Form Factors and Generalized Parton Distributions in Basis Light-Front Quantization
Lekha Adhikari, Yang Li, Xingbo Zhao, Pieter Maris, James P. Vary, Alaa Abd El-Hady
FForm Factors and Generalized Parton Distributionsin Basis Light-Front Quantization
Lekha Adhikari, ∗ Yang Li, † Xingbo Zhao, ‡ Pieter Maris, § James P. Vary, ¶ and Alaa Abd El-Hady ∗∗ Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, U.S.A. Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China. Physics Department, Zagazig University, Zagazig 44519, Egypt. (Dated: July 10, 2018)We calculate the elastic form factors and the Generalized Parton Distributions (GPDs) for fourlow-lying bound states of a demonstration fermion-antifermion system, strong coupling positronium( e ¯ e ), using Basis Light-Front Quantization (BLFQ). Using this approach, we also calculate theimpact-parameter dependent GPDs q ( x,(cid:126)b ⊥ ) to visualize the fermion density in the transverse plane( (cid:126)b ⊥ ). We compare selected results with corresponding quantities in the non-relativistic limit toreveal relativistic effects. Our results establish the foundation within BLFQ for investigating theform factors and the GPDs for hadronic systems. PACS numbers:
I. INTRODUCTION
Form Factors (FFs) are among the most important measurable quantities that provide information on the internalstructure of hadrons. Generalized Parton Distributions (GPDs) have been introduced as an additional tool to describehadronic substructures. Unlike Parton Distribution Functions (PDFs), which depend only on the momentum fraction x , GPDs also depend on the momentum transfer ∆. GPDs are defined as the non-forward matrix elements of thesame light-cone operators whose forward matrix elements (i.e. the expectation value) yield the PDFs [1]. In specifickinematic regions, GPDs yield the conventional FFs [2]. Therefore, GPDs are hybrid quantities having features incommon both with the FFs and with the PDFs. To gain a comprehensive understanding of physics that underliesFFs, one needs the decomposition of FFs with respect to the momentum fraction of the active parton (quark) thatabsorbs the photon. GPDs, by definition, provide FF decompositions evaluated at a given value of the invariantmomentum transfer t = ∆ . This feature, known as the GPD sum rules, allows one to calculate momentum-dissectedFFs using the same covariant current operator in Quantum Electrodynamics (QED) and Quantum Chromodynamics(QCD) [3, 4].Over more than a decade, there has been a strong interest in GPDs as many observables can be linked with ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] ∗∗ Electronic address: alaa [email protected] a r X i v : . [ nu c l - t h ] F e b them. Specifically, GPDs have been used extensively to investigate the total angular momentum of the quarks/gluonswithin a hadron, thus forming the foundation for the field of spin physics [2]. Moreover, they have been used tovisualize hadrons in three dimensions after performing suitable Fourier transforms [4, 5]. The resulting images areconveniently presented in a space where one dimension describes the light-cone momentum fraction ( x ) and the othertwo dimensions describe the transverse position ( (cid:126)b ⊥ ) of the parton (relative to the transverse center of momentum).These distributions in the transverse plane also preserve the partonic interpretations. Further details of other hadroniccorrelation functions can be found in Refs. [3–5]. Even though GPDs cannot be measured directly from experiments,they enter the Deeply Virtual Compton Scattering (DVCS) amplitude through convolution integrals. The real andimaginary part of the DVCS amplitude can be separated in experiments using the beam charge and beam spinasymmetry, respectively [6–11].Several investigations [12–18] have presented the FFs and GPDs for quark-antiquark bound states. For example,in Ref. [13], the pion GPDs and electromagnetic (e.m.) FFs have been calculated using two Light-Front (LF) phe-nomenological models (Mandelstam-inspired LF Model and LF Hamiltonian Dynamics model). Similarly, in Ref. [16],the e.m. FF is calculated for a two-fermion, pion-like system in the Breit frame.We are motivated by these previous works to evaluate the elastic FFs of a demonstration fermion-antifermionsystem, strong coupling positronium ( e ¯ e ), using the overlap integrals between light-front wavefunctions (LFWFs) inthe Drell-Yan frame. Within the same frame, we also calculate the GPDs using the same LFWFs that were usedto calculate the FFs. Positronium at strong coupling can be viewed as a prototype of quark-antiquark quarkoniumsystems, e.g., c ¯ c, b ¯ b . In the present work, we calculate the elastic FFs and GPDs for the leading Fock sector | e ¯ e (cid:105) . Ourcurrent FF and GPD results serve as prototypes for future applications to quarkonium systems that are solved in thesame (non-perturbative) bound-state framework [19].In this work, we adopt Basis Light-Front Quantization (BLFQ), a recently developed ab initio approach [20], tocalculate FFs and GPDs for four low-lying bound states of positronium 1 S (0 − + ), 1 S (1 −− ), 2 S (0 − + ), and2 P (0 ++ ). Here, states are identified with their non-relativistic quantum numbers (relativistic quantum numbers) N S +1 L J ( J P C ), where N is the principal quantum number , L is the total orbital angular momentum, S is the totalintrinsic spin, J is the total angular momentum, P is the parity and C is the charge conjugation. BLFQ is a non-perturbative approach for solving bound state problems in quantum field theory [20, 21]. It is a Hamiltonian-basedmethod [20] that combines the advantages of light-front dynamics [22] with recent advances in nuclear many-bodycalculations [23]. BLFQ has been successfully applied [24, 25] to the single electron problem in QED in order toevaluate the anomalous magnetic moment of the electron. Another recent work [26] has presented the electron GPDcalculations as a test problem for the BLFQ approach. Furthermore, the BLFQ approach has been extended totime-dependent strong external field problems such as non-linear Compton scattering [27, 28]. The FFs, GPDs, andimpact-parameter dependent GPDs for the bound states of positronium at strong coupling in the BLFQ approach arethe main results of this paper.We organize this paper as follows. In Sec. (II), FFs and GPDs are defined using the overlap integrals betweenLFWFs in relative coordinates. In Sec. (III), we briefly introduce the BLFQ approach, and in Sec. (IV), we present The relation between N , the principal quantum number, and n , the radial quantum number used in Particle Data Group, is N = n + L . our results for FFs, GPDs, and impact-parameter dependent GPDs. Finally, we present the summary and outlook inSec. (V). II. FORM FACTORS AND GENERALIZED PARTON DISTRIBUTIONS ON THE LIGHT FRONT
The elastic FFs are defined as [12, 13, 22, 29] I m J ,m (cid:48) J ( t ≡ ∆ ) (cid:44) P + (cid:104) ψ J ∗ m (cid:48) J ( P (cid:48) ) | j + (0) | ψ Jm J ( P ) (cid:105) , (1)where P and P (cid:48) are initial and final state momenta of the system, respectively, ∆ ≡ P (cid:48) − P is the momentumtransfer (we choose the Drell-Yan frame ∆ + = 0 , t ≡ ∆ = − (cid:126) ∆ ⊥ < j µ is the current operator, J is total angularmomentum of the system and m J is the total angular momentum projection for the system. For simplicity, the chargeof the electron e is excluded in the definition of the FFs.For J = m J = 0, the above relation directly produces the charge FF ( G C ). But for J = 1 in LF dynamics, dueto the light-front parity and the charge conjugation symmetries, we may define four independent helicity amplitudesusing the nine elastic FFs I m J ,m (cid:48) J with m J (and m (cid:48) J ) = 1 , , −
1. For example, conventional FFs such as the chargeFF ( G C ), magnetic FF ( G M ), and quadrupole FF ( G Q ) can be computed using these amplitudes [19, 30–35]. Forsimplicity, in the present study we limit the FF cases to I , ( t ) for J = 0 , J [33, 34].In the present work, we consider the limited case where the virtual photon couples only to the electron to cal-culate the helicity non-flip GPDs and corresponding FFs. Up to the leading Fock sector | e ¯ e (cid:105) , within the impulseapproximation, the elastic FFs using the Drell-Yan formula read F ( t ) (cid:44) I , ( t ) = (cid:88) λ e ,λ ¯ e (cid:90) dx e (cid:90) d (cid:126)k ⊥ ψ ∗ ( (cid:126)k (cid:48)⊥ , x e , λ e , λ ¯ e ) ψ ( (cid:126)k ⊥ , x e , λ e , λ ¯ e ) , (2)where (cid:126)k ⊥ and (cid:126)k (cid:48)⊥ = (cid:126)k ⊥ + (1 − x e ) (cid:126) ∆ ⊥ are the respective relative transverse momenta of the electron before and afterbeing struck by the virtual photon, x e ( x ¯ e ) is the longitudinal momentum fraction of the electron (positron) satisfying x e + x ¯ e = 1, (cid:126) ∆ ⊥ is the transverse component of the momentum transfer, and λ e ( λ ¯ e ) is the spin of the electron(positron). The LFWF ψ ( (cid:126)k ⊥ , x e , λ e , λ ¯ e ) is normalized according to (cid:88) λ e ,λ ¯ e (cid:90) dx e (cid:90) d (cid:126)k ⊥ (cid:12)(cid:12) ψ ( (cid:126)k ⊥ , x e , λ e , λ ¯ e ) (cid:12)(cid:12) = 1 (3)and for simplicity, we have suppressed the quantum numbers labeling ψ .The helicity non-flip GPDs in the region 0 ≤ x e ≤ H ( x e , ξ = 0 , t = − (cid:126) ∆ ⊥ ) = (cid:88) λ e ,λ ¯ e (cid:90) d (cid:126)k ⊥ ψ ∗ ( (cid:126)k (cid:48)⊥ , x e , λ e , λ ¯ e ) ψ ( (cid:126)k ⊥ , x e , λ e , λ ¯ e ) , (4)where ξ ≡ − ∆ + / ( P (cid:48) + + P + ) is the skewness parameter and in ∆ + = 0, ξ = 0.It is straightforward to extend our framework to helicity-flip GPDs, and (as mentioned before) to q ¯ q bound states,but for simplicity, we consider only four low-lying bound states of positronium here, for which there are well-establishedresults in the non-relativistic limit.In the Drell-Yan frame, the expressions for the GPDs (Eq. 4) are very similar to the expressions for FFs, exceptthat the longitudinal momentum fraction x e of the electron is not integrated over. Therefore, GPDs defined in Eq. 4are also known as “momentum-dissected FFs” and measure the contribution of the electron with momentum fraction x e to the corresponding FFs in Eq. 2.Now, referring to Ref.[4], the impact-parameter dependent GPDs are defined as the Fourier transform of the GPDswith respect to the momentum transfer ∆ q ( x ≡ x e ,(cid:126)b ⊥ ) = (cid:90) d (cid:126) ∆ ⊥ (2 π ) e − i(cid:126) ∆ ⊥ · (cid:126)b ⊥ H ( x, , − (cid:126) ∆ ⊥ ) . (5)Here, the momentum transfer (cid:126) ∆ ⊥ is the Fourier-conjugate to the impact parameter (cid:126)b ⊥ and (cid:126)b ⊥ corresponds to thedisplacement of the electron ( e ) from the transverse center of momentum of the entire system ( e ¯ e ). III. BASIS LIGHT-FRONT QUANTIZATION
The positronium bound-state problem is solved in a basis function approach on the light front [21]. In this approach,the longitudinal coordinate is confined in a box of length 2 L , − L ≤ x − ≤ + L , with anti-periodic boundary conditionfor fermions. The longitudinal momentum is discretized: k + = (2 j + 1) π/ L , j = 0 , , , · · · . As the QED Hamiltonianis block diagonal for different P + , we can fix it to be P + = 2 Kπ/ L , where K is a positive integer. The longitudinalmomentum fractions become, x = ( j + ) /K, j = 0 , , , · · · ( K − K represents the resolutionof the basis in the longitudinal direction. In the transverse direction, 2-dimensional (2D) harmonic oscillator (HO)functions are adopted as the basis. In terms of the dimensionless transverse momentum variable (cid:126)v ⊥ (= (cid:126)k ⊥ /b or (cid:126) ∆ ⊥ /b ),the ortho-normalized 2D HO basis function reads φ nm ( (cid:126)v ⊥ ) = (cid:115) n !( n + | m | )! π e imθ v | m | e − v / L | m | n ( v ) , (6)where v = | (cid:126)v ⊥ | , θ = arg (cid:126)v ⊥ , n and m are the (2D) radial and angular quantum numbers, L αn ( x ) is the associatedLaguerre polynomial, and b is the HO basis scale with dimension of mass.For the spin degrees of freedom, two quantum numbers λ e and λ ¯ e are used to label the helicities of the electronand positron, respectively. The momentum space LFWF used in Eq. 2 reads ψ ( (cid:126)k ⊥ , x, λ e , λ ¯ e ) = 1 b (cid:112) x (1 − x ) (cid:88) n,m (cid:104) n, m, x, λ e , λ ¯ e | ψ (cid:105) φ nm (cid:32) (cid:126)k ⊥ b (cid:112) x (1 − x ) (cid:33) , (7)where (cid:104) n, m, x, λ e , λ ¯ e | ψ (cid:105) is the LFWF in the BLFQ basis.The non-perturbative solutions for the LFWFs are provided by a recent BLFQ study [21]. Note that we haveconverted the non-perturbative solutions available in Ref. [21] from single-particle coordinates to relative coordinatesusing the Talmi-Moshinsky (TM) transformation [36]. Here, we exploit the fact that within the N max truncation (seebelow), the LFWFs preserve the factorization of the center of mass motion and the relative motion [37, 38]. n and m in the LFWFs (Eq. 7) are the quantum numbers in the relative coordinates.In order to make the numerical calculations feasible, the basis is made finite using truncation. In the relativecoordinate for the | e ¯ e (cid:105) Fock-sector, the truncation on the transverse degree of freedom is applied as follows:2 n + | m | + 1 ≤ N max . (8)In BLFQ, the total angular momentum J is only an approximate quantum number, due to the breaking of therotational symmetry by the Fock sector truncation and the basis truncation. However, the total angular momentumprojection for the system m J = m + λ e + λ ¯ e (9)is conserved in our system.Now, with the help of Eq. 7, the GPDs (Eq. 4) in the BLFQ basis read H ( x, , − (cid:126) ∆ ⊥ ) = 1 b x (1 − x ) (cid:88) n,n (cid:48) ,m,λ e ,λ ¯ e (cid:104) ψ | n (cid:48) , m, x, λ e , λ ¯ e (cid:105)(cid:104) n, m, x, λ e , λ ¯ e | ψ (cid:105) (10) × (cid:90) d (cid:126)k ⊥ φ ∗ n (cid:48) m (cid:18) (cid:126)k (cid:48)⊥ b (cid:112) x (1 − x ) (cid:19) φ nm (cid:18) (cid:126)k ⊥ b (cid:112) x (1 − x ) (cid:19) = 1 b x (1 − x ) (cid:88) n,n (cid:48) ,m,λ e ,λ ¯ e (cid:104) ψ | n (cid:48) , m, x, λ e , λ ¯ e (cid:105)(cid:104) n, m, x, λ e , λ ¯ e | ψ (cid:105)× (cid:90) d (cid:126)k ⊥ φ ∗ n (cid:48) m (cid:18) (cid:126)k ⊥ + (1 − x ) (cid:126) ∆ ⊥ b (cid:112) x (1 − x ) (cid:19) φ nm (cid:18) (cid:126)k ⊥ − (1 − x ) (cid:126) ∆ ⊥ b (cid:112) x (1 − x ) (cid:19) , (11)where b is the HO basis scale with dimension of mass. Note that in the last step, we have applied a shift in integrationvariables. Now, the integral over the product of the two HO functions with different arguments can be simplified byusing the TM coefficients for the 2D-HO functions [36] to reduce it to an integral over one HO function. Thus, onecan write H ( x, , − (cid:126) ∆ ⊥ ) = √ π (cid:88) n,n (cid:48) ,m,λ e ,λ ¯ e ( − n + n (cid:48) + | m | (cid:104) ψ | n (cid:48) , m, x, λ e , λ ¯ e (cid:105)(cid:104) n, m, x, λ e , λ ¯ e | ψ (cid:105) (12) × (cid:88) ν M N, ,ν, n,m,n (cid:48) , − m ( − ν φ ν (cid:18)(cid:114) − x x (cid:126) ∆ ⊥ b (cid:19) , where M N, ,ν, n,m,n (cid:48) , − m are TM coefficients used to separate the center of mass part and the relative part in the basisfunctions [36], N = n + n (cid:48) − ν + | m | , 0 ≤ ν ≤ n + n (cid:48) + | m | , φ nm ( (cid:126)v ⊥ ) is the 2D-HO basis function in momentumspace, see Eq. 6, and x = ( j + ) /K, j = 0 , , , · · · ( K − x , one can get the FFs in BLFQ basis, i.e. F ( t ) = (cid:90) dx H ( x, , − (cid:126) ∆ ⊥ ) ≈ K − (cid:88) j =0 K H (cid:18) j + 12 K , , − (cid:126) ∆ ⊥ (cid:19) , (13)where the approximation becomes exact in the continuum limit K → ∞ .Inserting Eq. 12 in Eq. 5, q ( x,(cid:126)b ⊥ ) in the BLFQ reads q ( x,(cid:126)b ⊥ ) = b √ π x (1 − x ) (cid:88) n,n (cid:48) ,m,λ e ,λ ¯ e ( − n + n (cid:48) + | m | (cid:104) ψ | n (cid:48) , m, x, λ e , λ ¯ e (cid:105)(cid:104) n, m, x, λ e , λ ¯ e | ψ (cid:105)× (cid:88) ν M N, ,ν, n,m,n (cid:48) , − m ( − ν (cid:101) φ ν (cid:18)(cid:114) x − x b(cid:126)b ⊥ (cid:19) , (14)where (cid:101) φ nm ( b(cid:126)b ⊥ ) is the Fourier transform of Eq. 6. - t [ a Β - ] - t F ( t ) [ a Β - ] (a) 1 S (0 − + ) with b = 0 . m e - t [ a Β - ] - t F ( t ) [ a Β - ] (b) 1 S (1 −− ) with b = 0 . m e
0. 1. 2. 3. 4. 5.0.0.10.2 - t [ a Β - ] - t F ( t ) [ a Β - ] (c) 2 S (0 − + ) with b = 0 . m e
0. 1. 2. 3. 4. 5.0.0.1 - t [ a Β - ] - t F ( t ) [ a Β - ] (d) 2 P (0 ++ ) with b = 0 . m e FIG. 1: − tF ( t ) vs − t for the four low-lying bound states of positronium with N max = 31, K = 61, m J = m (cid:48) J = 0, couplingconstant α = 0 .
3, and photon mass µ = 0 . m e . The “dotted line” represents the positronium FF calculations F ( t ) in BLFQbasis (Eq. 13) and “solid line” represents the FF calculations F NR(cid:96) =0 ( q ) from non-relativistic quantum mechanics (Eq. 15). Note t = − ∆ ⊥ and a b = 1 / ( αm e ) is the Bohr radius and b is the basis scale for the HO functions. IV. RESULTS AND DISCUSSION
We now present and discuss our results for FFs and GPDs obtained in the BLFQ approach beginning with the FFsfor the four low-lying bound states of positronium in Fig. 1. For FFs, the results are calculated with fixed photonmass µ = 0 . m e , where m e is the mass of the electron, and with N max = 31 , K = 61. The small photon mass µ wasintroduced as a regulator in the two-body effective interaction in Ref. [21]. The basis scale b is chosen to minimize theground-state energy at the given N max and K truncation for the given regulator µ and the given coupling constant α = 0 .
3. We present result in units of the Bohr radius a b = 1 / ( αm e ). We compare our positronium FFs calculated inBLFQ with the Non-Relativistic Quantum Mechanics (NRQM) FFs based on the multipole expansion of the one-bodycharge density. With suitable changes in the NRQM, we can adapt the one-body charge density calculated from thewave functions available for the hydrogen atom. N max = N max = N max = N max =
0. 1. 2. 3. 4. 5.0.0.10.2 - t [ a Β - ] - t F ( t ) [ a Β - ] FIG. 2: − tF ( t ) vs − t for 2 S (0 − + ) with different N max . The results are calculated at K = 61, m J = m (cid:48) J = 0, couplingconstant α = 0 . b = 0 . m e , and photon mass µ = 0 . m e . Note t = − ∆ ⊥ , a b = 1 / ( αm e ) is the Bohr radius, and b is thebasis scale for the HO functions. In NRQM, one can define non-relativistic FFs for different states by [39] F NR(cid:96) ( q ) ≡√ πi (cid:96) (cid:104) n, L, m L | j (cid:96) ( qr ) Y (cid:96) (ˆ r ) | n, L, m L (cid:105) = √ πi (cid:96) (cid:90) d r j (cid:96) ( qr ) ρ L ( (cid:126)r ) Y (cid:96) (ˆ r ) ( (cid:96) = 0 , , · · · , L ) , (15)where F NR(cid:96) ( q ) are the multipole FFs, (cid:126)q is the momentum transfer, n is the principal quantum number, L is thetotal orbital angular momentum ( (cid:126)L = (cid:126)J − (cid:126)S ) and m L is its magnetic projection, J is the total angular momentum, ρ L ( (cid:126)r ) (cid:44) ψ ∗ ( (cid:126)r ) ψ ( (cid:126)r ) = (cid:104) n, L, m L | (cid:126)r (cid:105)(cid:104) (cid:126)r | n, L, m L (cid:105) is the coordinate space one-body charge density, Y m(cid:96) (ˆ r ) is the sphericalharmonics, and j (cid:96) ( z ) is the spherical Bessel function of the first kind. In our present work, the FFs F NR(cid:96) ( q ) with (cid:96) = 0 are compared with the FFs F ( t ) in BLFQ basis (Eq. 13).In Fig. 1, one of the features that both FF calculations have in common is the formation of the nodes in N = 2 statesof positronium at lower | t | = | ∆ | . Furthermore, one can easily recognize the FF calculations in BLFQ (“dotted line”)are consistent with the FF calculations from the NRQM (“solid line”) at small momentum transfer. The differencesbetween the BLFQ and NRQM results for the FFs in Fig. 1 begin to be evident around | t | ∼ a − b for the nodelessstates and at | t | ∼ . a − b for the states with radial nodes. As we mentioned before for J = 1, the FF we havecalculated is I , .Here, we also investigate the convergence of the FF with respect to N max motivated by the observation that thecalculations in Ref. [21] showed that the mass spectrum is more sensitive to N max as compared to K or µ . Therefore,our FF and GPD results are calculated with fixed K and µ . Note that N max is the parameter governing the truncationin the transverse direction and we might surmise that the FF defined through momentum transfer in the transverseplane is therefore more sensitive to this truncation. Fig. 2 shows the convergence of the FF calculations with respectto N max for 2 S (0 − + ) keeping other regulators, K = 61, and µ = 0 . m e fixed. As may be expected, Fig. 2 suggeststhat the FF calculations have better N max convergence at lower | t | = | ∆ | since the higher momentum transfers probedetails of the charge density requiring higher HO basis states for accurate descriptions.Next, we present GPDs for the four low-lying bound states of positronium in Fig. 3 for fixed N max = 31 , K = 61 , (a) 1 S (0 − + ) with b = 0 . m e (b) 1 S (1 −− ) with b = 0 . m e (c) 2 S (0 − + ) with b = 0 . m e (d) 2 P (0 ++ ) with b = 0 . m e FIG. 3: 3D plot of helicity non-flip GPDs H ( x, ξ = 0 , t = − ∆ ⊥ ) (Eq. 12) for the four low-lying bound states of positroniumwith N max = 31, K = 61, m J = m (cid:48) J = 0, coupling constant α = 0 .
3, and photon mass µ = 0 . m e . Note a b = 1 / ( αm e ) is theBohr radius and b is the basis scale. and µ = 0 . m e . The parton distribution in each case shown in Fig. 3 is peaked at x = 0 . t -dependence of the positronium GPDs providesinsights into the non-perturbative structure of the system. It is interesting to note that the x -dependence changescharacter from low | t | to high | t | for some states as seen in Fig. 3(c) and Fig. 3(d). Furthermore, in momentum space,the decaying trend of the GPDs is more rapid with increasing | t | for N = 2 compared to N = 1. This signifies theradial extension of the former states is broader than that of the latter states (see the impact-parameter dependentGPDs below). This fact is consistent with NRQM because positronium in a radially excited state is more looselybound with a longer coordinate space tail to its wave function.The impact-parameter dependent GPDs q ( x,(cid:126)b ⊥ ) for our four selected positronium states are presented in Fig. 4for fixed N max = 31 , K = 61 , and µ = 0 . m e . The plots show the distribution of an electron carrying momentumfraction x in the transverse plane as a function of (cid:126)b ⊥ . As we know (cid:126)b ⊥ = (1 − x ) (cid:126)r ⊥ , where (cid:126)r ⊥ is the transverseseparation between the electron and the positron [1, 3, 4, 40], the distribution is generally asymmetric with respect to x = 0 . (cid:126)b ⊥ (cid:54) = 0. It is interesting to note the more complex landscapes appearing in Fig. 4(c) and Fig. 4(d) wherebi-modal distributions in x at low (cid:126)b ⊥ transform into single mode distributions in x at moderate (cid:126)b ⊥ . (a) 1 S (0 − + ) with b = 0 . m e (b) 1 S (1 −− ) with b = 0 . m e (c) 2 S (0 − + ) with b = 0 . m e (d) 2 P (0 ++ ) with b = 0 . m e FIG. 4: Impact-parameter dependent GPDs q ( x,(cid:126)b ⊥ ) (Eq. 14) for the four low-lying bound states of positronium with N max = 31, K = 61, m J = m (cid:48) J = 0, coupling constant α = 0 .
3, and photon mass µ = 0 . m e . Note a b = 1 / ( αm e ) is the Bohr radius and b is the basis scale. V. SUMMARY AND OUTLOOK
We have calculated the GPDs, FFs, and impact-parameter dependent GPDs for the model fermion-antifermionproblem of positronium at strong coupling in the BLFQ approach. We have compared FFs calculated in BLFQ withthose from the one-body density in momentum space in NRQM. They agree reasonably well in low-momentum transferregion, as may be expected. We have also studied the convergence with respect to N max for selective observables.Convergence is reasonably well-established in the low-momentum transfer region.We can extend the present work to higher spin states ( J ≥
1) and compute different physical FFs such as magneticFFs and quadrupole FFs. Such results can further be compared with those of quarkonium systems such as c ¯ c and b ¯ b that have been solved in the BLFQ framework [19]. Moreover, visualizing the simpler positronium system at strongcoupling in 3 dimensions provides us with benchmark cases that will help us better understand features that mayarise with the hadronic structure of mesons in the non-perturbative framework. Our ultimate goal is to apply thisnon-perturbative method to QCD to compute and study observables such as FFs and GPDs of the proton.0 VI. ACKNOWLEDGEMENTS
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