is spin up, the terms cancel. For p z = 0, one has the lowest lying state with energy For a charged pion the spin-dependent term cannot cancel the Landau term for both the quark and antiquark sectors. The lowest energy state is realised when the terms cancel for the quark flavour with the largest magnitude of electric charge.
Consider for example, the π + meson composed of ud . Here the lowest energy state is realised when the u quark is spin down, enabling cancellation of the spin-dependent and Landau terms. For p z = 0, the charged pion will hence have energy Note that as QCD interactions are not present, no energy is required to displace the quarks from each other and thus the magnetic polarisability vanishes.
In the absence of the Wilson background-field additive mass renormalisation, the charged pion energy becomes B. Clover correction Clover fermions are designed to remove O(a) artefacts arising in the Wilson action. Thus, the focus of this investigation is to determine whether the clover fermion action removes the O(a) field strength dependent additive mass renormalisation due to the Wilson term.
The clover fermion matrix is given by where a is the lattice spacing, m is the bare quark mass, ∇ is the covariant central finite difference operator, ∆ is the lattice Laplacian or Wilson term, and c cl is the coefficient of the clover term. We define as the clover term (note the sum is restricted to µ < ν to avoid double counting), where σ µν = i 2 [γ µ , γ ν ] and F µν is the clover discretisation of the lattice field strength tensor. In general, for QCD+QED calculations we need to consider the chromomagnetic and electromagnetic field strength contributions separately [12], In the free-field case, all the QCD links are set to 1 so the QCD field strength F qcd µν = 0. Here we consider only the electromagnetic field strength, and attempt to determine the appropriate value for the free parameter c cl , which is defined to be the coefficient of the electromagnetic clover term. Setting to be the average of the four elementary plaquettes, it is easy to show that the electromagnetic field strength tensor is constant in a uniform background field For convenience we define the lattice magnetic field strength as For a magnetic field in the z direction then the only nonzero field strength tensor entry is F 12 = B z such that σ·F becomes diagonal with entries ∓B L /2. The Wilson term is simply the lattice Laplacian, which effectively describes a scalar particle. In the free field case, this causes a shift in the critical mass given by such that the free field mass is adjusted by Hence, in a uniform magnetic field there is a corresponding Landau energy ∼ a 2 |qB|/2. The magnetic part of the clover term for O(a) improvement of the Wilson matrix is given by (21) This operator is diagonal and hence commutes with the dicretised Dirac operator and lattice Laplacian. The lowest eigenvalue for the clover term in a uniform background field is It is clear then that for small field strengths, if we set c cl = 1 to the tree-level value, the clover term will cancel the Landau shift induced by the Wilson term.

C. Tree-level results
Numerical results for the pion energies in a background magnetic field are shown in Fig. 1 for Wilson fermions and Fig. 2 for clover fermions. Here we have selected a hopping parameter value of κ = 0.12400, relative to the bare critical hopping parameter of κ cr = 1/8 ; in anticipation of exploring full QCD where the PACS-CS Sommer scale provides a = 0.0951 fm. This kappa value corresponds to a pion mass of 140 MeV. These calculations are performed on a lattice with anti-periodic boundary conditions in the time direction and fits accommodate both the forward and backward propagating states.
It is clear from Fig. 1 that the neutral pion energy closely follows that described by Eq. (10). The additive mass renormalisation due to the Wilson term is hence correctly described by Eqs. (1) and (10). The charged π + energy using Wilson fermions agrees with the analytic energy of Eq. (11), further validating the additive mass renormalisation of Eq. (1).
The results shown in Fig. 2 for the clover fermion action verify our calculation for the tree level coefficient correction. The energies of the two neutral pions, π 0 d and π 0 u depicted for clover-improved fermions do not show this additive mass renormalisation; they do not change as a function of field strength. Similarly, the charged pion results agree with Eq. (12) for the pion energy in the absence of the Wilson background-field additive mass renormalisation. Thus the clover term effectively removes the O(a) additive mass renormalisation of the Wilson term in the background magnetic field.

IV. QCD CASE
We have demonstrated that the addition of the tree level clover term in the case of the electromagnetic U (1) field provides a correction that removes the background field dependent additive mass renormalisation induced by the Wilson term. The next question that arises is whether this correction survives once QCD interactions are turned on, and what (if any) adjustments are needed to the clover coefficient.
To study this, we allow for the QCD and electromagnetic field strengths to have different clover coefficients, Here c sw is the coefficient of the traditional Sheikholeslami-Wohlert improvement term arising from the SU (3) QCD gauge field interactions, and c em is the coefficient of the clover term induced by the U (1) electromagnetic gauge field. We know that the c sw coefficient is renormalised by QCD. In particular, here we will be studyng clover fermions with a non-perturbatively improved (QCD) coefficient. To determine the appropriate value for c em in this instance, we will first consider the 'naive' case, and set c em = c sw such that the clover terms from the QCD and electromagnetic field strengths are treated in a uniform way. Having done this, we will look for the presence of magnetic field strength dependent artefacts in the pion mass, as was done for the QCD free-field case.
Once QCD interactions are present, the pion possesses an internal structure and mass which is dependent on the mass of the quarks in a more subtle way [16,17] than in Eqs. (10) and (11). There is now a complex interplay between the background field and QCD effects [9,18].
The energy of a relativistic pion with mass m π and charge qe in a magnetic field orientated in theẑ-direction is where β π is the magnetic polarisability, which is chargestate dependent. It is necessary to use this fully relativistic form of the energy for the pion rather than the Taylor expansion common in the literature [2,[4][5][6]9] as 2m/(E + m) differs substantially from one for the largest field strength [9]. At m π = 296 MeV this difference is as much as 22% and is significant when working with the increased precision of pion correlation functions.

A. Simulation Details
This work considers the 2 + 1 flavour dynamical gauge configurations provided by the PACS-CS [19] collaboration via the ILDG [20]. These configurations are founded on a nonperturbatively improved clover fermion action and Iwasaki gauge action [21]. Two values of the light quark hopping parameter κ ud = 0.13754, 0.13770 are considered corresponding to pion masses m π = 411 and 296 MeV [19]. The lattice spacing for each mass is set using the Sommer scale [19] with r 0 = 0.49 fm. The lattice volume is L 3 × T = 32 3 × 64 and the ensemble sizes are 449 and 400 configurations respectively. Unless otherwise mentioned, we focus on results from the lighter ensemble with κ ud = 0.13770 and m π = 296 MeV.
Two point correlation functions at three distinct nonzero background magnetic field strengths are calculated. To do this, propagators at ten non-zero field strengths are created. With reference to Eq. (6), the field strengths considered have k B = ±1, ±2, ±3, ±4, and ±6 in Eq. (6). These give physical field strengths of eB = ±0.087, ±0.174, ±0.261, ±0.348, and ±0.522 GeV 2 . Correlation functions are averaged over both positive and negative field strengths during analysis to provide an improved unbiased estimator.
We note that these configurations are electroquenched; the field exists only for the valence quarks of the hadron. To include the background field at configuration generation time is possible [22] but requires a separate hybrid-Monte-Carlo calculation for each field strength, which is prohibitively expensive. Fortunately, these effects are not relevant to the current investigation.
Three-dimensional spatial Gaussian smearing utilising stout-smeared links is applied at the source and a point sink is considered. This ensures that the pion ground state is well represented. Standard pseudoscalar interpo-lating fields χ =q γ 5 q are used, where the quark flavours are either uu or dd corresponding to π 0 u and π 0 d .

B. Energy Shifts for Wilson Fermions
To investigate the effect of the nonperturbatively improved clover coefficient in the full QCD calculations, the lowest-lying neutral-pion background-field energy is considered To determine how the known additive quark mass renormalisation of Eq. (1) effects the pion mass in full QCD, we commence with the consideration of the Gell-Mann-Oakes-Renner relation [16,23] where E Ω = −2 Ω|uu|Ω /f 2 π , f π is the pion decay constant and Ω|uu|Ω is the chiral condensate. As m 2 π ∝ m q , E Ω has a relatively weak quark mass dependence.
Using Eq. (1), we introduce a background-field dependent pion mass due to the Wilson additive mass renormalisation Because QCD effects will modify Eq. (1), we have introduced a coefficient ξ in Eq. (27) which in principle can be B-field dependent. For example, Bali et al. [12] investigated the change in the quark mass in Wilson-fermion QCD plus background magnetic-field simulations by examining the change in the critical hopping parameter as a function of magnetic field strength. For small external magnetic-field strengths the mass shift is an order of magnitude smaller from the free-field case and for their smallest field strengths, the sign of the shift differs. The dependence of Eq. (1) begins to emerge at large magnetic-field strengths as QCD effects become small. As discussed in the following subsection, our nonperturbatively improved cloverfermion results also display this order of magnitude suppression. However our survey of magnetic-field strengths does not reveal the Wilson-fermion sign change in the mass shift. Thus it is sufficient to treat ξ as a constant to be determined.
Studies [24,25] have indicated that E Ω changes slowly in an external magnetic field and on this basis, we consider the leading order approximation Finally the energy of a neutral pion in an external magnetic field using Wilson fermions is We note that terms linear in B from the magnetic field dependence of E Ω in Eq. (27) can combine with the linear term a ξ 2 q u/d eB to provide a contribution proportional to B 2 , thus contaminating the O(B 2 ) signal used to extract β π 0 . Ultimately, it is important to ensure this O(a) term is removed.
To explore the presence of additive quark-mass renormalisation, we focus on the quantity This energy shift can be constructed using the two correlator combinations where G(B, t) is the zero-momentum projected two-point correlation function. Upon taking the effective energy Eq. (31) yields E [π 0 ] (B) + m π 0 and Eq. (32) provides E [π 0 ] (B) − m π 0 . These effective energy shifts from R + and R − are then multiplied together to form the E 2 [π 0 ] (B) − m 2 π 0 energy shift of Eq. (30). We note that Eq. (32) is particularly helpful in isolating B-depdendent terms, as QCD contributions are correlated in the ratio of correlation functions and largely cancel.
Noting Eq. (30) has leading linear and quadratic terms in B we commence by considering the fit function Here k B is the quantisation number from the quantisation condition on magnetic field strength of Eq. (6). An estimate for the fit parameter c 1 can be obtained from Eq. (30) Recalling the Gell-Mann-Oakes-Renner relation at zero magnetic field provides E Ω (0) = m 2 π 0 /m q and drawing on the Wilson quark-mass relation where κ cr is the critical hopping parameter where the zero-field pion mass vanishes, Eq. (35) can be written Similarly, drawing on Eq. (30), c 2 is related to the magnetic polarisability As the u and d quarks are mass degenerate in our lattice QCD simulation, the neutral pion correlation functions with quark content uu or dd differ only by the strength of the quark flavour interactions with the external magnetic field. This allows the neutral pion correlation function to be evaluated at a larger range of field strengths than is possible for the π + , using the same propagators.

C. Nonperturbatively Improved Clover Fermions
The nonperturbatively improved clover fermion action used in the full QCD calculations differs from that used in the free-field calculations by the value of the clover coefficient c sw multiplying the O(a) clover term of the fermion action in Eq. (13).
The free-field simulations used the tree-level value c sw = 1 while the full QCD calculations use the nonperturbatively improved value c sw = 1.715 [19,26]. The extent to which this changes the cancellation of the additive mass renormalisation seen to occur in Fig. 2 is investigated using the energy shift defined above in Eq. (30).
The E [π 0 ] (B) − m π 0 energy shift is illustrated in Fig. 3  right; it is quite clear that it is easy to construct good plateau fits for this energy shift. This is in contrast to the E [π 0 ] (B) + m π 0 energy shift in Fig. 3 left, as the correlated QCD fluctuations between field strengths compound rather than cancel, making it difficult to fit constant plateaus. This difficulty in fitting constant plateaus reduces the fit parameter space considered as common plateau fits are required for both E   The energy shifts are fitted as a function of the fieldstrength quanta k B in Fig. 4. Recalling Eq. (30) has leading linear and quadratic terms in B we consider the fit function Under the assumption of the removal of additive mass renormalisation, we first consider fixing c 1 = 0 and using a c 2 k 2 B quadratic-only fit function. As illustrated in Fig. 4 this provides a poor description of the results and yields an unacceptable χ 2 dof = 6.1. Allowing for a nontrivial c 1 coefficient provides the linear + quadratic, c 1 k B + c 2 k 2 B fit of Eq. (39). This fit describes the lattice simulation results well with a χ 2 dof = 0.5. However, it also indicates the presence of additive mass renormalisation in the nonperturbatively improved clover fermion simulation.
To explore the necessity of this O(a) term linear in the magnetic field strength, a quadratic + cubic, c 2 k 2 B + c 3 k 3 B , fit is also considered. However, this model does not describe the simulation results, producing unacceptably high χ 2 dof values. Fits over four field strengths are also considered. This time, the acceptable fit requires linear, quadratic and cubic terms. Thus, the demand for an O(a) term associated with the presence of additive mass renormalisation is robust.

D. Expected Mass Renormalisation
These results indicate that the current treatment of the clover term is inadequate. In light of the success observed in the free field limit, we anticipate that the application of the nonperturbatively improved value c NP sw = 1.715 to both the QCD and the background field contributions to the clover term has spoiled the removal of additive mass renormalisation associated with the magnetic field.
If the tree-level value of c tree sw = 1 is the value required for the removal of O(a) errors, then we have over compensated by an amount This over compensation is relative to the standard Wilson action discrepancy of This over compensation factor can be incorporated into  (32) for E π 0 (B) + m π 0 (left) and E π 0 (B) − m π 0 (right) respectively using the BFC clover fermion action on the mπ = 296 MeV ensemble. The three smallest field strengths are illustrated. Shaded regions illustrate the fit windows selected through the consideration of the full covariance-matrix χ 2 dof , the extent of the fit window and the desire to select the same fit window for all effective-energy shifts.
the Wilson-fermion discussion of Section IV C through

V. BACKGROUND FIELD CORRECTION
Motivated by the analytic calculation in Section III B, and encouraged by the above result, in this section we investigate a new clover fermion action where the coefficients for the QCD and background magnetic field contributions to the field-strength tensor in Eq. (23) take different values. The electromagnetic clover coefficient takes the tree level value c em = 1, whilst the QCD clover coefficient retains its non-perturbative value of c NP sw = 1.715. In the following, we refer to this modified fermion action as the Background Field Corrected (BFC) form of the clover fermion action.

A. Additive Mass Renormalisation
The process of calculating correlation functions and forming energy shifts is repeated as detailed above in Eqs. (31) through (33). Figure 5 displays the new effective energy shifts and the associated fits.
In the absence of additive mass renormalisations, Eq. (30) simplifies to and therefore we consider fit functions of the form If the modified clover action has removed the Wilson additive mass renormalisation, the linear term of Eq. (44) will have a trivial coefficient. Indeed, fits without the linear term should describe the energy shifts well. Fits of Eq. (44) to results from the lowest three magnetic field strengths are illustrated in Fig. 6. As acceptable fits are obtained with the two leading terms, c 3 has been constrained to zero. The success of our BFC clover fermion action is reflected in the excellent description of the results using only a quadratic term in Eq. (44), in accord with the expectations of Eq. (43). Allowing for the possibility of a nontrivial value for c 1 , we find c 1 = (−5.3 ± 8.7) × 10 −4 GeV 2 , consistent with zero.
Drawing on the full range of five magnetic field strengths available to dd and uu pions, Fig. 7 illustrates fits of Eq. (44) to the BFC clover results. This time the results demand c 3 = 0 and thus a B 3 term is manifest. Again, the success of our BFC clover action in removing additive mass renormalisations is reflected in the excellent description of the results using only quadratic and cubic terms in Eq. (44), in accord with the expectations of Eq. (43). Allowing for the possibility of a nontrivial value for c 1 , we find c 1 = (1.1 ± 9.0) × 10 −4 GeV 2 , consistent with zero.
We note that similar results are observed for fits to the first four magnetic field strengths. The lattice results are described well by fits with c 1 constrained to zero, and allowing for a nontrivial value provides c 1 = (−6.  Here a term cubic in the magnetic field strength is required to describe the largest field strength.
terms lies between the fourth and fifth field strengths and therefore, only the three smallest field strengths are considered in determining the magnetic polarisability. In summary, the BFC clover fermion action has successfully removed the additive quark mass renormalisation due to the Wilson term. The key is to employ the tree-level value c em = 1 for uniform background field contributions to the electromagnetic clover term.

B. Magnetic Polarisability
Having demonstrated the removal of additive quarkmass renormalisation in the BFC clover action, the mag- netic polarisability can be determined without concern for the aforementioned O(a) contaminations entering the B 2 associated with the magnetic polarisability. The energy shift and fits performed in Section V A are used and the polarisability extracted from the coefficient c 2 of the quadratic term of Eq. (44). Comparing with Eq. (43) and drawing on the field quantisation condition of Eq. (5) with q = q d = −1/3, the magnetic polarisability is given by Results for the magnetic polarisability of the neutral pion from the O(a)-improved BFC clover action analysis of the lowest three magnetic-field strengths are reported in Table I. While the report of our analysis has focused on the light-quark hopping parameter of κ ud = 0.13770 corresponding to 296 MeV, results for m π = 411 are also reported.
As discussed in the previous section, we consider the first three field strengths to avoid the possibility of complications associated with non-trivial O(B 3 ) contributions. While the inclusion of a fourth point takes the magnetic polarisability from β π 0 = 0.54(7) to 0.52(5) × 10 −4 fm 3 , the small improvement in statistical uncertainty may be offset by an increase in systematic uncertainty associated with non-trivial O(B 3 ) contributions.

VI. CONCLUSIONS
In this paper the response of the pion to a uniform background magnetic field has been investigated. The existence of a field-strength dependent additive quarkmass renormalisation associated with Wilson fermions was confirmed. We performed an analytic calculation showing that at tree-level in the QCD-free case the clover term corrects for these spurious contributions.
When QCD interactions are included, a careful treatment of the QCD and electromagnetic clover terms in Eq. (23) is required. While the interactions of QCD require the nonperturbatively improved value of c sw = 1.715, it is essential to apply the tree-level value c sw = 1 to the background magnetic field contributions. With this treatment, it is possible to simultaneously remove the O(a) errors in the QCD contributions, and correct for the O(a) errors induced by the Wilson term in a background magnetic field. We refer to this modified fermion action as the Background Field Corrected clover fermion action.
With the suppression of O(a) errors associated with additive mass renormalisation in the BFC clover action, the magnetic polarisability of the neutral pion can now been determined. For the first time, a fully relativistic approach to the energy shift is used. Results are summarised in Table I and we anticipate these results will facilitate consensus within the field. β π 0 is positive, such that the energy of a pion in a magnetic field decreases. The extent to which this trend continues is of interest and our results give a hint that the O(B 3 ) contributions soften this trend.
Future work will approach the physical quark mass regime and interface with chiral perturbation theory studies [27]. It will also be important to directly include sea-quark-loop interactions with the background magnetic field to incorporate these contributions to the magnetic polarisability. However, this approach is prohibitively expensive. It requires a separate Monte-Carlo ensemble for each field strength considered and this will result in a loss of important QCD correlations between different magnetic field strengths.
An alternative approach is to separate the valence and sea-quark-loop contributions to the magnetic polarisability in effective field theory [7,9]. This will enable an accurate extrapolation of current lattice QCD results and an estimation of sea-quark-loop contributions to the magnetic polarisabilities of the pion.