Spatial image of reaction area from scattering I: SO_mu(2.1) algebra formalism PDF
Preview Spatial image of reaction area from scattering I: SO_mu(2.1) algebra formalism
Spatial image of rea tion area from s attering. I: SO (2.1) µ algebra formalism. ∗ M. V. Polyakov Institut fu(cid:4)r Theoretis he Physik II Ruhr-Universitaet Bo hum, NB6 D-44780 Bo hum, Germany † O.N. Soldatenko, A.N.Vall Department of Theoreti al Physi s, Irkutsk State University, Irkutsk, 664003 Russia ‡ A.A.Vladimirov Bogoliubov Laboratory of Theoreti al Physi s, JINR, 141980, Mos ow Region, Dubna, Russia 8 We develop general formalism of how to relate s attering amplitudes for ex lusive pro esses to 0 spatialimageoftargethadron. Morepre iselyweshowhowtodeterminethespatialdistributionof 0 outgoingparti lesinthespa eofso- allednearestapproa hparameter. Thisparameter hara terizes 2 theposition in the oordinate spa e thepla e where the(cid:28)nal parti le is produ ed. n a PACSnumbers: J 2 1 INTRODUCTION ] h Famous experiment by Hofstadter et al. [1℄ demonstrated that hadrons are not point-like parti les p - and they have non-trivial spatial stru ture. Re ently interest to the spatial images of hadrons was p revived with development of the theory of hard ex lusive pro esses, whi h an be systemati ally e h studied using formalism of Generalized Parton Distributions (GPDs) (for reviews see [2℄). It was [ demonstrated [3℄ that GPDs provide an information about spatial distribution of partons in the 3 transverse plane. v 7 In the present paper we develop general formalism of how to relate s attering amplitudes for 5 ex lusive pro esses to spatial image of target hadron. More pre isely we show how to determine 8 the spatial distribution of outgoing parti les in the spa e of so- alled nearest approa h parameter. 2 . This parameter hara terizes the position in the oordinate spa e the pla e where the (cid:28)nal parti le 8 0 is produ ed. 7 One-parti le states of a free relativisti parti le are de(cid:28)ned by the 10-parametri group of motion 0 M P µν ν : of Minkowski spa e, whi h is given by generators of the Poin are group: and . States an be v i spe i(cid:28)ed by hoosing one or another subgroup. It was found that for des ription of deep inelasti X pro esses it is useful to onsider the "in(cid:28)nite momentum frame" [4, 5℄. We remind brie(cid:29)y ne essary r a for this paper ingredients. The three following generators of the Poin are group: 1 1 B = (K +J ) , B = (K J ) , J , 1 1 2 2 2 1 3 √2 √2 − (1) with K = M , ǫ J = M , i i0 ijk k ij 2 satisfy the system of ommutation relations: [B ,B ] = 0 , 1 2 [J ,B ] = iB , 3 1 2 [J ,B ] = iB , 3 2 1 − [B2 +B2,J ] = 0 , (2) 1 2 3 [p ,B ] = iδ p+ , i j ij [p+,B ] = [p+,J ] = 0 . i 3 p p+ = 1 (p + p ) Here i is the operator of 3-momentum, √2 0 3 . The above ommutation relations B B J E(2) 1 2 3 imply that generators , and form the algebra. Casimir operator of this algebra is B2 = B2+B2 1 2 . Be ause of the last ommutation relation, this algebra an be realized in the three- p+ = const ~λ,p+ dimension momentum spa e restri ted by the ondition on states | i. These states satisfy equations: B2 ~λ,p+ = λ2 ~λ,p+ , | i | i B ~λ,p+ = λ ~λ,p+ , 1 1 | i | i B ~λ,p+ = λ ~λ,p+ , (3) 2 2 | i | i λ2 = λ2 +λ2 . 1 2 B B B2 p+ = const 1 2 Operators , , on the surfa e have ompli ated expression. However in the p /p 1 limiting ase of ⊥ ≪ the orresponding operators are given by simple expression: ∂ ∂ B = ip+ , B = ip+ . 1 2 − ∂p − ∂p (4) 1 2 Substituting these operators into equations (3), we obtain the following expression for the orre- sponding wave fun tion in momentum representation: ~ p~ λ p~ ,p+ ~λ,p+ = exp i ⊥ · . h ⊥ | i p+ ! (5) Now if we introdu e dimensional parameter ~ λ ~ b = , p+ ~ b than the wave fun tion (5) has the form of the eikonal kernel in the plane of impa t parameter . ~λ,p+ It allows us to interpret in the in(cid:28)nite momentum frame the eigenstates of the algebra (2) | i, as the states with (cid:28)xed spatial position in the transverse plane [6, 7℄. In the in(cid:28)nite momentum frame the transformation from the transverse momentum to the transverse position representation is redu ed to simple two-dimensional Fourier transformation (see eq. (5)). That is the transformation whi h relates the GPDs to the parton distributions in the spatial transverse plane [3℄. Note that su h interpretation is possible only in the in(cid:28)nite momentum spa e, in order to ensure the ondition p /p 1 t/⊥Q2 ≪1 . Fotr the hard ex lusive pro esses the latterQ 2ondition orresponds to the ondition ≪ , here is the 4-momentum transfer squared and is the hard s ale for the pro ess. First, t/Q2 in experiments the ratio an be not be not very small and orre tions to the eikonal pi ture 3 an be sizeable. Se ondly, the simple two-dimensional Fourier transform an not be used to obtain spatial distributions of partons for hard pro esses with large transverse momentum transfer (like hadron form fa tors, wide angle Compton s attering, et . [8℄) Therefore our aim here is to develop general formalism whi h would allow "spatial imaging" of hadrons in arbitrary referen e frame. We onsider another subalgebra of Poin are group. This is subalgebra of operators: 1 d = M p . i p2 ij j (6) p2 = const It is realized on the surfa e . Below we analyze in detail this algebra and use it for des ription and building one parti le states with de(cid:28)ned quantum numbers. OPERATOR OF THE NEAREST APPROACH r q q r r x(t) n A r r r d p d ^ A r O n Z A X~(t) ~q FIG. 1: A lassi al traje tory of an asymd~ptoti ally free parti le with momentum is pro eeded into rea tion area, is hara terized bya ve torof maximalapproa h tothesele t point"O". Inthelabor framethis pointisa enter oftarget, in the enter mass frame it is a meetingpointof beams. Let us onsider a traje tory of a free spinless parti le, whi h moves witharbitrary initial onditions (Fig.1) [9℄: q i X (t) = (t t )+ξ , i = 1,2,3 , i 0 i ~q2 +m2 − (7) ~q pm with - momentum of the parti le , - mass of the parti le. A distan e between the parti le and the oordinate origin "O" is D(t) = (X~(t)X~(t))12 . t = τ D(τ) τ At some moment of time the distan e has a minimal value. The time is de(cid:28)ned by the extremum ondition: d 1 d D(t) = X (t) X (t) = 0 . i i dt t=τ D(t) dt t=τ (8) (cid:12) (cid:16) (cid:17)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 4 Using (7) , we obtain: ~ (~uξ) τ = t , 0 − u2 q i (9) u = . i ~q2 +m2 t = τ D(t) p Substituting into , we get the following expression for omponents of the ve tor of the d i nearest approa h : u i ~ d = X (τ) = (~uξ)+ξ . i i −u2 i (10) L , k = 1,2,3 k Let us introdu e the operator of orbital momentum , in usual way: L = ε ξ q . k klm l m (11) Then the expression (10) an be rewritten as: 1 1 d = ε q L = M q . i q2 ijk j k q2 ij j (12) X~(t) We see that for ea h lassi al traje tory of an asymptoti ally free parti le, whi h moves out ~ ~q d from the rea tion area with momentum , we an (cid:28)nd the ve tor of the nearest approa h of the traje tory to the point "O". We an interpret this ve tor as an e(cid:27)e tive oordinate of the parti le reation region. ~ d Note that for the pro ess reversed in time, the ve tor orresponds to the impa t parameter. The impa t parameter is useful for geometri al interpretation of s attering whi h is based on the eikonal approximation for the s attering amplitude. In the eikonal approximation the states with de(cid:28)nite impa t parameter of the initial parti le are de(cid:28)ned as semi- lassi al approximation of states with de(cid:28)nite orbital momentum. The impa t parameter representation is a good toll for studying di(cid:27)ra tive pro esses, see e.g [10℄. However for the small value of impa t parameter semi- lassi al approa h is not valid. It is espe ially learly seen from the un ertainty prin iple: the states with (cid:28)nitemomentumandin(cid:28)nitesimalimpa tparameterareforbidden. So,therearesigni(cid:28) antquantum S e(cid:27)e ts in the region of small impa t parameters. Also a al ulation of matrix elements of -matrix needs onsist~ent quantum-me hani al de(cid:28)nition of states. In this ase it is a state with a de(cid:28)ne d value of the . QUANTUM DESCRIPTION OF THE NEAREST APPROACH PARAMETER. Using a standard quantization pro edure based on the anoni al ommutation relations: [ξ q ] = ıδ , i j ij 5 we get a system of ommutation relations: 1 d = (ε q L iq ) , i q2 ijk j k − i d = (d )+ , [d q2] = 0 , i i i [d L ] = iε d , i j ijk k − [L L ] = iε L , i j ijk k i [d d ] = − ε L , i j q2 ijk k (13) i [d q ] = iδ q q , i j ij − q2 i j 2i [d ,q q ] = iδ q +iδ q q q q , i j k ij k ik j − q2 i j k .................... et . d L SO(3,1) q2 = const i j We see that operators and form algebra of group on the sphere . Corre- sponding Casimir operator of this algebra be omes a number: 1 1 Cˆ = d2 L2 . − q2 ≡ q2 d , d , L 1 2 3 Operators form a nontrivial subalgebra: i [d d ] = L , 1 2 −q2 3 [d1L3] = id2 , (14) − [d L ] = id . 2 3 1 SO(2,1) This is algebra of group. Its properties are investigated and presented in details in mono- graph [11℄. This group is non ompa t and it has three series of unitary representations in spa e of quadrati ally integrated fun tions: general, non ompa t and dis reet. Casimir operator of this algebra is: 1 Kˆ = d2 L2 , d2 = d2 +d2 , [d Kˆ] = 0 , [d Kˆ] = 0 . ⊥ − q2 3 ⊥ 1 2 1,2 3 6 (15) Kˆ We interpret the eigenvalue of this operator in the ontinuous spe trum as a quantum-me hani generalization of a squared e(cid:27)e tive distan e of the reated parti le, as it was de(cid:28)ned above. Here z we note, that relations (14) suppose a de(cid:28)nite hoi e of the dire tion. We link this dire tion with p~ A dire tion of the momentum of proje tile parti le . A realizationof the algebra (14) is onne ted losely to the geometri alproperties of the transverse momentum spa e - the spa e on whi h this algebra is realized. Let us onsider unitary transforma- d d 1 2 tions of the momentum spa e generated by the generators and [12℄: q = e ip~d~q eip~d~ , i = 1,2 , i′ − · i · ~p = p~n , ~n = (cosϕ , sinϕ) , ~n d~= n d +n d , < p < + . (16) 1 1 2 2 · −∞ ∞ 6 p q If we di(cid:27)erentiate this relations in respe t to , we get the following Lee equation for i′: dq′ ~n ~q′ i = n · q′ , i = 1,2 , dp i − q2⊥ i (17) ′ q (p = 0) = q . i i This equation an be easily integrated: q + qsinh(p)+~n ~q cosh(p) ~n ~q n q′ = i q · ⊥ q − · ⊥ i , i = 1,2 . i (cid:16) cosh(p)+ (~n·q~⊥) sinh(p) (cid:17) (18) q q q ~κ ~p It is onvenient to introdu e new parameter related to the parameter by: p ~κ = q~n tanh , q (19) (cid:18) (cid:19) then we have: (~κ~q)(1 B) (~κ~q) 1 q = 1+ − κ +Bq 1+ − ~q ~κ , i′ κ2 i i q2 ≡ ⊕ (20) (cid:16) (cid:17)(cid:16) (cid:17) (cid:2) (cid:3) with κ2 B = 1 . s − q2 SO(2,1) We an see from (20) , that the motion of the transverse momentum spa e is the nonlinear transformation. The operation ⊕ is not a group operation. Be ause of a produ t of two su h operations ontains not only the resulting operation ⊕, but also a rotation. But the operation ⊕ ds has a set of group properties. In parti ular, transformations (20) posses the invariant interval : ds2 = G dq dq , ik i k q2 q q i k G = δ + , i,k = 1,2 , ik q2 ik q2 (21) 3 (cid:18) 3 (cid:19) q2 = q2 q2 . 3 − ⊥ G ik Ri i tensor orresponding to the metri tensor has the form: 1 q q i k R = δ + . ik −q2 ik q2 (22) 3 (cid:18) 3 (cid:19) The orresponding s alar urvature is: 2 R = . −q2 (23) ~q We see that the transver~κse momentum p~κla2ne wqh2i h is obtaine by transformations (20) from (cid:28)xed ⊥ by hanging parameter in the region | | ≤ , is the spa e with onstant negative urvature. q q > 0 3 3 Let us now show that transformations (20) does not hange the sign of , i.e. forward ( ) q < 0 3 and ba kward ( ) semi-spheres are invariant subspa es. Applying the transformation (16) to q 3 we get another Lee equation: dq′ ~n ~q′ 3 = · q′ , q′(p = 0) = q . dp − q2⊥ 3 3 3 (24) 7 Integrating it we obtain: ~n ~q′ ′ q = q exp · dp . 3 3 − q2⊥ (25) (cid:26) Z (cid:27) On the other hand, from (18) we an on lude that: qsinh(p)+~n ~q cosh(p) ~n ~q′ = q · ⊥ q . · ⊥ cosh(p)+ ~n·q~⊥ sinh(p) (26) q q q Substituting this expression to (25) and integrating it we arrive at: 1 ′ ~κ ~q −1 κ2 2 q3 = q3 1+ q·2⊥ 1− q2 . (27) (cid:18) (cid:19) (cid:18) (cid:19) Combining this expression and relation (20), (cid:28)nally we get: q′ q q 3 = 3 = 3 = . q2 q′2 q2 q2 q3 inv (28) − − | | ⊥ ⊥ p p The onsequen e of this invarian e is appearan e of the signature in amplitudes ( ross-se tions) C whi h orresponds to reating of the parti le to the forward or ba kward semi-sphere. At the end of this se tion we note that nonlinear transformations (20) an be linearized in the u ,u ,u 0 1 2 three-dimension spa e on hyperboloid. Indeed, if we introdu e new variables : q ~q u = (u0,u1,u2) , u0 = q , ~u = q⊥ , u2 = u20 −u21 −u22 = 1 , (29) 3 3 than the transformations (20) an be written in these variables as: ′ u = Λ(~κ)u , (30) Λ(~κ) with the transformation matrix : κ κ 1 2 1 q q 1 κ 1 B 1 B Λ(~κ) = 1 B + − κ2 − κ κ B q κ2 1 κ2 1 2 (31) κ 1 B 1 B 2 − κ κ B + − κ2 q κ2 1 2 κ2 2 1 κ2 2 Det(Λ(~κ)) = 1 , B = 1 . − q2 (cid:18) (cid:19) SO (2,1) µ REPRESENTATION OF ALGEBRA IN THE TRANSVERSE MOMENTUM SPACE Letus obtaingroup basi fun tions, whi h realizestates witha de(cid:28)ne valueof theCasimiroperator L 3 (15) and de(cid:28)nite value of the third proje tion of orbital momentum . For that we introdu e spheri al oordinates in the momentum spa e: q = qsinθcosϕ , 1 8 q = qsinθsinϕ , 2 q = qcosθ . 3 d , L i 3 Using an expli it expression for operators , we obtain: i ∂ sinϕ ∂ d = cosϕcosθ sinθcosϕ , 1 q ∂θ − sinθ ∂ϕ − (32) (cid:16) (cid:17) i ∂ cosϕ ∂ d = sinϕcosθ + sinθsinϕ , 2 q ∂θ sinθ ∂ϕ − (cid:16) (cid:17) ∂ L = i . 3 − ∂ϕ The Casimir operator is: 1 ∂2 cosθ ∂ cos2θ ∂2 Kˆ = − cos2θ + 3cosθsinθ + 2cos2θ . q2 ∂θ2 sinθ − ∂θ sin2θ ∂ϕ2 − (33) ! (cid:16) (cid:17) Let us onsider a system of equations: KˆΨ = b2Ψ , L Ψ = mΨ . 3 (34) m = 0 To make analysis simpler we onsider (cid:28)rst spe ial ase of . For this ase we have only one ordinary di(cid:27)erential equation: d2f (1 x2)2 df (1 x2)2 + − +4q2b2f = 0 , − dx2 x dx (35) with 1 θ Ψ(θ) = f(θ), x = tan . cosθ 2 (36) θ = π/2 This equation an be redu ed to the Riemann equation. The point is the singular point of 0 θ < π/2 equation (35). Regular and ontinuous solution for the region ≤ is the so alled ones fun tion: 1 1 Ψ(θ) = P , 0 θ < π/2 1/2+iµ cosθ − cosθ ≤ (cid:18) (cid:19) (37) µ = q2b2 1/4 1/2 , q2b2 1/4 , − ≥ P (z) (cid:0) (cid:1) ν with (cid:21) the Legendre fun tions. π/2 θ < π To (cid:28)nd the solution in the ba kward semi-sphere ≤ , we note that equation (35) is x 1/x cosθ cosθ invariant under the transformation → , whi h orresponds to transformation → − . π < θ π So, regular and ontinuous solution in the region 2 ≤ has the following form: 1 1 π Ψ(θ) = P , < θ π . 1/2+iµ −cosθ − −cosθ 2 ≤ (38) (cid:18) (cid:19) Finally, ombining (37) and (38) we obtain: q q Ψ(~q ) = P , q = qsinθ ,0 θ π . 1/2+iµ ⊥ q2 q2 − q2 q2 ! ⊥ ≤ ≤ (39) − − ⊥ ⊥ p p 9 Kˆ There is an important physi al onstraint on the spe trum of the operator : 1 b2 . ≥ 4q2 (40) This onstraint follows from the Heisenberg un ertainty prin iple (cid:21) a parti le with (cid:28)xed (cid:28)xed energy an not be reated in the region of the oordinate transverse plane of arbitrary small radius. m = 0 The generalization for ase 6 is simple, the solution of Eqs. (34) has the form: 1 1 Ψ(θ,ϕ) = P|m| eimϕ , 0 θ π , 0 ϕ 2π , cosθ 1/2+iµ cosθ ≤ ≤ ≤ ≤ − | | (cid:18)| |(cid:19) (41) µ = q2b2 1/4 1/2 , q2b2 1/4 , m = 0, 1, 2,... . − ≥ ± ± (cid:0) (cid:1) The set of fun tions (41) forms omplete and normalizable basis. The de omposition of fun tions m = 0 in this basis is known as Fo k-Meller expansion and for ase it was investigated by V.A.Fo k [13℄. SO (2,1) µ For the s attering problem, we are interested in, only the plane wave on group are relevant. These plane waves orrespond to the main series of irredu ible representations on the group fun tion spa e, this series is generated by ⊕ operation (20). The situation is similar to a SO(3,1) problem of building of a relativisti on(cid:28)gurational spa e on Lorentz group . In our ase the role of oordinate operators isplayed by motiongenerators inthe momentumspa e. The momentum E2 q2 = m2 spa e is represented on positive part of the two-sheet hyperboloid − . This problem was solved in Refs. [14, 15℄. It was shown, that the relativisti on(cid:28)gurational spa e onjugate to the momentum spa e in Fourie-transformation mean. In our ase the role of the unitary transformation x q kernel from to -spa e is played by the plain waves on hyperboloid, so alled Shapiro's fun tions SO(3,1) [16℄. These fun tions implements the main series of unitary representation group. ψ(~q ) We do not gi~ve detailed al ulations, we just dis uss main step. Let the state ⊥ diagonalizes (~nd ) ~n ~p A an operator ⊥ , with is a ve tor perpendi ular to the momentum : ~ (~nd ) ψ(~q ) = ψ(~q ) . ⊥ ⊥ onst ⊥ (42) Now, using the transformation (20) we obtain that fun tion q2 f(~q ) = 1 ψ(~q ) ⊥ ⊥ s − q2 ⊥ (43) satis(cid:28)es the following fun tional equation: f(~q ~κ) = f(~q ) f(~κ) . ⊥ ⊕ ⊥ · (44) ~ exp i~q b Usual plain waves ⊥ · orrespond to translations of the Eu lidean plain for the ase when (cid:16) (cid:17) ⊕ operation is just a sum operation. ~κ Obviously, the relation (44) is not satis(cid:28)ed for arbitrary , but only for lines of a oordinate κ = κ (t) i i grid in the momentum spa e. Let be a parametri representation of those lines. We an assume without loss of generality that: dκ κ (t = 0) = 0 , i = n , n2 = 1 , i = 1,2 . i i dt (45) (cid:18) (cid:19)t=0 10 κ = 0 i Expanding the equation (44) in the vi inity of the point and using that: dq′ q q i = δ i k , ik dκ − q2 (46) (cid:18) k(cid:19)~κ=0 f(~q ) we obtain the following equation for ⊥ : ∂f (δ x x )n = 2iβf , ik i k k − ∂x (47) i with q ∂f 1 i x = , = iα , β = (α~ ~n) , i,k = 1,2 . i k q x − 2 · (cid:18) k (cid:19)~x=0 It is a di(cid:27)erential equation in partial derivative of the (cid:28)rst order. Its solution is determined up to ζ arbitrary fun tion of the s alar de(cid:28)ned as: 1 x2 ~n ~q ζ = − , z = · ⊥ . (1 z)2 q − β Tψ(h~qe )onstant is arbitrary as well. This arbitrariness isKˆ(cid:28)xed by the ondition that the fun tion ⊥ must be the eigenfun tion of the Casimir operator : Kˆψ(~q ) = b2ψ(~q ) . ⊥ ⊥ (48) f Eventually this gives the orresponding equation for the fun tion : ∂2f ∂f b2q2 (x x δ ) +2~x f = 0 . i k ik − ∂x ∂x · ∂~x − 1 x2 (49) i k − The system of equations (47) and (49) is equivalent to the system (42) and (48). Its solution gives us the basis fun tions of plane waves type. Let us introdu e the notation: ψ(~q ) ξ(q~ ,~µ) , ⊥ ≡ ⊥ ~n ~µ = ~nµ µ with - the two-dimension elementary ve tor, introdu ed in equation (42), . was de(cid:28)ned in (37). The (cid:28)nal result is the following: 1 +iµ q q ~n q~ −2 ξ(q~ ,~µ) = − · ⊥ ⊥ q2 q2 q2 q2 ! − − ⊥ ⊥ 1 = up0(u n)−2+ipµ ; (u n) = u0 ~u ~n · · − · 1 +iµ −2 (50) = u u u2 1cos(ϕ θ) , 0 0 − 0 − − (cid:20) q (cid:21) q~ = (q cosϕ , q sinϕ) , ~n = (cosθ,sinθ) , ⊥ ⊥ ⊥ n = (1,~n) , n2 = n2 ~n 2 = 0 , 0 − n with - a three dimensional light-like ve tor on hyperboloid. f ψ From the relation (43) between and it follows that: 1 f f(q~ ,~µ) = (u n)−2+iµ . ≡ ⊥ · (51)
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