Magnetic polarisability of the nucleon using a Laplacian mode projection
AADP-20-6/T1116
Magnetic polarisability of the nucleon using a Laplacian mode projection
Ryan Bignell, ∗ Waseem Kamleh, and Derek Leinweber
Special Research Centre for the Subatomic Structure of Matter (CSSM),Department of Physics, University of Adelaide, Adelaide, South Australia 5005, Australia (Dated: May 20, 2020)Conventional hadron interpolating fields, which utilise gauge-covariant Gaussian smearing, areineffective in isolating ground state nucleons in a uniform background magnetic field. There isevidence that residual Landau mode physics remains at the quark level, even when QCD interac-tions are present. In this work, quark-level projection operators are constructed from the SU(3) × U(1) eigenmodes of the two-dimensional lattice Laplacian operator associated with Landau modes.These quark-level modes are formed from a periodic finite lattice where both the background fieldand strong interactions are present. Using these eigenmodes, quark-propagator projection opera-tors provide the enhanced hadronic energy-eigenstate isolation necessary for calculation of nucleonenergy shifts in a magnetic field. The magnetic polarisability of both the proton and neutron iscalculated using this method on the 32 ×
64 dynamical QCD lattices provided by the PACS-CSCollaboration. A chiral effective-field theory analysis is used to connect the lattice QCD resultsto the physical regime, obtaining magnetic polarisabilities of β p = 2 . (cid:0) +13 − (cid:1) × − fm and β n = 2 . (cid:0) +15 − (cid:1) × − fm , where the numbers in parentheses describe statistical and systematicuncertainties. PACS numbers: 13.40.-f, 12.38.Gc, 12.39.Fe
I. INTRODUCTION
The magnetic polarisability describes the response ofa system of charged particles to an external magneticfield. The study of nucleon polarisabilities is an area ofkey experimental and theoretical interest [1–6], and themagnetic polarisabilities are key quantities in this area.Measurement of magnetic polarisabilities is difficult [7, 8]and improvement in experimental measurements is evi-dent in recent years [6, 9, 10]. Lattice QCD can play animportant role in making predictions in this area.The uniform background field method has been usedsuccessfully to calculate magnetic moments [11–14] ofhadrons and the magnetic polarisability of neutral parti-cles such as the neutral pion [15, 16] and neutron [17, 18].Herein, we present new lattice QCD techniques that en-able an investigation of the proton polarisability in anaccurate manner.An uniform external magnetic field is induced throughthe introduction of an exponential U (1) phase factoron the gauge links across the lattice. This externalfield changes the energy of the nucleon according to theenergy-field relation [11, 17–22] E ( B ) = m + (cid:126)µ · (cid:126)B + (2 n + 1) | qe B | m − π β B + . . . (1)where the nucleon has mass m and magnetic mo-ment and magnetic polarisability (cid:126)µ and β respectively.The Landau energy term [23] is proportional to | qe B | .There is in principle an infinite tower of Landau levels, ∗ [email protected] (2 n + 1) | qe B | / m for n = 0 , , , . . . which poses anadditional complication for charged hadrons such as theproton.While the extraction of the magnetic polarisabilityseems simple − simply fit the linear and quadratic co-efficients of the energies of Eq. (1) as a function of fieldstrength − this approach is problematic as the magneticpolarisability appears at second-order in the energy of thenucleon [17, 18, 20–22]. The contribution of the mag-netic polarisability to the nucleon energy is necessarilysmall compared to the overall energy of the particle ifthe energy expansion of Eq. (1) is to have small O (cid:0) B (cid:1) contributions.Three-dimensional gauge-covariant Gaussian smear-ing [24] on the quark fields at the source has been shownto efficiently isolate the nucleon ground state in pureQCD calculations [25, 26]. This is not the case when auniform background field is present; the magnetic fieldbreaks three-dimensional spatial symmetry and intro-duces electromagnetic perturbations into the dynamicsof the charged quarks, thus altering the physics present.When QCD interactions are absent, each quark will havea Landau energy proportional to its charge. In the pres-ence of QCD these quarks hadronise, such that in theconfining phase the Landau energy will correspond tothat of the composite hadron.It is clear that in the confining phase the effects ofthe QCD and magnetic interactions compete with eachother. Previous studies have demonstrated that Landauphysics remains relevant even when QCD interactions arepresent [17, 27]. This leads us to the idea of using quarkoperators on the lattice that capture both of these forces.In particular, we have the freedom of choosing asymmet-ric source and sink operators in order to construct cor-relation functions that provide better overlap with theenergy eigenstates of the nucleon in a background mag- a r X i v : . [ h e p - l a t ] M a y netic field.The two-dimensional U (1) Laplacian is associated withthe Landau modes of a charged particle in a magneticfield. In our previous study of the neutron [17] weconsidered a quark sink projection based on these two-dimensional U (1) eigenmodes on gauge-fixed QCD fields.In the present study we explore the use of a projector de-rived from the eigenmodes of the two-dimensional latticeSU(3) × U(1) Laplacian operator. The use of a fullygauge-covariant eigenmode-projected quark sink whichencapsulates both QCD and Landau level physics elim-inates the need for gauge fixing. We find the use of theSU(3) × U(1) modes as a quark projection operator tobe effective in isolating the ground state of the proton inan external magnetic field, enabling an accurate deter-mination of the proton magnetic polarisability.The presentation of this research is as follows. Sec-tion II briefly describes our implementation of a uniformmagnetic field. Section III describes the process by whichthe magnetic polarisability can be extracted from nucleontwo point correlation functions while Section IV describesthe smeared source and SU(3) × U(1) projected sink usedto isolate the nucleon ground states at non-zero magneticfield strengths. The results at several quark masses arepresented in Section V, and these are used to inform thechiral extrapolations to the physical regime of SectionVI. Section VII summarises conclusions.
II. BACKGROUND FIELD METHOD
The following background field method is used to in-troduce a constant magnetic field along a single axis.This technique is derived first in the continuum wherea minimal electromagnetic coupling is added to form thecovariant derivative D µ = ∂ µ + i qe A µ , (2)where qe is the charge of the fermion field and A µ isthe electromagnetic four potential. On the lattice, theequivalent modification is to multiply the QCD gaugelinks by an exponential phase factor U µ ( x ) → U µ ( x ) e ( i a qe A µ ( x )) . (3)As (cid:126)B = (cid:126) ∇ × (cid:126)A , a uniform magnetic field along the ˆ z axisis obtained via B z = ∂ x A y − ∂ y A x . (4)To give a constant magnetic field of magnitude B in the+ˆ z direction on the lattice we exploit both A x and A y .Throughout the lattice we set A x = − B y . To maintainthe constant magnetic field across the ˆ y edge of the latticewhere periodic boundary conditions are in effect, we set A y = + B N y x along the ˆ y boundary y = N y . Thisthen induces a quantisation condition for the uniform magnetic field strength [22] qe B a = 2 π kN x N y . (5)Here a is the lattice spacing, N x and N y are the spatialdimensions of the lattice, and k an integer specifying thefield strength in terms of the minimum field strength.In this work the field quanta k is in terms of the chargeof the down quark, i.e., k = k d and q = − / q d e B a = 2 π k d N x N y . (6)Hence a field with k d = 1 will be in the − ˆ z direction.The magnetic field experienced by a baryon is defined tobe k B = − k d . III. MAGNETIC POLARISABILITY
The naive process of fitting to Eq. (1) as a functionof field strengths is not a viable method with which toextract the magnetic polarisability. Instead a ratio ofcorrelation functions is constructed to isolate the energyshift in a manner enabling correlated QCD fluctuationsto be reduced. To form this ratio, we define the spin-fieldantialigned two-point correlation function G (cid:24)(cid:23) ( B ) = G (+ s, − B ) + G ( − s, + B ) . (7)and the spin-field aligned correlator by G (cid:24)(cid:22) ( B ) = G (+ s, + B ) + G ( − s, − B ) . (8)Here spin-up/down is represented by (+ s/ − s ) respec-tively and the magnetic field orientation along the spinquantisation direction, ˆ z , by ( ± B ). These spin-field an-tialigned and aligned correlators form an improved unbi-ased estimator for the required correlation functions asthey are averages over the required spin and field combi-nations.The ratio required to isolate the magnetic polarisabil-ity of Eq. (1) draws on Eqs. (7) and (8) along with thespin-averaged zero-field correlator G (0 , t ) R ( B, t ) = G (cid:24)(cid:22) ( B, t ) G (cid:24)(cid:23) ( B, t ) G (0 , t ) . (9)The zero-field correlator subtracts the mass term fromthe total energy of the anti-aligned and aligned contribu-tions while the contribution from the magnetic momentterm of Eq. (1) is removed by the product of the spin-fieldantialigned and aligned correlators. This product yieldsan exponent of the sum of the aligned and antialignedenergy shifts δE (cid:24)(cid:22) ( B ) + δE (cid:24)(cid:23) ( B )2 = | qe B | M − π β | B | + O (cid:0) B (cid:1) . (10)Thus the desired energy shift is δE ( B, t ) = 12 1 δ t log (cid:18) R ( B, t ) R ( B, t + δ t ) (cid:19) (11) → | qe B | M − π β | B | + O (cid:0) B (cid:1) , (12)for large Euclidean time. For the neutrally charged neu-tron, | qe B | = 0 and hence the Landau level term van-ishes, providing direct access to the polarisability. Forthe proton this term makes an important contributionand we investigate its magnitude in a variety of fits. A. Fitting
For a charged hadron such as the proton, the energyshift for the polarisability is as specified by Eq. (12) whichhas a term linear in B and a term quadratic in B . Assuch, an appropriate fit as a function of field strength hasboth a linear and a quadratic dependence δE ( k d ) = c k d + c k d (13)where we fit as a function of k d , the integer magneticflux quanta in Eq. (6). c and c are fit parameters withthe units of δE ( k d , t ). This is in contrast to the neutronwhere only a quadratic term is required [17].As c is a free parameter, fitting in this manner allowsthe charge of the proton to be non-unitary. Thus, we alsoconsider the linear constrained energy shift δE ( k d ) − | e B | M = c k d = − π β | B | + O (cid:0) B (cid:1) . (14)Here the known linear term has been explicitly sub-tracted with q = 1 such that only a term quadratic in B is fitted. This constrains the charge of the proton to q p = 1.The quantisation condition of Eq. (6) provides β = − c α q d a (cid:18) N x N y π (cid:19) , (15)where α = 1 / . . . is the fine structure constant.The energy shifts which are fit in Eq. (14) must bewell determined at all field strengths. We apply the sin-gle state ansatz, requiring that a constant plateau fit canbe found as a function of Euclidean lattice time t . Cor-relations between adjacent Euclidean time slices are con-sidered through the use of the full covariance matrix χ dof which is estimated via the jackknife method [28]. The re-sulting fit as a function of field strength must then alsofit the energy shifts with an acceptable χ per degree offreedom.Fit windows were kept consistent across the three non-zero field strengths considered where possible. Wherethis proved difficult, the fit windows between fieldstrengths were allowed to vary in a monotonic manner,with the lowest field strength having the longest fit re-gion. Through this process we ensure that the lowestlying state is isolated for each energy shift. IV. QUARK OPERATORS
The use of asymmetric source and sink operators en-able the construction of nucleon correlation functionswhich have greater overlap with the lowest energy eigen-states of the nucleon in a background magnetic field. Thisis important as the energy shift required for the magneticpolarisability is small compared to the total energy. Thesignal becomes disguised by noise at late Euclidean time.The quark source is constructed using three-dimensional gauge-invariant Gaussian smearing [24] as iscommon practice in lattice QCD [25, 26] while the quarksink uses the SU(3) × U(1) eigenmode quark-projectionmethod discussed below.
A. SU(3) × U(1) eigenmode projection
For a charged particle in a uniform magnetic field thelattice Landau levels are eigenmodes of the (2D) U (1)Laplacian. Here we wish to construct a fully gauge-covariant quark sink projection operator that encom-passes QCD as well as the electromagnetic potential. Todo so, we calculate the low-lying eigenmodes | ψ i (cid:105) of thetwo-dimensional lattice Laplacian∆ (cid:126)x,(cid:126)x (cid:48) = 4 δ (cid:126)x,(cid:126)x (cid:48) − (cid:88) µ =1 , U µ ( (cid:126)x ) δ (cid:126)x +ˆ µ,(cid:126)x (cid:48) + U † µ ( (cid:126)x − ˆ µ ) δ (cid:126)x − ˆ µ,(cid:126)x (cid:48) , (16)where U µ ( (cid:126)x ) are the full SU(3) × U(1) gauge links asapplied in the full lattice QCD calculation. We can thendefine a projection operator by truncating the complete-ness relation 1 = (cid:88) i =1 | ψ i (cid:105) (cid:104) ψ i | . (17)In the pure U (1) case, the lowest Landau level on the lat-tice has a degeneracy equal to the magnetic flux quanta | k | given in Eq. 5, providing a natural place to truncatethe above sum. The introduction of the QCD interac-tions into the Laplacian causes the U (1) modes associatedwith the different Landau levels to mix, such that it is nolonger possible to clearly identify the modes associatedwith the lowest Landau level at small field strengths. In-stead, we simply choose a fixed number n > | k | modes toproject. This truncation has a similar effect to perform-ing (2D) smearing, by filtering out the high frequencymodes. Indeed, we find that for small values of n theprojected hadron correlator becomes noisy, just as it doeswhen performing large amounts of sink smearing.The eigenmode truncation is chosen to be sufficientlylarge so as to avoid introducing large amounts of noiseinto the two-point correlation function, but also smallenough to place a focus on the low-energy physics rele-vant to isolating the magnetic polarisability. Truncationat 32, 64 and 96 modes are investigated in a mannersimilar to that at the source. While 32 modes was not E ( B , t ) ( G e V ) k d = 1 E ( B , t ) ( G e V ) k d = 2
16 18 20 22 24 26 28 30 t E ( B , t ) ( G e V ) k d = 3 FIG. 1. The proton energy shift δE ( k d , t ) of Eq. (11) forthree field strengths using 64 (orange diamond) and 96 (bluesquare) eigenmodes in the projection operator of Eq. (19).The m π = 0 .
702 GeV ensemble is shown. effective, it can be observed in Figure 1 that both 64and 96 eigenmodes produce consistent behaviour in theproton energy shift.Due to the two-dimensional nature of the Laplacian,the low-lying eigenspace is calculated independently foreach ( z, t )-slice on the lattice. Consequently, we can in-terpret the four-dimensional coordinate space represen-tation of an eigenmode (cid:104) (cid:126)x, t | ψ i, (cid:126)B (cid:105) = ψ i, (cid:126)B ( x, y | z, t ) , (18)as selecting the two-dimensional coordinate space rep-resentation ψ i, (cid:126)B ( x, y ) from the eigenspace belonging tothe corresponding ( z, t )-slice of the lattice. The four-dimensional coordinate space representation of the pro-jection operator follows, P n ( (cid:126)x, t ; (cid:126)x (cid:48) , t (cid:48) ) = n (cid:88) i =1 (cid:104) (cid:126)x, t | ψ i, (cid:126)B (cid:105) (cid:104) ψ i, (cid:126)B | (cid:126)x (cid:48) , t (cid:48) (cid:105) δ zz (cid:48) δ tt (cid:48) . (19)The Kronecker delta functions in the definition above en-sure that the outer product is only taken between eigen-modes from the same subspace (i.e. the projector actstrivially on the ( z, t ) coordinates).This projection operator is then applied at the sink tothe quark propagator in a coordinate-space representa-tion as S n (cid:16) (cid:126)x, t ; (cid:126) , (cid:17) = (cid:88) (cid:126)x (cid:48) P n ( (cid:126)x, t ; (cid:126)x (cid:48) , t ) S (cid:16) (cid:126)x (cid:48) , t ; (cid:126) , (cid:17) . (20)
16 18 20 22 24 26 28 30 t E ( G e V ) FIG. 2. Proton zero-field effective masses from smearedsource to SU(3) × U(1) eigenmode projected sink correla-tors using various levels of covariant Gaussian smearing atthe source on the m π = 0 .
702 GeV ensemble. The source isat t = 16. We select n = 96 modes for our analysis.Using the SU(3) × U(1) eigenmode quark-projectionoperator and a tuned smeared source produces nucleoncorrelation functions at non-trivial field strengths wherethe proton is in the QCD ground state and the n = 0lowest lying Landau level approximation is justified. Theenergy shifts required by Eqs. (11) and (12) display goodplateau behaviour, as exhibited in Figures 1 and 6. B. Source Smearing
Whilst we attempt to encapsulate the quark-levelphysics of the electromagnetic interaction at the sink, weuse a smeared source to provide a representation of theQCD interactions with the intent of isolating the QCDground state. A broad range of smearing levels are ex-amined at zero field strength, B = 0 in order to do this.The effective mass at B = 0 was investigated for eachensemble and the smearing which produces the earliestonset of plateau behaviour is chosen. This B = 0 effectivemass is shown in Figure 2 for m π = 0 .
702 GeV where theoptimal smearing of 150 sweeps is chosen. On the lightestensemble considered, at m π = 0 .
296 GeV as shown inFigure 3, the choice is not as obvious. In this case the fullset of correlation functions at each finite field-strength isrun for each smearing and these results examined.This reveals a particularly interesting problem withlarge amounts of smearing which can be seen in the anti-aligned energy shown in Figure 4. The anti-aligned en-ergy is examined in preparation its use in the energyshift ratio of Eq. (9) and has spin and magnetic fieldanti-aligned as in Eq. (7).When the source is excessively smeared, the larger fieldstrengths couple preferentially to higher Landau levelsrather than the lowest. This is evident in how the 350
16 18 20 22 24 26 28 t E ( G e V ) FIG. 3. Proton zero-field effective masses from smearedsource to SU(3) × U(1) eigenmode projected sink correla-tors using various levels of covariant Gaussian smearing atthe source on the m π = 0 .
296 GeV ensemble. The source isat t = 16.
16 18 20 22 24 26 28 t E ( G e V ) FIG. 4. Proton k d = 2 anti-aligned effective energies fromsmeared source to SU(3) × U(1) eigenmode projected sinkcorrelators using various levels of covariant Gaussian smearingat the source on the m π = 0 .
296 GeV ensemble. The sourceis at t = 16. sweeps effective energy differs from the other smearingsin both value and slope. This difference is close to thedifference between Landau levels for the proton, i.e. for E n ( B ) ∼ m + | e B | (2 n + 1) , (21)the difference can be determined by considering the rela-tivistic energy difference E ( B ) − E ( B ) = 2 | e B | . Thedifference of squares can be factored as( E ( B ) − E ( B )) ( E ( B ) + E ( B )) = 2 | e B | . (22)Defining ∆ E ( B ) = E ( B ) − E ( B ) as the energy dif-ference visible in Figure 4, we obtain a quadratic form∆ E ( B ) (∆ E ( B ) + 2 E ( B )) − | e B | = 0 . (23) Recalling that the field strength experienced by the pro-ton is related to that of the down quark by k B = − k d ,the appropriate values for Figure 4 are E ( k d = 2) ∼ . | e B ( k d = 2) | ∼ .
522 GeV , and the energy difference between these two Landau lev-els is ∆ E ( k d = 2) ∼ .
46 GeV. This is consistent withthe difference between smearings visible in Figure 4.The smeared source examination is followed at each ofthe quark masses where for masses m π = 702, 570, 411,296 MeV, optimal smearings of N sm = 150, 175, 300, 250respectively are obtained.An advantage of using the U (1) × SU (3) Laplacianprojector is that it is well defined at zero magnetic fieldstrength, where the U (1) field is equal to unity. Thismeans that the fluctuations at finite B and B = 0 arestrongly correlated, such that they cancel out when tak-ing the ratio of the correlators in Eq. (9), providing an im-proved signal in comparison to the U (1) projection. Thisimprovement does come at a computational cost, as the U (1) × SU (3) Laplacian eigenmodes must be calculated onevery configuration. Using the SU(3) × U(1) eigenmodequark-projection operator and a tuned smeared sourceproduces nucleon correlation functions at non-trivial fieldstrengths where the proton is in the QCD ground stateand the n = 0 lowest lying Landau level approximationis justified. This is demonstrated by the energy shifts re-quired by Eqs. (11) and (12), which display good plateaubehaviour as exhibited in Figures 1 and 6. C. Hadronic Landau Projection
The SU(3) × U(1) eigenmode projection technique de-fined above is relevant to Landau effects at the quark-level. As the proton is a charged hadron it will alsoexperience Landau level behaviour in a magnetic field.The Landau-level physics at the hadronic level is easierto capture due to the colour-singlet nature of the pro-ton. We can simply use the eigenmodes of the U (1)Laplacian, with a well defined degeneracy for the low-est Landau level [17, 29]. Below we describe in detail theprescription for the hadronic Landau level projection.To study hadronic two-point correlation functionsin the zero-field case one calculates the momentum-projected correlator G ( (cid:126)p, t ) = (cid:88) (cid:126)x e − i (cid:126)p · (cid:126)x (cid:68) Ω (cid:12)(cid:12)(cid:12) T (cid:110) χ ( (cid:126)x, t ) ¯ χ (cid:16) (cid:126) , t (cid:17)(cid:111) (cid:12)(cid:12)(cid:12) Ω (cid:69) , (24)where χ and ¯ χ are appropriate interpolating fields.This standard approach of a three-dimensional Fourierprojection is not appropriate for the proton when theuniform background magnetic field is present. The pres-ence of the background field causes the energy eigen-states of the charged proton to no longer be eigenstates FIG. 5. Lowest-lying U(1) eigenmode probability densities ofthe lattice Laplacian operator in a constant background mag-netic field oriented in the ˆ z direction are plotted as a functionof the x, y coordinates. As k B = − k d , the degenerate eigen-modes for the sixth quantised field strength relevant to theproton for k d = 2 are displayed in a linear combination thatmaximises the overlap of the first mode with the source. Theorigin is at the centre of the x − y plane. of the p x , p y momentum components. Hence, we insteadproject the x, y dependence of the two-point correlatoronto the lowest Landau level, ψ (cid:126)B ( x, y ), explicitly, andalso select a specific value for the z component of mo-mentum, G (cid:16) p z , (cid:126)B, t (cid:17) = (cid:88) r ψ (cid:126)B ( x, y ) e − i p z z × (cid:68) Ω (cid:12)(cid:12)(cid:12) T (cid:110) χ ( r, t ) ¯ χ (cid:16) (cid:126) , t (cid:17)(cid:111) (cid:12)(cid:12)(cid:12) Ω (cid:69) . (25)In the continuum limit, the lowest Landau mode has aGaussian form, ψ (cid:126)B ∼ e −| qe B | ( x + y ) / . However, in afinite volume the periodicity of the lattice causes the wavefunction’s form to be altered [21, 30]. As such, we insteadcalculate the lattice Landau eigenmodes using the two-dimensional U(1) gauge-covariant lattice Laplacian in ananalogous way to Eq. (16) [17, 29]. Here U µ contains onlythe U (1) phases appropriate to the background magneticfield quantised on the lattice.The correlator projection is then onto the spacespanned by the degenerate modes ψ i, (cid:126)B associated with the lowest lattice Landau level available to the proton G (cid:16) p z , (cid:126)B, t (cid:17) = (cid:88) r n (cid:88) i =1 ψ i, (cid:126)B ( x, y ) e − i p z z × (cid:68) Ω (cid:12)(cid:12)(cid:12) T (cid:110) χ ( r, t ) ¯ χ (cid:16) (cid:126) , t (cid:17)(cid:111) (cid:12)(cid:12)(cid:12) Ω (cid:69) . (26)The degeneracy of the lowest-lying Landau mode is givenby the magnetic-field quanta | k d | .In evaluating Eq. (26), we also consider the case offixing n = 1, such that only the first eigenmode havingthe best overlap with the source is considered. Assumingthe source ρ ( x, y ) = δ x, δ y, is located at the origin, theoverlap with the first mode i = 1 is optimised througha rotation of the U (1) eigenmode basis that maximisesthe value of | (cid:104) ρ | ψ i =1 , (cid:126)B (cid:105) | . An optional phase can beapplied so that ψ i =1 , (cid:126)B (0 ,
0) is purely real at the sourcepoint.In most cases the results are almost indistinguishableand we proceed with n = | k d | . The only exception isfor the ensemble with m π = 0 .
411 GeV. Here the i = 1mode alone provides superior results. As discussed inSection II, for the proton k B = − k d and therefore thedegeneracy is n = | k d | . Figure 5 illustrates the sixdegenerate modes associated with k d = 2.More generally, this hadronic eigenmode-projected cor-relator offers superior isolation of the ground state for theproton [21] and is crucial for the identification of constantplateaus in the energy shift of Eq. (11). D. Simulation Details
The 2 + 1 flavour dynamical gauge configurations pro-vided by the PACS-CS [31] collaboration through theILDG [32] are used in this work. These configurationsspan a variety of masses, allowing a chiral extrapolationto be performed. A non-perturbatively improved cloverfermion action and Iwasaki gauge action provide a phys-ical lattice spacing of a = 0 . r = 0 .
49 fm. The de-tails of each of these ensembles, including the pion massand statistics used can be found in Table I. Fixed bound-ary conditions in the time direction are used and thesource placed at N t / e B = ± . ± . ± . ± . ± .
522 GeV ;corresponding to k d = ± , ± , ± , ± , ± TABLE I. Lattice simulation parameters with corresponding statistics used. κ ud a (fm) m π (GeV) L × T m π L N src ×
64 11.6 50.13727 0.1009 0.570 32 ×
64 9.4 40.13754 0.0961 0.411 32 ×
64 6.4 40.13770 0.0951 0.296 32 ×
64 4.5 7 field is known analytically; a tree-level contribution of F Bµν from the background field can be included in theclover term of the fermion action, avoiding the non-perturbative improvement coefficient, C SW . This re-moves the non-physical magnetic-field induced additivemass renormalisation due to the Wilson term of thefermion action.The configurations used are electro-quenched; themagnetic field exists only for the valence quarks of thehadron. While it is possible to include the backgroundfield on each configuration [35] this requires a separateMonte Carlo simulation for each field strength and sois prohibitively expensive. Performing separate calcula-tions would also remove the correlated QCD fluctuationsbetween finite-field and zero-field correlation functions,reducing the efficacy of Eq. (9). V. RESULTS
The formalism for extracting the magnetic polarisabil-ity of the nucleons using lattice QCD and the backgroundfield method has now been established. Hadronic Lan-dau level projected correlation functions are computed atseveral non-zero field strengths using a specialised SU(3) × U(1) quark sink. Thus hadronic as well as quark levelLandau energy level effects are considered in order to iso-late the ground state energy of the nucleon in an externalmagnetic field. A tuned smeared source provides a goodrepresentation of the QCD ground state.The effectiveness of this approach is visible in Fig-ures 6 and 7 where the energy shift required to ex-tract the magnetic polarisability of the proton is plot-ted for the m π = 0 .
570 GeV and m π = 0 .
296 GeV en-sembles respectively. The effective energy shifts displaygood plateau behaviour across all three non-zero fieldstrengths. This is a common feature across the heavierthree quark masses considered. Good plateaus can befound at all three field strengths.At the lightest quark mass considered, the third fieldstrength does not present good plateau behaviour. Assuch only the first two field strengths are considered.The fit function of Eq. (14) is applied to the non-zeroenergy shifts produced by Eq. (11). The constant Eu-clidean time fits to Eq. (11) are selected using a strict χ dof criteria, where we require χ dof ≤ .
2. This ensuressingle-state dominance in the energy shift.In determining the optimal Euclidean-time fit windowsto use, each possible fit window across all three field
16 18 20 22 24 26 28 30 32 t E ( B )( G e V ) k d = 1, dof =1.0, len =7 k d = 2, dof =0.43, len =7 k d = 3, dof =0.15, len =7 FIG. 6. The magnetic polarisability effective energy shift, δE ( B, t ) of Eq. (11) for the m π = 0 .
570 GeV proton as afunction of Euclidean time (in lattice units), using a smearedsource and the SU(3) × U(1) quark-eigenmode projectiontechnique. Results for field strengths k d = 1 , , χ dof are also illus-trated.
16 18 20 22 24 26 28 t E ( B )( G e V ) k d = 1, dof =0.33, len =5 k d = 2, dof =1.2, len =5 FIG. 7. The magnetic polarisability effective energy shift, δE ( B, t ) of Eq. (11) for the m π = 0 .
296 GeV proton as afunction of Euclidean time (in lattice units), using a smearedsource and the SU(3) × U(1) quark-eigenmode projectiontechnique. Results for field strengths k d = 1 and 2 are shown.The selected fits for this ensemble and the χ dof are also illus-trated. k d E ( G e V ) m =0.702 GeV dof =0.9609 k d E ( G e V ) m =0.570 GeV dof =1.099 k d E ( G e V ) m =0.411 GeV dof =0.6708 k d E ( G e V ) m =0.296 GeV dof =0.3273 FIG. 8. Constrained quadratic fits of the energy shift to thefield quanta at each quark mass for the proton. strengths is considered. Where all the Euclidean-timeplateau and field-strength dependent fits are acceptable,magnetic polarisability values are calculated from thequadratic coefficient c of Eq. (14).Where this fitting process yields no or only a smallnumber of acceptable fit windows, we also allow theEuclidean-time fit window at each field strength to vary.These fits must still have a common fit end point butthe fit start point is allowed to increase in a monotonicmanner with increasing field strength. The smallest fieldstrength must have the longest fit window, the secondfield strength the second longest and similarly for furtherfield strengths. This fitting process expands the fit win-dow parameter space available and is particularly helpfulat lighter quark masses.At each pion mass considered, the linearly constrainedquadratic fits of Eq. (14) are determined. The final fitsare presented in Figure 8. In every case, the full covari-ance matrix based χ dof indicates an acceptable fit withthe charge of the proton constrained to one.We have also assumed higher-order terms of the expan-sion of Eq. (1) are negligible. In order to check the valid-ity of this assumption, a linearly constrained quadratic +quartic fit incorporating a c k d term is performed. Wefind this quartic term provides no additional informa-tion, and similar magnetic polarisabilities are observed.When the unconstrained linear + quadratic fit is consid-ered; the linear coefficient c produces a charge value q in agreement with one.This is the first time that Euclidean-time plateau fitshave been successfully constructed for the proton’s mag-netic polarisability energy shift. Thus the SU(3) × U(1)quark level eigenmode projection technique is effective inisolating the energy shifts required to access the magneticpolarisability of the proton to be well determined.A similar method is followed for the neutron. This timea standard Fourier transform at the hadron sink projectsto zero momentum, and as the neutron is overall chargeless the subtraction process of Eq. (14) is not required.
TABLE II. Magnetic polarisability values for the neutron andproton at each quark mass considered. The number of sourcesconsidered for each quark mass varies as described in Table I.The numbers in parentheses describe statistical uncertainties. m π (GeV) β n (cid:0) fm × − (cid:1) n χ dof β p (cid:0) fm × − (cid:1) p χ dof
20 25 30 35 t E ( B )( G e V ) k d = 1, dof =1.0, len =6 k d = 2, dof =0.42, len =5 k d = 3, dof =0.59, len =4 FIG. 9. The magnetic polarisability effective energy shift, δE ( B, t ) of Eq. (11) for the m π = 0 .
570 GeV neutron as afunction of Euclidean time (in lattice units), using a smearedsource and the SU(3) × U(1) quark-eigenmode projectiontechnique. Results for field strengths k d = 1 , , χ dof are also illus-trated. Figures 9 and 10 display representative Euclidean-timefits to the effective energy shifts of the neutron, analogousto Figures 6 and 7 for the proton.The final quadratic fits for the neutron are displayed inFigure 11. From the quadratic term of the fit in Eq. (14),the magnetic polarisability can be found using Eq. (15).Polarisability results for both the proton and neutron aresummarised in Table II.For the neutron field-strength dependent fit on the m π = 0 .
702 GeV ensemble, only the first two non-zerofield strengths are used. Fitting to the third requires theadditional quartic term and produces a magnetic polar-isability value that agrees with a single quadratic fit tothe first two field strengths. The neutron polarisabilityenergy shift at the largest field strength considered suf-fers from the same signal-to-noise problem as that of theproton and is hence not used to extract the magneticpolarisability.The neutron magnetic polarisabilities obtained hereinare in good agreement with those obtained in Ref. [17] onthe same ensembles. There, a U (1)-based Landau-modeprojection technique was applied to Landau-gauge fixedquark propagators. We find the SU(3) × U(1) eigenmode
16 18 20 22 24 26 28 30 t E ( B )( G e V ) k d = 1, dof =0.58, len =5 k d = 2, dof =0.74, len =5 FIG. 10. The magnetic polarisability effective energy shift, δE ( B, t ) of Eq. (11) for the m π = 0 .
296 GeV neutron as afunction of Euclidean time (in lattice units), using a smearedsource and the SU(3) × U(1) quark-eigenmode projectiontechnique. Results for field strengths k d = 1 and 2 are shown.The selected fits for this ensemble and the χ dof are also illus-trated. k d E ( G e V ) m =0.702 GeV dof =0.8522 k d E ( G e V ) m =0.570 GeV dof =0.8754 k d E ( G e V ) m =0.411 GeV dof =0.7422 k d E ( G e V ) m =0.296 GeV dof =0.9046 FIG. 11. Quadratic fits of the energy shift to the field quantaat each quark mass for the neutron. quark projection technique to be similarly successful inisolating the neutron ground state in a background mag-netic field. Now, for the first time, a unified method forextracting both proton and neutron magnetic polarisabil-ities has been presented. It is anticipated that the mesonsand hyperons will also be tractable using this approach.
VI. CHIRAL EXTRAPOLATION
To connect lattice results to the physical regime, chiraleffective-field theory ( χEF T ) provides a powerful tool.This analysis is a generalisation of Ref. [36] with modifi-cations arising from the consideration of both the protonand neutron.The chiral expansion considered is
B BBM
FIG. 12. The leading-order meson loop contribution to themagnetic polarisability of the nucleon.
B BB M FIG. 13. The next to leading-order meson loop contributionto the magnetic polarisability of the nucleon, allowing transi-tions to nearby strongly coupled baryons. β B (cid:0) m π (cid:1) = (cid:88) M β M B (cid:0) m π , Λ (cid:1) + a (Λ) + a (Λ) m π + (cid:88) M, B (cid:48) β M B (cid:48) (cid:0) m π , Λ (cid:1) + O (cid:0) m π (cid:1) , (27)where a (Λ) and a (Λ) are residual series coefficients [37]constrained by our infinite volume corrected lattice QCDresults and Λ is a renormalisation scale. The leading-order loop contributions β M B and β M, B (cid:48) are shown inFigures 12 and 13. Figure 13 allows for transitions of thebaryon B to nearby strongly coupled baryons, B (cid:48) , withmass splitting ∆, via a meson, M , loop whereas Figure12 does not encounter any baryon mass-splitting effects.In the heavy-baryon approximation [38] appropriatefor a low energy expansion, these have integral forms [36] β M B (cid:0) m π , Λ (cid:1) = e π π f π χ M B (cid:90) d k (cid:126)k u ( k, Λ) (cid:16) (cid:126)k + m M (cid:17) , (28)0and for ∆ = m B (cid:48) − m B (cid:54) = 0 β M, B (cid:48) (cid:0) m π , Λ (cid:1) = e π π f π χ M B (cid:48) (cid:90) d k u ( k, Λ) × ω (cid:126)k ∆ (cid:0) ω (cid:126)k + ∆ (cid:1) + (cid:126)k (cid:16) ω (cid:126)k + 9 ω (cid:126)k ∆ + 3 ∆ (cid:17) ω (cid:126)k (cid:0) ω (cid:126)k + ∆ (cid:1) , respectively. Here ω (cid:126)k = (cid:113) (cid:126)k + m M is the energy carriedby the meson M which has three-momentum (cid:126)k , f π = 92 . u ( k, Λ) is a dipoleregulator u ( k, Λ) = 1 (cid:16) (cid:126)k / Λ (cid:17) , (29)which ensures that only soft momenta flow through theeffective-field theory degrees of freedom.The renormalised low-energy coefficients of the chiralexpansion are formed from the residual series coefficients a (Λ), a (Λ) and the analytic contributions of the loopintegrals [39] which also depend on Λ. The full detailsof the renormalisation procedure are provided in the Ap-pendix of Ref. [39]. The standard coefficients for fullQCD, χ M B and χ M B (cid:48) reflect photon couplings to theintermediate meson.The loop integral of Eq. (28) for β M B contains theleading nonanalytic contribution proportional to 1 /m M .For finite B (cid:48) − B mass splitting, ∆ = m B (cid:48) − m B ,the loop integral of Eq. (29) accounts for transitionsto nearby strongly coupled baryons B (cid:48) and contributesa nonanalytic logarithmic contribution proportional to( − / ∆) log ( m M / Λ), to the chiral expansion.Here we consider mesons M = π and η for nucleontransitions with B = n or p for the integral of Eq. (28).While the total charge of these mesons is zero, it is im-portant to consider their contributions in assessing thecontribution of sea-quark-loops. The η (cid:48) -meson is muchheavier and thus its contribution is suppressed and safelyneglected.We consider transitions to the baryons B (cid:48) = Σ , Λ , ∆and Σ ∗ with mesons M = π, η and K . These transitionsare accounted for by Eq. (28) with the appropriate masssplittings and couplingsOur lattice QCD results are electroquenched - theydo not include contributions of photon couplings to dis-connected sea-quark loops of the vacuum. Disconnectedsea-quark loops form part of the full meson dressing of χEF T and thus it is necessary to model the correctionsassociated with their absence in the lattice QCD calcula-tions. Hence the standard coefficients for full QCD χ M B and χ M B (cid:48) are altered to account for partial quenchingeffects [40] as explained in Ref. [36] for the neutron. Theproton is briefly discussed below while the neutron fol-lows the analysis in Ref. [17, 36].
A. Partially quenched χ EFT
In order to model the corrections to account for par-tial quenching effects, the contribution of each quark-flowdiagram is separated into ’valence-valence’, ’valence-seaand ’sea-sea’ contributions. Each of these describes thecoupling of the two photons to the valence or sea quarksavailable in the intermediate state mesons. All possiblequark-flow diagrams for the p → p π channel are con-structed in Figure 14 without attaching external photonsto the meson. As there is baryon no mass splitting, thisis an example of Figure 12.As Figures 14a and 14b have both sea and valencequark lines of the intermediate meson, photon lines maybe attached to the valence or sea-quark lines of the inter-mediate meson. Hence they may contribute to all threesectors. This is in contrast to Figure 14c which containsonly valence quarks and hence contributes only to thevalence-valence sector. The contributions to each sectoris proportional to the quark charges, i.e. for Figure 14bthe leading non-analytic term of the chiral expansion hascoefficents χ v − v ∝ q u , (30) χ v − s ∝ q u q u (31) χ s − s ∝ q u , (32)where Eq. (31) reflects the two orderings of the photoncouplings available.While the sum of the valence-valence, valence-sea andsea-sea contributions is zero for this process due to theneutrality of the π meson; the valence-sea and sea-seaterms are not present in the lattice QCD simulation andhence must be accounted for.The sea-sea disconnected sea-quark-loop flow for Di-agram 14a can be isolated by temporarily replacing thedown-quark loop with a strange quark [41]. This providesa coupling strength χ diag ( b ) s − s ∝ q d χ K Σ + = q d D − F ) . (33)Repeating the above procedure for the up-quark loop ofDiagram 14b one finds χ diag ( c ) s − s ∝ q u (cid:0) χ K + Λ + χ K + (cid:1) = q u (cid:18)
13 (3 F + D ) + ( D − F ) (cid:19) . (34)The components of the p → p π channel which havea disconnected sea-quark loop have been identified andhence the sea-sea contributions have been calculated.The same process may be performed for the valence-seacontributions. As the total contribution is known fromstandard χ PT, the remaining valence-valence contribu-tion which includes the connected quark-flow diagram ofDiagram 14(a) is also known. All such channels for theintegral processes described by Eq. (27) are investigated1 dddu uu ud d (a) The down quark loop diagram where the two photons cancouple to valence-valence, sea-sea or valence-sea quarks. uuuu ud du u (b) The up quark loop diagram where the two photons can coupleto valence-valence, sea-sea or valence-sea quarks. uuuu ud du u (c) The quark-flow diagram where the two photons can couple onlyto valence quarks.
FIG. 14. Decomposition of the process p → p π into its possible one-loop quark-flow diagrams. The configuration of the twophoton couplings to the valence and/or sea quarks determines the coefficients of partially quenched chiral perturbation theory. using the diagrammatic procedure described above for p → p π .Having performed this procedure for each relevantchannel, the coefficients used when fitting the latticeQCD results reflect the absence of the disconnected sea-quark-loop contributions and can be determined by sub-tracting the valence-sea and sea-sea contributions fromthe total contribution in Tables III and IV for the pro-ton and neutron respectively. The numerical value of thecoefficients can be found in Table V where the standardvalues of g A = 1 .
267 and C = 1 .
52 with g A = D + F andthe SU(6) symmetry relation F = D are used.The regulator mass, Λ = 0 .
80 GeV [37, 42–45] is chosenin anticipation of accounting for the missing disconnectedsea-quark-loop contributions in the lattice QCD calcu-lations. This regulator mass enables corrections to thepion cloud contributions associated with missing discon-nected sea-quark-loop contributions as it defines a pioncloud contribution to masses [37], magnetic moments [43]and charge radii [42]. The nucleon core contribution isinsensitive to sea-quark-loop contributions at this regu-lator mass [46].To consider the effect of the finite-volume of the lat-tice, we replace the continuum integrals of the chiral ex-pansion with sums over the momenta available on the periodic lattice. It is important to note that the latticevolume is slightly different on each of the four lattice en-sembles used due to our use of the Sommer scale. Toproduce inite-volume corrected (FVC) results, β F V Cv − v , wetake the difference between these sums and the contin-uum integrals β F V Cv − v (cid:0) m π (cid:1) = β lat.v − v (cid:0) m π (cid:1) − (cid:88) M β M BSUM (cid:0) m π , Λ F V (cid:1) + (cid:88) M, B (cid:48) β M B (cid:48)
SUM (cid:0) m π , Λ F V (cid:1) + (cid:88) M β M B (cid:0) m π , Λ F V (cid:1) + (cid:88) M, B (cid:48) β M B (cid:48) (cid:0) m π , Λ F V (cid:1) , (35)where we note that we are correcting for finite-volumeonly and hence the coefficients used in evaluating thesesums and integrals reflect only valence-valence contribu-tions. The finite-volume corrections should be indepen-dent of the value of the regulator parameter, Λ F V , as longas Λ
F V is sufficiently large. Here we choose Λ
F V = 2 . χP T analysis is that the lead-ing and next-to-leading non-analytic terms of the chi-2 TABLE III. Chiral coefficients for the leading-order loop integral contributions for the proton.
Process Total Valence-sea Sea-sea p → N πp → p π (cid:0) q u q u (cid:0) χ K + Σ + χ K + Λ (cid:1) + 2 q d q d χ K Σ + (cid:1) (cid:0) q u (cid:0) χ K + Σ + χ K + Λ (cid:1) + q d χ K Σ + (cid:1) p → p η (cid:0) q u q u (cid:0) χ K + Σ + χ K + Λ (cid:1) + 2 q d q d χ K Σ + (cid:1) (cid:0) q u (cid:0) χ K + Σ + χ K + Λ (cid:1) + q d χ K Σ + (cid:1) p → p η (cid:48) (cid:0) q u q u (cid:0) χ K + Σ + χ K + Λ (cid:1) + 2 q d q d χ K Σ + (cid:1) (cid:0) q u (cid:0) χ K + Σ + χ K + Λ (cid:1) + q d χ K Σ + (cid:1) p → n π + χ π + n q u q d (cid:0) χ K + Σ + χ K + Λ (cid:1) q d (cid:0) χ K + Σ + χ K + Λ (cid:1) p → p + π − q d q u χ K Σ + q u χ K Σ + p → Σ Kp → (cid:0) Σ , Λ (cid:1) K + χ K + Σ + χ K + Λ q u q s (cid:0) χ K + Σ + χ K + Λ (cid:1) q s (cid:0) χ K + Σ + χ K + Λ (cid:1) p → Σ + K q d q s χ K Σ + q s χ K Σ + p → ∆ πp → ∆ π + χ π + ∆ q u q d χ K + Σ ∗ q d χ K + Σ ∗ p → ∆ ++ π − χ π − ∆ ++ q d q u χ K Σ ∗ + q u χ K Σ ∗ + p → ∆ + π (cid:0) q u q u χ K + Σ ∗ + 2 q d q d χ K Σ ∗ + (cid:1) (cid:0) q u χ K + Σ ∗ + q d χ K Σ ∗ + (cid:1) p → ∆ + η (cid:0) q u q u χ K + Σ ∗ + 2 q d q d χ K Σ ∗ + (cid:1) (cid:0) q u χ K + Σ ∗ + q d χ K Σ ∗ + (cid:1) p → ∆ + η (cid:48) (cid:0) q u q u χ K + Σ ∗ + 2 q d q d χ K Σ ∗ + (cid:1) (cid:0) q u χ K + Σ ∗ + q d χ K Σ ∗ + (cid:1) p → Σ ∗ Kp → Σ ∗ K + χ K + Σ ∗ q u q s χ K + Σ ∗ q s χ K + Σ ∗ p → Σ ∗ + K q d q s χ K Σ ∗ + q s χ K Σ ∗ + ral expansion are model-independent predictions of chiralperturbation theory. The leading source of uncertaintycomes from the higher order terms in the expansion. Weprovide an estimation of the uncertainty in these termsthrough variation of the regulator parameter Λ over awide range. B. Analysis
The extrapolation to the physical regime requires thatthe residual series coefficients a (Λ) and a (Λ) are con-strained by fitting to the finite-volume corrected latticeresults a (Λ) + a (Λ) m π = β F V Cv − v (cid:0) m π (cid:1) − (cid:88) M β M Bv − v (cid:0) m π , Λ (cid:1) − (cid:88) M, B (cid:48) β M B (cid:48) v − v (cid:0) m π , Λ (cid:1) , (36)where the regulator parameter Λ takes the value Λ = 0 . m π . Valence-sea and sea-sealoop integral contributions are accounted for by using thechiral coefficients for the “total” process. A physical ex-trapolation is produced by setting m π = m physπ = 0 . β p = 2 . × − fm where the numbers in parentheses represents the sta-tistical uncertainty. By considering the variation of theregulator parameter over the broad range 0 . ≤ Λ ≤ . β p = 2 . (cid:0) +13 − (cid:1) × − fm . Figure 15 highlights a comparision of the chiral extrap-olation prediction produced herein with a selection of re-cent experimental measurements. Excellent agreementis seen between the experimental measurements and theresult obtained herein. This highlights the utility of thequark projection technique and partially quenched chiraleffective field theory used herein. It validates our un-derstanding of QCD behind their development and use.The chiral expansion of Eq. (27) may also be used toguide future lattice QCD calculations at a range of lat-tice volumes by using the discretised sum forms of thecontinuum integrals with either valence-valence or totalintegral coefficients. Figure 16 shows finite volume ex-trapolations of β p with total full QCD coefficients for arange of lattice volumes, 3 . ≤ L s ≤ . m π L s ≥ .
4. Here the 7 . ∼ TABLE IV. Chiral coefficients for the leading-order loop integral contributions for the neutron.
Process Total Valence-sea Sea-sea n → N πn → n π (cid:0) q u q u χ K + Σ − + 2 q d q d (cid:0) χ K Σ + χ K Λ (cid:1)(cid:1) (cid:0) q u χ K + Σ − + q d (cid:0) χ K Σ + χ K Λ (cid:1)(cid:1) n → n η (cid:0) q u q u χ K + Σ − + 2 q d q d (cid:0) χ K Σ + χ K Λ (cid:1)(cid:1) (cid:0) q u χ K + Σ − + q d (cid:0) χ K Σ + χ K Λ (cid:1)(cid:1) n → n η (cid:48) (cid:0) q u q u χ K + Σ − + 2 q d q d (cid:0) χ K Σ + χ K Λ (cid:1)(cid:1) (cid:0) q u χ K + Σ − + q d (cid:0) χ K Σ + χ K Λ (cid:1)(cid:1) n → p π − χ π − p q d q u (cid:0) χ K Σ + χ K Λ (cid:1) q u (cid:0) χ K Σ + χ K Λ (cid:1) n → n − π + q u q d χ K + Σ − q d χ K + Σ − n → Σ Kn → (cid:0) Σ , Λ (cid:1) K q d q s (cid:0) χ K Σ + χ K Λ (cid:1) q s (cid:0) χ K Σ + χ K Λ (cid:1) n → Σ − K − χ K + Σ − q u q s χ K + Σ − q s χ K + Σ − n → ∆ πn → ∆ π (cid:0) q u q u χ K + Σ ∗ − + 2 q d q d χ K Σ ∗ (cid:1) (cid:0) q u χ K + Σ ∗ − + q d χ K Σ ∗ (cid:1) n → ∆ η (cid:0) q u q u χ K + Σ ∗ − + 2 q d q d χ K Σ ∗ (cid:1) (cid:0) q u χ K + Σ ∗ − + q d χ K Σ ∗ (cid:1) n → ∆ η (cid:48) (cid:0) q u q u χ K + Σ ∗ − + 2 q d q d χ K Σ ∗ (cid:1) (cid:0) q u χ K + Σ ∗ − + q d χ K Σ ∗ (cid:1) n → ∆ + π − χ π − ∆ + q d q u χ K Σ ∗ q u χ K Σ ∗ n → ∆ − π + χ π + ∆ − q u q d χ K + Σ ∗ − q d χ K + Σ ∗ − n → Σ ∗ Kn → Σ ∗ K q d q s χ K Σ ∗ q s χ K Σ ∗ n → Σ ∗ − K + χ K + Σ ∗ − q u q s χ K + Σ ∗ − q s χ K + Σ ∗ − TABLE V. SU (3) flavour coupling coefficients for the chiral effective field theory analysis. The header row indicates theintermediate baryon species in the meson-baryon loop dressing. Through conservation of quark flavour, one can identify thebaryon which is being dressed. Σ ∗ ∆ Σ Λ Nχ K + Σ ∗ − C χ π − ∆ + C χ K + Σ − D − F ) χ K Λ 13 ( D + 3 F ) χ π + n D + F ) χ K Σ ∗ C χ π + ∆ − C χ K Σ ( D − F ) χ K + Λ 13 ( D + 3 F ) χ π − p D + F ) χ K Σ ∗ + C χ π + ∆ C χ K + Σ ( D − F ) χ K + Σ ∗ C χ π − ∆ ++ C χ K Σ + D − F ) point to be β n = 2 . (cid:0) +15 − (cid:1) × fm , where the numbers in parentheses represent statisticaland systematic errors respectively. This value is in verygood agreement with the value obtained in our earlierwork of Ref. [17] where the U (1) Landau eigenmodeprojection technique is used with a chiral extrapolationto obtain β n = 2 . × fm . This agree-ment indicates the success of both the SU (3) × U (1) and U (1) eigenmode quark projection techniques. Figure 17presents a comparison of β n to recent experimental mea-surements where good agreement is also observed.In an identical manner to the proton, Figure 18presents finite-volume extrapolations with full QCD co-efficients as a guide to future lattice QCD calculations.For both the proton and neutron, the leading and next-to-leading non-analytic terms of the chiral expansion give rise to a significant enhancement of the magnetic polar-isabilities. In the case of the neutron, this chiral contri-bution reverses the trend observed on the lattice. Nev-ertheless, the coefficients of these non-analytic terms aremodel independent and well known. Figures 16 and 18highlight the volume dependence of these contributionsand indicate a significant challenge to directly observethese effects in future lattice QCD calculations. Herewe note that the infinite volume value for the neutron isgreater than that of the L s = 7 . ∼ O (cid:0) a (cid:1) corrections are expected to be small relativeto the uncertainties already presented. Indeed Ref. [54]4 m ( GeV ) p ( f m ) Inf. Vol. FV Corr.
EFT
Prediction Experiment: PDG Experiment: Beane Experiment: Blanpied Experiment: MacGibbon Experiment: McGovern Experiment: Olmos de León Experiment: Pasquini
FIG. 15. The magnetic polarisability of the proton, β p , fromthe chiral effective field analysis herein ( χEF T Prediction).Lattice results of this work, finite-volume corrected (FVCorr.) with total QCD coefficients are compared with ex-perimental measurements via an infinite-volume (Inf. Vol)chiral extrapolation with total QCD coefficents. The errorbar at the physical point reflects systematic and statisticaluncertainties added in quadrature. Experimental results fromthe PDG [9], McGovern et al. [10], Beane et al. [48], Blanpied et al. [8], Olmos de Le´on et al. [49], MacGibbon et al. [7] andPasquini et al. [6] are offset for clarity. m ( GeV ) p ( f m ) FV: 3.0 fm FV: 4.0 fm FV: 5.0 fm FV: 6.0 fm FV: 7.0 fm Inf. Vol. Lattice PointsFV Corr.
FIG. 16. Finite volume (FV) extrapolations of β p with totalfull QCD coefficients appropriate for fully dynamical back-ground field lattice QCD simulations. The infinite-volumecase relevant to experiment is also illustrated. Both finite-volume valence-valence lattice QCD results (Lattice Points)and their finite-volume (FV Corr.) total QCD corrected val-ues are illustrated. The Lattice Points are offset for clarity. estimates a < .
5% error at the lattice spacing used inthis study.Our conservative estimates for the systematic error dueto the continuum limit extrapolation have a negligibleeffect on the final error when added in quadrature. m ( GeV ) n ( f m ) Inf. Vol. FV Corr.
EFT
Prediction Experiment: PDG Experiment: Kossert Experiment: Myers Experiment: Griesshammer
FIG. 17. The magnetic polarisability of the neutron, β n fromour chiral effective field analysis ( χEF T Prediction). Latticeresults of this work, finite-volume corrected (FV Corr.) withtotal QCD coefficients are compared with experimental mea-surements via an infinite-volume (Inf. Vol) chiral extrapola-tion with total QCD coefficents. The error bar at the physicalpoint reflects systematic and statistical uncertainties added inquadrature. Experimental results from Kossert et al. [50, 51],the PDG [9], Myers et al. [52] and Griesshammer et al. [53]are offset for clarity. m ( GeV ) n ( f m ) FV: 3.0 fm FV: 4.0 fm FV: 5.0 fm FV: 6.0 fm FV: 7.0 fm Inf. Vol. Lattice PointsFV Corr.
FIG. 18. Finite volume (FV) extrapolations of β n with totalfull QCD coefficients appropriate for fully dynamical back-ground field lattice QCD simulations. The infinite-volumecase relevant to experiment is also illustrated. Both finite-volume valence-valence lattice QCD results (Lattice Points)and their finite-volume (FV Corr.) total QCD corrected val-ues are illustrated. The Lattice Points are offset for clarity. C. Magnetic polarisability difference β p − β n The difference between the magnetic polarisability ofthe proton and the neutron can provide a test of Reggeondominance [56, 57]. When Reggeon dominance is as-sumed, the difference of magnetic polarisabilities canbe predicted using chiral perturbation theory techniques5 m ( GeV ) p n ( f m ) Inf. Vol. FV Corr. This work Experiment: PDG Reggeon Dom. : GHLR 2015
FIG. 19. The difference of the proton and neutron magneticpolarisabilities β p − β n . Lattice results of this work, finite-volume corrected (FV Corr.) with full QCD coefficients arecompared with experimental measurements via an infinite-volume (Inf. Vol) chiral extrapolation. The error bar at thephysical point (This Work) reflects systematic and statisticaluncertainties added in quadrature. Experimental results fromthe PDG [9] and a Reggeon dominance prediction from Gasser et al. [56] are offset for clarity. and Baldin sum rules [53, 56, 58].We calculate the correlated difference between themagnetic polarisability of the proton and the neutronat each quark mass and then extrapolate to the physicalregime using the formalism already discussed. In takingthe difference, u - d symmetry in the leading loop-integralcoefficients of the chiral expansion in full QCD cause thecontributions to cancel, leaving a simple linear extrapola-tion of our infinite-volume and full QCD corrected latticeresults. The resulting prediction at the physical point is β p − β n = 0 .
80 (28) (4) × fm , where both statistical and systematic errors are indicatedrespectively. The central value differs from the differencebetween the extrapolated values discussed above due tothe removal of rounding errors at each stage of the calcu-lation. Figure 19 shows an extrapolation of all four latticemass results to the physical regime with comparison tothe PDG value [9] and a result derived using Reggeondominance [56]. VII. CONCLUSIONS
The magnetic polarisabilities of the proton and neu-tron have been calculated using asymmetric operators atthe source and sink. Gauge invariant Gaussian smearingat the source encodes the dominant QCD dynamics whilethe SU(3) × U(1) eigenmode quark projection techniqueis used at the sink to encapsulate the low-lying quark-level Landau physics resulting from the presence of theuniform magnetic field. At the hadronic level, it is crucial to use a Landau wavefunction projection onto the proton two point correlationfunction as the proton is charged and hence experiencesLandau level physics in a uniform magnetic field. Thecombination of these techniques has enabled constantplateau fits to be found in the magnetic polarisabilityenergy shift of the proton for the first time.Furthermore, using the QCD gauge-covariant SU(3) × U(1) projection we are simultaneously able to pro-duce magnetic polarisability energy shifts correspondingto both the neutron and proton ground states in a uni-form background field. This represents a significant ad-vance over the previous gauge-fixed U (1) quark-level pro-jection used to study the neutron polarisability.Connection with experimental results in the phys-ical regime is achieved through the use of heavy-baryon chiral effective field theory and lattice QCDsimulations at several pion masses. The resultingtheoretical prediction for the magnetic polarisabilityof the proton is β p = 2 . (cid:0) +13 − (cid:1) × − fm and β n = 2 . (cid:0) +15 − (cid:1) × − fm for the neutron. Thesepredictions are built upon ab initio lattice QCD simu-lations using effective-field theory techniques to accountfor disconnected sea-quark-loop contributions, the finitevolume of the periodic lattice and an extrapolation to thelight quark masses of nature. These theoretical predic-tions are in good agreement with current experimentalmeasurements and pose an interesting challenge for in-creased experimental precision.While we are necessarily in the confining phase ofQCD, due to the small (cid:126)B field strengths required for theperturbative energy expansion; from the success of theSU(3) × U(1) eigenmode projected quark sink techniqueit is clear that the external magnetic field has a signifi-cant effect on the distribution of the quarks within thenucleon.Our lattice results are electroquenched, they do not di-rectly incorporate the sea-quark-loop contributions fromthe magnetic field. Future work would require a separateMonte Carlo ensemble for each value of B considered andas such is prohibitively expensive due to a loss of QCDcorrelations. Another avenue that could be considered isto investigate the relativistic corrections to the energy-field expansion of Eq. (1). Here, improvements in latticeprecsion will be required in order to succesfully fit theenergies ( E + M ) and construct the relativistic energyshift.It will be particularly interesting to extend this workto the case of hyperons. There the increased mass of thestrange quark will illustrate differences between Σ + and p or Ξ and n polarisabilities and give first insights intothe environment sensitivity of quark-sector contributionsto baryon magnetic polarisabilities.6 ACKNOWLEDGMENTS
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