Spin Densities and Chiral Odd Generalized Parton Distributions

Transcription

1 Spin Densities and Chiral Odd Generalized Parton Distributions Harleen Dahiya Dr. B.R. Ambedkar National Institute of Technology, Jalandhar, PUNJAB XVI International Conference on Hadron Spectroscopy Newport News, Virginia, USA September 13-18, 2015 H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

2 Overview 1 Introduction 2 Generalized Parton Distributions 3 Covariant Model 4 Chiral Odd GPDs 5 Spin Densities 6 Results 7 Summary and Conclusions H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

3 Introduction Introduction Internal Structure: The knowledge of internal structure of baryons in terms of quark and gluon degrees of freedom in QCD provides a basis for understanding more complex, strongly interacting matter. At high energies, (α s is small), QCD can be used perturbatively. At low energies, (α s becomes large), one has to use other methods such as eective Lagrangian models. Knowledge has been rather limited because of connement and it is still a big challenge to perform the calculations from the rst principles of QCD. One of the most outstanding problem of particle physics is to unravel the internal structure of hadrons such as proton and neutron in terms of their fundamental quark and gluon degrees of freedom. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

4 Introduction Light Front QCD (LFQCD) is an ab initio approach to strongly interacting system. Like perturbative and lattice QCD, it is directly connected to the QCD Lagrangian, but it is a Hamiltonian method, formulated in Minkowinski space rather than Euclidean space. The essential ingredient is Dirac's front form of Hamiltonian dynamics, where one can quantize the theory at xed light-cone time τ = t + z/c rather than ordinary time t. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

5 Introduction Light Front Variables The variables x + = x 0 + x 3 and x = x 0 x 3 are the light-front time and longitudinal space variables respectively. Transverse variable x = (x 1, x 2 ). We dene the longitudinal momentum k + = k 0 + k 3 and light-front energy k = k 0 k 3. Dispersion Relation: For a free massive particle k 2 = m 2 leads to k + 0 and the dispersion relation k = k2 +m2 k +. Features: (1) There is no square root factor. (2) The dependence of the energy k on the transverse momentum k is just like in the nonrelativistic dispersion relation. (3) For k + positive (negative),k is positive (negative). This fact has several interesting consequences. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

6 Generalized Parton Distributions Generalized Parton Distributions An essential tool to investigate hadron structure is the study of Deep Inelastic Scattering. From the parton densities one can extract the distribution of longitudinal momentum carried by the quarks, antiquarks and gluons and their polarizations. The distribution of partons in the plane transverse to the direction in which the hadron is moving is still not known. The role orbital angular momentum in making up the total spin of a nucleon still remains unanswered. GPDs are physical observables which can provide deep insight about the internal structure of the nucleon and more generally, non-perturbative QCD. GPDs provide a 3-D picture of the partonic nucleon structure and encode information on the distribution of partons both in the transverse plane as well as in the longitudinal direction. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

7 Generalized Parton Distributions GPDs are much richer in content about the hadron structure than ordinary parton distributions. GPDs allows us to access partonic congurations with a given longitudinal momentum fraction, but also at specic location (transverse) inside the hadron. GPDs depend on three variables x, ζ, t. x is the fraction of momentum transfer. ζ gives the longitudinal momentum transfer. t is the square of the momentum transfer in the process. Several experiments such as H1 collaboration, ZEUS collaboration and xed target experiments at HERMES have nished taking data on DVCS. Generalized Parton Distributions can be accessed through deep exclusive processes such as Deeply Virtual Compton Scattering. DVCS reaction γ + p γ + p has extraordinary senstivity to fundamental features of the proton's structure and the GPDs reduce to ordinary parton distributions in the forward limit of zero momentum transfer. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

8 Generalized Parton Distributions GPDs have been classied into two types: the chiral even GPDs where quark helicity does not ip (H, E, H, Ẽ) and the chiral odd GPDs which include quark helicity ip (H T, E T, H T, Ẽ T ). The GPDs can be dened from the quark-quark proton correlator function as follows: dz Φ Γ Λ,Λ(x,, P) = + z p, Λ ψ( z 2 )Γψ( z 2 ) p, Λ z + =0,z =0, 2π eixp where Γ = γ +, γ + γ 5, iσ i+ γ 5 (i = 1, 2), with target spins Λ, Λ and momenta p, p. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

9 Generalized Parton Distributions Chiral Odd GPDs For the chiral odd case, we take Γ = iσ i+ γ 5. The correlator can be parameterized as Φ iσi+ γ 5 Λ,Λ (x, ζ, t) = U(P, Λ ) (iσ i+ H T (x, ζ, t) + γ+ i + γ i E T (x, ζ, t) 2M + P+ i + P i HT (x, ζ, t) + γ+ P i P + γ i ) Ẽ M 2 T (x, ζ, t) U(P, Λ). 2M The four momentum light-cone components in a asymmetric frame can be dened as: P = (P +, M 2 ) ( P +, 0, P = (1 ζ)p +, M ) (1 ζ)p +,, = ( ζp +, (1 ζ/2)m /2 (1 ζ)p +, ), t = t 0 2 (1 ζ), t 0 = ζ2 M 2 (1 ζ), where is the square momentum transfer in the process and ζ is the skewness parameter. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

10 Generalized Parton Distributions GPDs in Impact Parameter Space ζ plays the role of the Bjorken variable in deeply virtual Compton scattering ζ = Q2 2P q. For ζ = 0 GPDs can be represented in impact parameter space. Fourier transform (FT) of the GPDs with respect to the transverse momentum transfer at zero skewness ζ gives the distribution of partons in the transverse position or impact parameter space. b is introduced conjugate to giving the distribution of the quarks in the transverse plane. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

11 Generalized Parton Distributions GPDs in impact parameter space ζ = 0 implies that the momentum transfer is completely in the transverse direction. H(x, b ) = 1 d 2 (2π) 2 H(x, 0, 2 )e i b, E(x, b ) = 1 (2π) 2 d 2 E(x, 0, 2 )e i b. b= b is the impact parameter measuring the transverse distance between the struck parton and the center of momentum of the hadron. With the help of impact parameter dependent parton distribution functions (ipdpdfs) one can conclude the transverse position of partons in the transverse plane. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

12 Covariant Model Covariant Model A covariant model has been used to evaluate the quark-proton helicity amplitudes. The formalism is based upon the dissociation of the initial proton into a quark and a xed mass system (diquark). The helicity amplitude f ΛΛ Λ γ 0 for the deep virtual meson production can be introduced with the helicities of the virtual photon and the initial proton being Λ γ, Λ and the helicities of the pion and the proton being 0, Λ respectively. The helicity amplitude fλ ΛΛ ΛΛ γ 0 into gλ γ 0 (x, ζ, t, Q2 ) and A Λ λ,λλ(x, ζ, t) can be decomposed into hard part and soft part as follows f ΛΛ Λ γ 0 (ζ, t) = λ,λ g ΛΛ Λ γ 0(x, ζ, t, Q 2 ) A S Λ λ,λλ(x, ζ, t). The convolution integral is given by 1 ΛΛ dx, the term g ζ+1 Λ γ 0 (hard part) describes the partonic subprocess γ + q π 0 + q and quark-proton helicity amplitude A S Λ λ,λλ (soft part) contains the GPDs. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

13 Covariant Model Continue.. In order to obtain the explicit contributions from the up and down quarks by using the SU(4) symmetry of the proton wavefunction, we have considered both the scalar (spin-0) and the axial-vector (spin-1) congurations for the diquark. We have also studied the spin densities for the up and down quarks in this model for monopole, dipole and quadrupole contributions for unpolarized and polarized quarks in unpolarized and polarized proton. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

14 Covariant Model Spin-0 Scalar Diquark For the case of spin-0 scalar diquark, the amplitude can be written as follows A 0 Λ λ,λλ = d 2 k φ Λ λ (k, P )φ Λλ (k, P), where the vertex functions can be dened as u(k, λ)u(p, Λ) φ Λ,λ (k, P) = Γ(k), k 2 m 2 φ Λ,λ (k, P ) = Γ(k ) U(P, Λ )u(k, λ ) k 2 m 2. The Γ(k) give the scalar coupling at proton-quark-diquark vertex and can be dened as Γ(k) = g s k 2 m 2 (k 2 M 2 Λ )2, where g s is a coupling constant. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

15 Covariant Model The vertex structures for spin-0 diquark are given as φ ++(k, p) = A(m + Mx), φ + (k, p) = A(k 1 + ik 2 ), φ (k, p) = φ ++ (k, p), φ + (k, p) = φ + (k, p). H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

16 Covariant Model Spin-1 Axial-Vector Diquark For the case of spin-1 axial-vector diquark, the amplitude can be expressed as A 1 Λ λ,λλ = d 2 k φ µ Λ λ (k, P ) ɛ λ µ ɛ λ ν φ ν Λλ(k, P), λ where λ is the diquark helicity. The vertex functions in this case can be dened as φ ν Λ,λ(k, P) = Γ(k) u(k, λ)γ5 γ µ U(P, Λ) k 2 m 2, φ µ Λ,λ (k, P ) = Γ(k ) U(P, Λ )γ 5 γ µ u(k, λ ) k 2 m 2. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

17 Covariant Model Continue.. The explicit vertex structures for spin-1 diquark are given as where A = 1 x φ + ++(k, p) = A k 1 ik 2 1 x, φ ++(k, p) = A (k 1 + ik 2 )x, 1 x φ + + (k, p) = 0, φ + (k, p) = A(m + Mx), φ + +(k, p) = A(m + Mx), φ +(k, p) = 0, Γ(k) k 2 m 2, k 2 m 2 = M 2 x x 1 x M 2 x m 2 k2 1 x and k i = k i (1 x) i (i = 1, 2). Here M x is the invariant mass of the diquark and we have taken it at a xed value. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

18 Chiral Odd GPDs The chiral odd GPDs can now be expressed in terms of the helicity amplitudes where τ[2 H T (x, 0, t) + E T (x, 0, t)] S = A S ++,+ + A S +,, H T (x, 0, t) S = A S ++, + A +,+, τ 2 HT (x, 0, t) S = A S +,+, Ẽ T (x, 0, t) S = 0, τ = t0 t 2M. We can obtain the avor structure of the GPDs using the SU(4) symmetry of the proton wavefunction as follows F u T = 3 2 F 0 T 1 6 F 1 T, F d T = 1 3 F 1 T, where F q T = { H T, Ẽ T, H T, E T } (q = u, d). H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

19 Chiral Odd GPDs The term E T + 2 H T gives the deformation in the center of momentum frame due to spin-orbit correlation and can be dened in terms of impact parameter space as follows E T (x, b 2 ) + 2 H T (x, b 2 ) = d 2 e ib (E T (x, 0, t) + 2 H T (x, 0, t)). We can compute E T + 2 H T for for the cases of scalar diquark spin-0 and axial-vector spin-1 and the results for S = 0 and S = 1 components can be expressed as (2 H T (x, 0, t) + E T (x, 0, t)) 0 = 2M d 2 gs 2 (1 x) 5 (Mx + m) 2 k, x L 2 1 L2 2 (2 H T (x, 0, t) + E T (x, 0, t)) 1 = 2M d 2 ga 2 (1 x)4 k, x L 2 1 L2 2 where L 1 = k 2 M 2 x(1 x) + M 2 x x + M 2 Λ(1 x), L 2 = k (1 x) 2 M 2 x(1 x) + M 2 x x + M 2 Λ(1 x). H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

20 Chiral Odd GPDs x 0.1 x 0.2 x x 0.1 x 0.2 x Figure: ( E T + 2 H T ) q (q = u, d) as function of b for xed values of x for the up (left panel) and down (right panel) quarks. The magnitude increases as the value of the b decreases and the distribution peaks are highest at b = 0 which is the center of momentum. As we move away from the center of momentum towards larger values of b, the density of partons decreases. As the value of x increases the magnitude of decreases. The dierence between the magnitudes of 2 H T + E T is more towards the lower values of b. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

21 Spin Densities Spin Densities The three dimensional densities ρ(x, b, λ, Λ) which gives the probability to nd a quark with momentum fraction x and transverse position b with lightcone helicity λ(= ±1) and longitudinal polarization Λ(= ±1) can be dened as ρ(x, b, λ, Λ) = 1 2 [H(x, b2 ) + b j ɛ ji S i M E (x, b 2 ) + λλ H(x, b 2 )]. H describes the density of unpolarized quarks in the unpolarized proton. H reects the dierence in density of quarks with helicity either being equal or opposite to the proton helicity. E describes a sideways shift in the unpolarized parton density when the proton is transversely polarized. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

22 Spin Densities Spin Densities ρ(x, b, s, S ) gives the probability to nd a quark with momentum fraction x with transverse position b and transverse spin s in the proton with transverse spin S. We have ρ(x, b, s, S ) = 1 2 [H(x, b2 ) + s i S i( H T (x, b 2 ) 1 4M 2 b H ) T (x, b 2 ) + b j ɛ ji ( ) S i E (x, b 2 ) + s i [E T (x, b 2 ) + 2 H M T (x, b 2 )] + s i (2b i b j b 2 δ ij ) S j H T (x, b 2 )]. M 2 It receives contribution from the monopole H 1, dipole s 2 2 ib j (E T + 2 H T /M) and quadrupole 1s 2 is i (bx 2 by 2 ) H T /M 2 terms. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

23 Spin Densities The chiral even terms H(x, 0, t) and E(x, 0, t) for S=0 and S=1 diquarks are respectively expressed as ( ( H 0 1 (x, 0, t) = (Mx + m) 2 (1 x) 4 I 3 + (1 x) 4 I1 + I x 2 (M 2 x(1 x) Mx 2 x MΛ(1 2 x) m 2 (1 x) (1 ) x)2 2 )I 3 2 ( H 1 2(1 x)2 I1 + I 2 (x, 0, t) = + x 2 I 3 (M 2 x(1 x) M 2 x x M 2 Λ(1 x) (1 x)2 2 2 E 0 (x, 0, t) = 2M(Mx + m)(1 x) 4 I 3, E 1 (x, 0, t) = 2M(Mx + m)(1 x) 4 I 3. Here )), 1 (1 α)dα 1 αdα 1 I 1 = π, I 0 D 2 2 = π 0 D, I α(1 α)dα 2 3 = π, 0 D 3 D = α(1 α)(1 x) 2 2 M 2 x(1 x) + Mx 2 x + MΛ(1 2 x). H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

24 Spin Densities The FT for dierent contributions are expressed as 1 2 s i b j (E T (x, b ) + 2 H T (x, b )) M = s i b j 2M 2 d 2π J 1( b)(e T (x, 0, t) + 2 H T (x, 0, t)), 1 ( H 2 s T (x, b ) ) i S i H T (x, b ) b 4M 2 = 1 ( d 2 s i S i 2π J 0( b)h T (x, 0, t) d 4M 2 2π ( J 1 ( b) + 1 ) 2 b(j 0( b) J 2 ( b)) HT (x, 0, t), 1 2 s i S i (b 2 i b 2 H j ) T (x, b ) M 2 = 1 2M 2 s i S i (b 2 i b 2 d ( j ) 1 ) 2π 2 2 J 0 ( b) H T (x, 0, t), 1 2 S E (x, b ) 1 i b j = M 4πM S i b j 2 J 1 ( b)e(x, 0, t)d, 1 2 S j b i E (x, b ) = 1 4π S j b i 2 J 1 ( b)e(x, 0, t)d. ) 1 ( S j b i E (x, b ) s i b j (E T (x, b ) + 2 H T (x, b ) ) = 1 2 M 4π S j b i 2 J 1 ( b)e(x, 0, t)d + 1 S i b j 2 d J 1 ( b)(e T 4π M (x, 0, t) + 2 H T (x, 0, t)) where J 0 ( b), J 1 ( b), J 2 ( b) are the Bessel functions of rst kind. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

25 Results Figure: The monopole contribution H/2 for the unpolarized quarks in the unpolarized proton for the up (left panel) and down (right panel) quarks. The distribution for the up quark is more spread as compared to the distribution of the down quark and is almost twice. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

26 Results E E Figure: The dipole contribution 1s 2 ib j (E T +2 H T )/M for the transversely polarized quarks in the unpolarized proton for the up (left panel) and down (right panel) quarks. The distribution has a reection symmetry along the ŷ direction and all orientations are equally probable in the positive and negative ŷ direction. The density obtained for the up quark is however greater than the density obtained for the down quark. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

27 Results Figure: The sum of monopole H/2 and dipole contribution 1 2 s ib j (E T + 2 H T ) for the up (left panel) and down (right panel) quarks. The distortion towards the +ve y-axis for the up quark. A comparatively smaller distortion is observed for the down quark. The dipole contribution introduces a large distortion transverse to both the quark spin and the momentum of the proton suggesting that quarks also have a transverse component of orbital angular momentum and a large value of rst moment of E T + 2 H T. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

28 Results E E Figure: The dipole contribution 1 2 S ib j E for the unpolarized quarks in the transversely polarized proton for the up (left panel) and down (right panel) quarks. The dipole contribution is twice as larger for the up quark as compared to the down quark and the distribution is more spread over the and plane for the up quark than the down quark. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

29 Results Figure: The sum of monopole H/2 and dipole contribution 1 2 S ib j E /M for the unpolarized quarks in the transversely proton for the up (left panel) and down (right panel) quarks. Distortion is obtained and is larger for the up quark than for the down quark. This is basically due to the presence of the E term which already has a large magnitude for the up quark. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

30 Results Figure: The monopole contribution 1 2 s is i (H T b H T /4M 2 ) for the quarks in the proton polarized in the same direction for the up (left panel) and down (right panel) quarks. The sign ips for up and down quarks. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

31 Results E E Figure: The quadrupole contribution 1 2 s is i (b 2 i b 2 j ) H T /M 2 for the quarks in the proton polarized in the same direction for the up (left panel) and down (right panel) quarks. The opposite sign for the quadrupole 1 2 s is i (b 2 i b 2 j ) H T /M 2 term is due to the sign dierence in the up and down quark's x dependence of H T and H T as predicted by the model. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

32 Results E E Figure: The dipole contribution 1S 2 jb i E /M for ˆx polarized quarks in a proton polarized in ŷ direction for the up (left panel) and down (right panel) quarks. The distortion due to the dipole contribution 1 2 S y E for ˆx polarized quarks when the proton is transversely polarized in the ŷ direction gets rotated with respect to the results for dipole contribution 1 2 S ib j E for the unpolarized quarks in the transversely polarized proton. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

33 Results ) Figure: The total dipole 1[S 2 jb i E s i b j (E T + 2 H T )/M] for ˆx polarized quarks in a proton polarized in ŷ direction for the up (left panel) and down (right panel) quarks. The total dipole contribution is obtained from the dipole contribution 1s 2 ib j (E T + 2 H T )/M for the transversely polarized quarks in the unpolarized proton with the additional factor of S j b i E. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

34 Results E E Figure: The quadrupole contribution s i S j b i b j H T /M 2 for ˆx polarized quarks in a proton polarized in ŷ direction for the up (left panel) and down (right panel) quarks. The quadrupole term s i S j b i b j H T /M 2 for the up quark is well spread over the plane whereas for the down quark the distribution is spread in almost half of the region as compared to the up quark. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

35 Summary and Conclusions When unpolarized quarks are present in the unpolarized proton the density distributions for the monopole H and dipole 1 2 s ib j (E T + 2 H T ) terms are found to be larger for the up and down quarks and when we take the contributions from both the terms, the density distribution gets distorted in the plane. A sign ip for the up and down quarks is observed for the polarized quarks in the polarized proton for the monopole 1 2 s is i (H T b H T /4M 2 ) and the quadrupole 1 2 s is i (b 2 i b 2 j ) H T /M 2 contributions which is due to the dierent sign obtained for the H T and H T in the model. The ˆx-polarized quarks in the ŷ-polarized proton and the spin distribution is rotated with respect to the results obtained for unpolarized quarks in unpolarized proton. The shift obtained here is however in the same direction which leads to the same sign of the magnetic moment of the up and down quarks. The spin densities provide a complete description of the spin structure of the nucleon and its relation with TMDs could be tested in future experiments. H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36

36 Summary and Conclusions Thank You H Dahiya (NITJ) Chiral Odd GPDs HADRON / 36