Spatial image of reaction area from scattering.II: On connection between the differential cross-sections in transverse momentum and in nearest approach parameter
N. Bobrovskaya, M. V. Polyakov, A. N. Vall, A. A. Vladimirov
aa r X i v : . [ h e p - ph ] J a n Spatial image of rea tion area from s attering. II :On onne tion between the di(cid:27)erential ross-se tions in transverse momentum and in nearest approa hparameter.N.Bobrovskaya and A.N.Vall ∗ Department of Theoreti al Physi s, Irkutsk State University, Irkutsk, 664003 RussiaM.V.Polyakov † Institut fur Theoretis he Physik II Ruhr-Universitaet Bo hum, NB6 D-44780 Bo hum, GermanyA.A.Vladimirov ‡ Bogoliubov Laboratory of Theoreti al Physi s, JINR, 141980, Mos ow Region, Dubna, RussiaThe onne tion between di(cid:27)erential ross se tions of parti le C reation on transverse momentumand on nearest approa h parameter ~b is investigated in the ontext of formalism SO (2 , algebra.Where parameter ~b hara terizes parti le reation area. This distribution tightly on erned withspatial stru ture of parti les intera tion and allows intuitive physi interpretation. It is shown thatarea of large transverse momentum arries in the main ontribution to distribution fun tion on b in ba kward semisphere in interval / ≤ bq ∼ , where q is momentum of the parti le C . Theleft border of inequation de(cid:28)nes by Geizenberg un ertainty relation on parameters "momentum -radius of lo alization area" . The spatial stru ture of reation area in transverse momentum planeis a set of dis rete axially-symmetri al (for spinless parti les C ) zones. The re eived onne tionbetween ross se tions on ~q ⊥ and ~b is exa t and don't onne t with any model. So ability appears toanalyze the spatial stru ture of target using experimental data of parti le C angle distribution. Itwas re eived an exa t relation between < b ± > and < cos θ ± > for any A + B → C + D pro essesin enter of parti les A and B mass frame, where average is going on orresponding di(cid:27)erential ross-se tions. It is shown that quantum-me hani onstrain on b spe trum ( b q > ~ / ) bringsto onstrain < cos θ ± > > / . As appli ation it is onsidered the pro ess γ + p → π + p at photonenergy E γ = 5 Gev.PACS numbers:DIFFERENTIAL CROSS-SECTION ON THE TRANSVERSE MOMENTUM OF PARTICLE C In this paper we will reprodu e in detail the derivation of onne tion between the di(cid:27)erential ross-se tion inparti le C momentum and S -matrix element [1℄. As in the framework of the same s heme the similar al ulation forthe ross-se tion on nearest approa h parameter b [2℄ will be made.Let us examine the pro ess A + B → C + D for two physi al ases. In the (cid:28)rst ase parti le C is reated with(cid:28)xed momentum ~q , the se ond, parti le C is reated in state | ~µ, q, ǫ > . The di(cid:27)erential ross-se tion on transversemomentum of the parti le C is dσ ± dΩ ~q = 1 n n T V | ~u | dN ± dΩ ~q . (1)Here ~u is relative speed of initial parti les, n , n are densities of parti les A and B , ( ± ) means that the parti le C s atters to the forward and ba kward half-sphere ( z -axis is dire ted along the momentum of parti le A ), and N isthe total number of parti les C and D reation events in all spa e (volume V ) during in(cid:28)nite time ( T ). This is N = Z d ~q d ~q | < ~q ; ~q | ˆ F | in > | = X ǫ = ± N ( ǫ ) == X ǫ = ± Z d ~q q dq dΩ ~q | < ~q ⊥ , ǫ q q − q ⊥ ; ~q | ˆ F | in > | , (2)where | in > = (2 π ) n / n / a + ( ~p ) a + ( ~p ) | > ≡ (2 π ) n / n / | ~p ; ~p > . The reation operators are related to parti les A and B , operator ˆ F is onne ted with S -matrix by relation S = I + i ˆ F .Translational invarian e allows present the matrix element in the form: < f | ˆ F | in > = δ (4) ( q in − q f ) < f | ˆ A | in > , (3)therefore: N = T V (2 π ) X ǫ = ± Z d ~q q dq dΩ ~q δ (4) ( q + q − P ) ×× | < ~q ⊥ , ǫ q q − q ⊥ ; ~q | ˆ A | in > | == T V (2 π ) X ǫ = ± Z q dq dΩ ~q δ ( E q + q ( ~q − ~P ) + m D − P ) ×× | < ~q ⊥ , ǫ q q − q ⊥ ; ~q = ~P − ~q | ˆ A | in > | . (4)Here, the four-momentum P = P + P . In (4) the δ - fun tion determines the energy shell for parti le C rea tion : (cid:26) E q + q ( ~q − ~P ) + m D − P ) (cid:27) q =˜ q = 0 . (5)Let us de(cid:28)ne: E D = q ( ~q − ~P ) + m D λ ( ~q ⊥ ) = ( qE q E D q P − ( ~q · ~P ) E q ) q =˜ q . (6)Then the following relation is valid: δ ( E q + q ( ~q − ~P ) + m D − P ) = λ ( ~q ⊥ ) δ ( q − ˜ q ) . (7)Integrating (4) over q we get: N = T V (2 π ) X ǫ = ± Z dΩ ~q q λ ( ~q ⊥ ) ×× | < ~q ⊥ , ǫ q q − q ⊥ ; ~q = ~P − ~q | A | in > | , q = ˜ q , q = | ~q | . (8)So, we obtain a relation for parti le C transverse momentum distribution: dN ± dΩ ~q = T V (2 π ) q λ ( ~q ⊥ ) | < ~q ⊥ , ± q q − q ⊥ ; ~q = ~P − ~q | A | in > | , q = ˜ q . (9)Let us de(cid:28)ne: A ( ǫ ) ( ~q ) = < ~q ⊥ , ǫ q q − q ⊥ ; ~q = ~P − ~q | A | ~p ; ~p > , q = ˜ q . (10)Then (9) takes a view: dN ± dΩ ~q = (2 π ) T V n n q λ ( ~q ⊥ ) | A ( ± ) ( ~q ) | , q = ˜ q . (11)The di(cid:27)erential ross-se tion is resulted by substitution (11) to relation (1).Here we mark an important noti e. An angle dependents in ross se tion dσ ± dΩ ~q appears not only through variable ~q ⊥ , but in general ase also through ˜ q . In the frame of parti les A and B entre of mass ( .m.f.) the parti le C energyde(cid:28)ned only through square of invariant mass s = ( p A + p B ) . And it is E ⋆C = 12 √ s ( s + m C − m D ) . (12)So: ˜ q = p ⋆C = q ( E ⋆C ) − m C . And E ⋆C isn't depend for s attering angle (the star means .m.f.). The situation hanges in the labor frame (l.f.). Inthis ase there are riti al parameter κ whi h is [3℄ κ = √ s · p ⋆C m C · p A . In the ase κ > the onne tion between parti le C momentum and s attering angle has a single meaning and hasa view: ˜ q = p C = √ sE ⋆C p A cos( θ ) + ( E A + m A )[ s ( p ⋆C ) − m C p A sin ( θ )] / ( E A + m B ) − p A cos ( θ ) , (13)where < θ < π is s attering angle in l.f. (there isn't the maximum angle limitation). We will ba k to dis uss thisexpression below.THE DIFFERENTIAL CROSS-SECTION ON NEAREST APPROACH PARAMETER ~b IN C.M.F.Now we dis uss the ase, when the parti le C reates in state | ~µ, q, ǫ > . Choosing for state | out > the orrespondingbasis, we obtain following relation for the total number of C and D parti les reation events [2℄: N = (2 π ) X ǫ = ± Z d ~q q dq dΩ ~µ | < ~µ, q, ǫ ; ~q | ˆ F | in > | . (14)As distin t from the previous ase, the state < ~µ, q, ǫ ; ~q | is not a state with a de(cid:28)ne momentum. So the parti le C ross-se tion in this state an't be expressed through the matrix element < ~µ, q, ǫ ; ~q | ˆ F | in > . To (cid:28)nd dσ ± dΩ ~µ wewill use a state expansion on states with the (cid:28)xed parti le C transverse momentum [2℄: < ~µ, q, ǫ ; ~q | = 1(2 π ) Z ¯ ξ ( ~q ⊥ , ~µ ) < ~q ⊥ , ǫ q q − q ⊥ ; ~q | dΩ ~q . (15)Substituting this expansion to (14) we obtain: N = 1(2 π ) X ǫ = ± Z d ~q q dq dΩ ~µ dΩ ~q dΩ ~k ¯ ξ ( ~q ⊥ , ~µ ) ξ ( ~k ⊥ , ~µ ) ×× F ( ~q ⊥ , ~q ) ¯ F ( ~k ⊥ , ~q ) , | ~q | = | ~k | = q , (16)where we use a notation: < ~q ⊥ , ǫ q q − q ⊥ ; ~q | ˆ F | in > = F ( ~q ⊥ , ~q ) . (17)Let us make an identi al transformation in (16): F ( ~q ⊥ , ~q ) = [ F ( ~q ⊥ , ~q ) − F ( ~k ⊥ , ~q )] + F ( ~k ⊥ , ~q ) , ¯ F ( ~k ⊥ , ~q ) = [ ¯ F ( ~k ⊥ , ~q ) − ¯ F ( ~q ⊥ , ~q )] + ¯ F ( ~q ⊥ , ~q ) . Then: N = 1(2 π ) Re X ǫ = ± Z d ~q q dq dΩ ~µ dΩ ~q dΩ ~k ¯ ξ ( ~q ⊥ , ~µ ) ξ ( ~k ⊥ , ~µ ) | F ( ~q ⊥ , ~q ) | −− π ) X ǫ = ± Z d ~q q dq dΩ ~µ dΩ ~q dΩ ~k ¯ ξ ( ~q ⊥ , ~µ ) ξ ( ~k ⊥ , ~µ ) ××| F ( ~q ⊥ , ~q ) − F ( ~k ⊥ , ~q ) | , | ~q | = | ~k | = q . (18)The se ond term in (18) is turned to zero, owing to the ompleteness of the system of basi fun tions ξ ( ~k ⊥ , ~µ ) [2℄ : Z dΩ ~µ ¯ ξ ( ~q ⊥ , ~µ ) ξ ( ~k ⊥ , ~µ ) ∼ δ ( ~k ⊥ − ~q ⊥ ) . Finally we obtain: N = 1(2 π ) Re X ǫ = ± Z d ~q q dq dΩ ~µ dΩ ~q dΩ ~k ¯ ξ ( ~q ⊥ , ~µ ) ξ ( ~k ⊥ , ~µ ) | F ( ~q ⊥ , ~q ) | . (19)Su h representation for N is universal in the meaning that it follows both di(cid:27)erential ross-se tions in terms of ~q ⊥ and also in terms of ~µ . This allows mark out singular fa tors in N , onne ted with in(cid:28)nite time and volume. Turningto the matrix elements from ˆ F to ˆ A and integrating over momentum ~q , we get: N = V T (2 π ) Re X ǫ = ± Z q dq dΩ ~µ dΩ ~q dΩ ~k ¯ ξ ( ~q ⊥ , ~µ ) ξ ( ~k ⊥ , ~µ ) ×× δ ( E q + q ( ~q − ~P ) + m D − P ) | A ( ǫ ) ( ~q ) | . (20)If we integrate this orrelation over the parameter ~µ and momentum q then we automati ally obtain relation (11).But we will integrate over momentum ~k ⊥ . So we have: N = V T (2 π ) Re X ǫ = ± Z q dq dΩ ~µ dΩ ~q κ ( µ ) ¯ ξ ( ~q ⊥ , ~µ ) ×× δ ( E q + q ( ~q − ~P ) + m D − P ) | A ( ǫ ) ( ~q ) | , (21)where [4℄ κ ( µ ) = Z ξ ( ~k ⊥ , ~µ ) dΩ ~k = 2 π ch ( πµ ) √ π | Γ( iµ + ) | = √ π (cid:12)(cid:12)(cid:12) Γ( iµ (cid:12)(cid:12)(cid:12) . (22)Further integrating over q in relation (21) we obtain: N = V T (2 π ) Re X ǫ = ± Z dΩ ~µ dΩ ~q κ ( µ ) ¯ ξ ( ~q ⊥ , ~µ ) q λ ( ~q ⊥ ) | A ( ǫ ) ( ~q ) | , q = ˜ q . (23)Taking into a ount (11) we an rewrite this expression in the following form: N = 1(2 π ) Re X ǫ = ± Z dΩ ~µ dΩ ~q κ ( µ ) ¯ ξ ( ~q ⊥ , ~µ ) dN ( ǫ ) dΩ ~q , q = ˜ q . (24)This relation for the total number of events N is right in any frame of parti les A and B . Crossing from N to di(cid:27)erentialdistribution demands (cid:28)xation of frame. If the initial state set in .m.f, then ˜ q don't depend of the s attering angle.And in this ase we have for the di(cid:27)erential distribution on ~µ : dN ± dΩ ~µ = 1(2 π ) κ ( µ ) Re Z dΩ ~q ¯ ξ ( ~q ⊥ , ~µ ) dN ± dΩ ~q , q = ˜ q , (25)where ˜ q = ( s + m C − m D ) s − m C . There are q = ˜ q everywhere below (rea tion surfa e).Using relations between µ and bµ = ( b q −
14 ) / , dΩ ~µ = q tanh( πµ ) d~b , d~µ = µ dµ dϕ, d~b = b db dϕ , (26)we get a distribution on ~b . So we have for di(cid:27)erential ross-se tions: dσ ± dΩ ~µ = 1(2 π ) κ ( µ ) Re Z dΩ ~q ¯ ξ ( ~q ⊥ , ~µ ) dσ ± dΩ ~q ,dσ ± dΩ ~q = (2 π ) q λ ( ~q ⊥ ) | ~u | | A ( ǫ ) ( ~q ) | ,λ ( ~q ⊥ ) = qE q E D q P − ( ~q · ~P ) E q ,E D = q ( ~q − ~P ) + m D . (27)The matrix element ˆ A is onne ted with ˆ S -matrix by relation < f | S | in > = < f | in > + iδ (4) ( q f − q in ) < f | ˆ A | in > .Now, let us integrate dσ ± dΩ ~µ (27) over the dire tion of ve tor ~µ : dσ ± dµ = 12 π µth ( πµ ) κ ( µ ) Z dΩ ~q ( q p q − q ⊥ P − + iµ ( q p q − q ⊥ ) dσ ± dΩ ~q ) . (28)Here we used an integral representation of one fun tion [5℄: π Z dϕ ( u − q u − ϕ − θ )) − / µ = 2 πP − / µ ( u ) . (29)Subsequent transformation of di(cid:27)erential ross-se tion on ~µ onne t with turning to hyperboli variables: u = ( u , u , u ) , u = q p q − q ⊥ , ~u = ~q ⊥ p q − q ⊥ , u = u − u − u = 1 (30)In this variables the di(cid:27)erential volume is: dΩ ~q = d~q ⊥ q p q − q ⊥ = d~uu = du dϕu (31)where ϕ is the azimuth angle of ve tor ~q ⊥ .Taking into a ount that the di(cid:27)erential ross-se tion dσ ± dΩ ~q does not depend on ϕ we obtain: dσ ± dµ = µ tanh( πµ ) κ ( µ ) ∞ Z du u P − + iµ ( u ) (cid:18) dσ ± dΩ ~q (cid:19) , (32)where the angular part of the ross-se tion in the right part of integral is expressed through the variable u . Finallywe represent the di(cid:27)erential ross-se tion on µ through the di(cid:27)erential ross-se tion on s attering angle. Let θ and ϕ be axial and azimuth angles of the momentum ~q . Let us turn to integrating over this angles in the expression (28).For this we noti e that for arbitrary fun tion f ( ~q ) the following integral relation is right: Z f ( ~q ) d ~q = Z q dq dΩ f ( ~q ) = X ǫ = ± Z q dq dΩ ~q f ( ~q ⊥ , ǫ q q − q ⊥ ) , (33) dΩ = sin θdθdϕ , dΩ ~q = 1 q p q − q ⊥ d ~q ⊥ . From this follows that: Z dΩ f ( ~q ) = X ǫ = ± Z dΩ ~q f ( ~q ⊥ , ǫ q q − q ⊥ ) , (34)or in terms of θ and ϕ angles Z dΩ ~q f ( ~q ⊥ , q = + q q − q ⊥ ) = Z dz π Z dϕf ( ~q ⊥ , q = qz ) , Z dΩ ~q f ( ~q ⊥ , q = − q q − q ⊥ ) = Z − dz π Z dϕf ( ~q ⊥ , q = qz ) , (35) z = cos θ , f ( ~q ⊥ , q = qz ) ≡ f ( ~q ) . So relations between variables are qz = ( + p q − q ⊥ forward half-sphere , − p q − q ⊥ ba kward half-sphere . , q ⊥ = q p − z . We turn to integrating over angles in the integral (28). We obtain: dσ + dµ = 12 π µ tanh( πµ ) κ ( µ ) Z dzz P − / iµ ( 1 z ) dσdz ,dσ − dµ = 12 π µ tanh( πµ ) κ ( µ ) Z − dz | z | P − / iµ ( 1 | z | ) dσdz . (36)where dσdz is the di(cid:27)erential ross-se tion on the osine of s attering angle of parti le C in the A + B → C + D pro ess.Here is taken into a ount that dσ ± dΩ ~q do not depend of variable ϕ , therefore dσ ± dΩ ~q = 12 π dσdz . Unifying relations (22) , ( 27) , ( 36) and taking into a ount that: π ) Z κ ( µ ) ¯ ξ ( ~q ⊥ , ~µ )dΩ ~µ = 1 , we obtain the norm of the di(cid:27)erential ross-se tion on the total ross-se tion σ ( AB → CD ) X ǫ = ± ∞ Z (cid:18) dσ ( ǫ ) dµ (cid:19) dµ = Z − (cid:18) dσdz (cid:19) dz = NT V n n | ~u | = σ ( AB → CD ) . (37)As follows from (27), in ontrast to dσ ± dΩ ~q the di(cid:27)erential ross-se tion dσ ± dΩ ~µ is not positively sign determined on all µ interval. Contribution of negative value area of dσ ± dΩ ~µ redu es e(cid:27)e tively to de reasing of the total event number ofparti le C reation. So, this spatial area we an interpret as area where taking pla e an absorption of parti les C .But the total number of asymptoti states with de(cid:28)ne µ regulates by the relation (37).APPLICATION TO SIMPLE MODELS dσd Ω IN C.M.F.As an example we dis uss an one-parti le ex hange in t - hannel, for elasti A + B → A + B s attering. Corresponding ross se tion as fun tion of u has a polar view: dσd Ω = N α ( t − M ) = α N (2 q ) z − z ) = α N (2 z q ) u ( u − εz ) , (38)a. - @ Π b q D €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ q Σ - H AB ® CD L d Σ - €€€€€€€€€€€€€ db b. R R I II III
FIG. 1: a)Distribution fun tion on b in the model of one parti le ex hange. bq < / (cid:21) area forbidden by un ertainty relation, / < bq . √ (cid:21) area of reation parti les C , bq & √ (cid:21) area of absorption parti les C . b)The zone stru ture of plain ~b in themodel of one parti le ex hange. Zone I (cid:21) area forbidden by un ertainty relation, zone II (cid:21)area of reation parti les C , zone III(cid:21) area of absorption parti les C . z = 1 . here z = 1 + 2 M sλ ( s, m A , m B ) = 1 + M q ,M (cid:21) mass of ex hange parti le, λ ( x, y, z ) = ( x + y + z − xy − xy − yz ) (cid:21) wellknown the triangle fun tion, α (cid:21)the oupling onstant, N = (2 π ) q ˜ λ ( ~q ⊥ ) | ~u | = (2 π ) ( s − ( m − m ) ) s . In this ase for the ross-se tion σ ± we obtain: σ ± = α N (2 q ) πz ( z ∓ , (39)where σ ( ε ) = ∞ Z (cid:18) dσ ( ǫ ) dµ (cid:19) dµ = R (cid:0) dσdz (cid:1) dz , ε = 1 R − (cid:0) dσdz (cid:1) dz , ε = − . (40)The integral (32) with su h ross se tion is omputed analyti ally. The normalize distribution on nearest approa hparameter takes a view: σ ε dσ ε db = q b πµ ) κ ( µ )cosh( µπ ) z − εz " P iµ − / (cid:16) − εz (cid:17) + ε p z − P iµ − / (cid:16) − εz (cid:17) , (41)where ε = ± , and P iµ − / ( x ) (cid:21) asso iated ones fun tion. The plot of (41) is shown on Fig.1a.Sin e z > so the argument of ones fun tions belong to segment [ − , . The ones fun tion is positive with su hargument (and asso iated ones fun tions with m > too). So the ross-se tion is positive de(cid:28)ned on ε = 1 . On ε = − the ~b -plane divides into zones with radiuses R , R and R Fig.1b.Here R = ~ / q de(cid:28)nes a border of forbidden area, where Geizenberg un ertainty relation is broken (phase spa eof parti le C is less then allowable). Area R < b < R de(cid:28)nes spatial area of parti le C reation. And area b > R is area where absorption of parti les is taken pla e relating to equality (37). At that, if we symboli ally take densityof distribution I ( b ) ∼ in area R < b < R , when density in area b > R would be ∼ − . It is made onditionalupon that exponential de rease of I ( b ) in the ba kward sphere determines by radius R ∼ /q i.e. I ( b ) ∼ exp( − πbq ) at bq ≫ .The left border of R -zone is de(cid:28)ned by zero of expression (41). In the area of small values of parameter M s the ross-se tion (41) turn into zero on µ = µ = √
72 + 2 2 √ ǫ + 221 √ ǫ + O ( ǫ ) , ǫ = 2 M sλ ( s, m , m ) ∼ M s . In the area M s ≪ the ross-se tion takes a view: σ − dσ − db = q b tanh( πµ ) κ ( µ )cosh( µπ ) " − µ O ( M s ) , (42) σ + dσ + db = q b tanh( πµ ) κ ( µ )cosh( µπ ) " cosh( µπ ) π + O ( M s ) . (43)CONNECTION BETWEEN < b > AND THE CROSS-SECTION ON TRANSVERSE MOMENTUM OFPARTICLE C IN C.M.F OF A AND B PARTICLESLet us al ulate the an average value of nearest approa h parameter square < b ± > . We have by de(cid:28)nition: < µ ± > = 1 σ ± Z ∞ µ dσ ± dµ dµ , (44)where σ ± was de(cid:28)ned in (40). So: < b ± > = 1 q (cid:16) < µ ± > + 14 (cid:17) , q = ˜ q . (45)Substituting the representation for dσ ± dµ from (32), into (44) and taking into a ount that ( ompare with (22) and(29)): κ ( µ ) = 2 π Z ∞ P iµ − / ( x ) dxx , we obtain < b ± > = 2 π ˜ q σ ± Z ( µ + 14 ) P iµ − / ( x ) P iµ − / ( u ) dσ ± d Ω dxx du u d Ω µ . (46)The di(cid:27)erential equation on ones fun tions P iµ − / ( u ) with argument u an be represented in following form [5℄: ( µ + 14 ) P iµ − / ( u ) = (cid:16) − ( u − d du − u ddu (cid:17) P iµ − / ( u ) . The left part of this expression went as underintegral fa tor into relation (46) and its substitution allows us integrateover µ . From the ompleteness relation of ones fun tions follow [2℄: Z ∞ P iµ − / (cid:0) x (cid:1) P iµ − / (cid:0) u (cid:1) d Ω µ = δ ( x − u ) . (47)Next, we an integrate over variable x . After some transformations we obtain: < b ± > = 2 π ˜ q σ ± Z u dσ ± d Ω du u . (48)Crossing in this relation from the variable u to the s attering angle θ , we obtain: < b ε > = q σ + Z cos θ dσ + d cos θ d cos θ , ε = 12 q σ − Z − cos θ dσ − d cos θ d cos θ , ε = − . (49)So, (cid:28)nally we have: < b ± > = 8 sλ ( s, m C , m D ) < cos θ ± > . (50)where < cos θ ε > = σ + Z z dσdz dz , ε = 11 σ − Z − z dσdz dz , ε = − , z = cos θ , and < b ± > is de(cid:28)ned by relation (44).It follows: < µ ± > = 2 < cos θ ± > − . (51)The parameter µ is real number, so µ > . From it and (51) follows an important physi al inequality: < cos θ ± > > . (52)Let us dis uss the nature of this inequality. It follows from the reality of parameter µ = p b q / ~ − / , where ~ (cid:21) Plank onstant. Here b is an eigen value of Kazimir's operator on SO (2 , -group [2℄. Spe trum of this operatorin Hilbert spa e of states satis(cid:28)es ondition b q > ~ / , and that is provide a reality of parameter µ . By itself thisinequality has an quantum nature and show us the fa t that parti le C an not reates in phase spa e less when itallows by Geizenberg's un ertainty relation.We noti e on the fa t that relation (50) between < b ± > and < cos θ ± > right for any pro ess A + B → C + D in enter of mass frame A and B parti les.In the model with one-parti le ex hange (38) it is easy to obtain that < cos θ ± > = ± z ( z ∓
1) ln (cid:16) z ∓ z (cid:17) ∓ z + 2 z . So it follows: < b ± > = 1 q " ± z ( z ∓
1) ln (cid:16) z ∓ z (cid:17) ∓ z + 4 z . Let us analyze this expressions as fun tions of parameter z = 1 + M / q (relation (38)). The analyze shows that < cos θ ± > hanges in borders: < < cos θ + > , . . < cos θ − > < , when z hanges in interval from to ∞ .0DIFFERENTIAL CROSS SECTION ON NEAREST APPROACH PARAMETER ~b IN L.F.As it was note in the previous se tion, in the labor frame of B parti le energy and momentum of C parti le dependof s attering angle (13). Consequen e of this fa t is appearan e on the rea tion plane q = ˜ q as the parameter µ dependen e from the s attering angle, if we use relation (26) for the transition to the parameter b . SO µ (2 , algebraimplements on basis fun tions, for whi h the parameter µ is natural variable. Completeness and orthogonality ofbasis are appeared in terms of it. If we take µ in relation (24) as an independent parameter, then di(cid:27)erential rossse tion dσdµ is de(cid:28)ned unambiguously and doesn't depend of frame. But at the same time the physi al meaning ofthis parameter isn't lear, only as phase spa e measure ( b, q ⊥ ) . In .m.f. transition in the integral (24) from µ to b does not hange anything onsiderably, and di(cid:27)erential distributions on µ and on b are equivalent. But in l.f. thedi(cid:27)erential distribution hanges radi ally. Let us ross in the integral (24) to the variable ~b using relations (26). Thanwe have: N = 1(2 π ) Re X ǫ = ± Z d~b Z dΩ ~q ϑ [ b − R (˜ q )] q tanh( πµ ) κ ( µ ) ¯ ξ ( ~q ⊥ , ~µ ) dN ( ǫ ) dΩ ~q , (53)here µ = ( b q −
14 ) / , R (˜ q ) = ~ q , ϑ ( x ) − dis ontinuous fun tion , q = ˜ q , and ˜ q = p C de(cid:28)nes in (13)From this it follows for the di(cid:27)erential ross se tion on ~b : dσ ( ǫ ) d~b = 1(2 π ) Re X ǫ = ± Z dΩ ~q ϑ [ b − R (˜ q )] q th ( πµ ) κ ( µ ) ¯ ξ ( ~q ⊥ , ~µ ) dσ ( ǫ ) dΩ ~q . (54)Integrating left and right part over the dire tion of ~b ve tor and rossing to angle variables θ and ϕ of ~q ⊥ ve tor weobtain the result, whi h is analogous to relations (36): dσ + db = b π Z dzz ϑ [ b − R (˜ q )] q tanh( πµ ) κ ( µ ) P − / iµ ( 1 z ) dσdz ,dσ − db = b π Z − dz | z | ϑ [ b − R (˜ q )] q tanh( πµ ) κ ( µ ) P − / iµ ( 1 | z | ) dσdz . (55)From omparison of (55) and (36) it follows that di(cid:27)eren es between distributions on the parameter b in .m.f. andin l.f. generates by di(cid:27)eren e of the dependen e of parti le C momentum from kinemati variables s and t for thistwo ases. From the expli it expression ˜ q = p C (36) it follows that it is a (cid:29)uent fun tion of angle and with ertaina ura y it is possible ross from ˜ q to some midvalue in integrals (55). For example, we re eive that on the ba kwardsemisphere: ˜ q ( z ) ≃ ˜ q (¯ z ) , z = cos( θ ) , ¯ z = − . , and it isn't depend on angle. Then the distribution (55) oin ides with the distribution (36) at ˜ q ( z ) ≃ ˜ q (¯ z ) . Moredetailed stru ture of parti le C reation area re(cid:29)e ts in a generalized distribution fun tion: d σdb dz = b π · ϑ [ b − R (˜ q )] q tanh( πµ ) κ ( µ ) 1 | z | P − / iµ ( 1 | z | ) dσdz , − < z < , (56) q = ˜ q = p C ( z ) , µ = ( b q −
14 ) / . It gives us a tomographi pi ture of partial integrals over z to the spatial distribution dσdb in the whole interval of b .It is similar to Wigner fun tion [6℄ From it we an get expressions for the di(cid:27)erential ross se tion on q ⊥ (11), fordi(cid:27)erential ross se tion on b (36, 55) and norm relations (37).1a. x-0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 d s / d x / ndf χ ± ± ± ± -76.5 p4 113.6 ± -223.8 p5 544.9 ± ± ± -1367 p8 1437 ± -2038 / ndf χ ± ± ± ± -76.5 p4 113.6 ± -223.8 p5 544.9 ± ± ± -1367 p8 1437 ± -2038 ’ANDERSON 76 PR D14, 679’ b. - Σ€€€€€€€€€€€€€€€ q db exp @ Π€€€€€ D FIG. 2: a)Di(cid:27)erential ross se tion of pro ess γ + p → π + p , x = cos θ b) Di(cid:27)erential ross se tion on b for pro ess γ + p → π + p - = = = Θ=Π Π€€€€€€€€€€
Π€€€€€€€€€€ Π€€€€€€€€€€ Π€€€€€€ Σ€€€€€€€€€€€€€€€€€ db d Θ I IIIII
FIG. 3: Generalize distribution fun tion in plane ( b, z ) for the pro ess γ + p → π + p , E γ = 5 GeV , q ≃ MeVAs an illustration of represented method, let us onsider pro ess of neutral π -meson photoprodu tion on proton.There is experimental data on angle distribution of π in the interval − . < z = cos θ < (ba kward semisphere)at photon energy E γ = 5 GeV (Fig.3a)[7℄. At su h energy and in this angle interval the meson energy depends ofangles weakly. And we re eive in relations (56) π momentum as its midvalue q ≃ MeV. There is the generalizedistribution fun tion at urrent parameters (Fig.4). The area I is de(cid:28)ned by ondition b < R = ~ / q . This islower border of parameter b physi al values, it is onsequent of Geizenberg un ertainty relation in phase spa e ( b, q ) . R = 1 . · − m at urrent values. The area II is parti le C reation area. In this area the distribution fun tionis positive. Finally, the area III is area where the distribution fun tion is negative. Contribution of this area to thetotal number of parti le C reation events is negative. So, as it was dis ussed, this area is interpreted as the parti le C absorption area. The total number of events in unit time is regulate a ording to (37).Fig.3b is distribution on b integrated over available angle interval − . < z = cos θ < : dσdb = Z − . (cid:20) d σdb dz (cid:21) dz . Evidently to the graphi , parti le C reation areas are also divide to zones. Numeri al al ulation at urrent parametersgives us the following values of zone radiuses. The (cid:28)rst zone - ( R < b < R ), where R = 5 . · − m. These ond zone - ( R < b < R ), where R = 1 . · − m , R = 2 . · − m. The third zone - ( b > R ) , where R = 3 . · − m and so on.The ross se tion fast falls with b growing. If we take the maximum value of ross se tion in the (cid:28)rst zone as unit,when the maximum value in the (cid:28)rst absorption zone ( bq ∼ ) would be equal to . · − , and the maximum in the2se ond reation zone would be . · − . Here we represented estimates of the spatial distribution starting from somemidvalue of π energy. In fa t we have to do exa t al ulation in (56) of µ angle dependen e, using orrespondingrelation for π momentum. CONCLUSIONSo, in the paper it was relieved the exa t relation between the di(cid:27)erential ross se tion on the dete t parti lemomentum and the distribution fun tion on out going nearest approa h parameter of this parti le. This fun tiondes ribes a distribution of matter in rea tion area and allows understand the nature of its omposite parts on hadroni s ale. This onne tion has a di(cid:27)erent view in the enter of mass frame of A and B parti les and in laboratory frame ofparti le B . On the simple model it was shown that rea tion area divides on reation and absorption areas of parti le C . It was re eived a relation between < b ± > and < cos θ ± > in enter of parti les A and B mass frame, whereaverage is going on orresponding di(cid:27)erential ross-se tions. This relation is exa t and right for any A + B → C + D pro esses. There is a bottom bound on < cos θ ± > also follows from it.Only ompleteness of states | ~µ, q, ǫ > in one parti le Fo k spa e was used for (cid:28)nding onne tion between rossse tions.It is re eived the generalized distribution on plane ( b, z ), and the di(cid:27)erential ross se tions and the norm followsfrom that. As an illustration it was onsiderated a real pro ess of photoprodu tion π on the proton, and it was donevaluation of the main hara teristi s of spatial distribution.ACKNOWLEDGEMENTSIt is our pleasure to thank prof. I.B.Khriplovi h for onstru tive dis ussion and riti al remarks. This investigationhas been supported in part by grant President of Russian Federation for support of leading s ienti(cid:28) s hools (NSh-5362.2006.2) (N.B & A.N.V), by the Russian Foundation for Fundamental Resear h (A.A.V), and by the Heisenberg-Landau Program of JINR grant No. 07-02-91557 (A.A.V) ∗ Ele troni address: vallirk.ru † Ele troni address: Maxim.Polyakovtp2.ruhr-uni-bo hum.de ‡‡