Table Of Content

Theory of hadron decay into baryon-antibaryon final state 2 1 0 2 Yu.A.Simonov n a State Research Center J Institute of Theoretical and Experimental Physics, 0 1 Moscow, 117218 Russia ] h January 11, 2012 p - p e h Abstract [ 2 The nonperturbative mechanism of baryon-antibaryon production v duetodoublequarkpair(qq¯)(qq¯)generation insideahadronisconsid- 5 4 ered and the amplitude is calculated as matrix element of the vertex 5 operator between initial and final hadron wave functions. The vertex 5 operator is expressed solely in terms of first principle input: current . 9 quark masses, string tension σ and α . In contrast to meson-meson 0 s production via single pair generation, in baryon case a new entity 1 1 appears in the vertex: the vacuum correlation length λ, which was : v computed before through string tension σ. As an application elec- i troproduction of Λ Λ¯ was calculated and an enhancement near 4.61 X c c GeV was found in agreement with recent experimental data. r a 1 Introduction The baryon-antibaryon ( ¯) final states in hadron reactions are rather typ- BB ¯ ical phenomena, e.g. in charmonium decays pp¯, ΛΛ etc. channels are sig- nificant [1]. In e+e collisions the ¯ final states are carefully studied and − display in many cases (Λ+Λ ,ΛΛ¯,pBp¯B) a nontrivial behavior near the corre- c −c sponding thresholds, see [2] for a review and references. In B decays the produced pp¯ pairs were observed with near-threshold enhancements [3]. In 1 c c Λc Λc ✬ u ✬ u x d x d y ✉ y ✉ ✉u¯ d¯ ✉u¯ d¯ Λ¯c Λ¯c ✫c¯ ✫ c¯ Figure 1: Double quark-pair creation at points x,y in the field of heavy quarks cc¯. this paper we consider a rather general type of reactions, when a quarkonia state(QQ¯)decaysinto ¯,where (¯)containquarkQ(antiquarkQ¯). From BB B B dynamical point of view, the simplest case is the OZI allowed decay of heavy quarkonium into ¯ pair of heavy-flavor baryons, e.g. ψ(nS) Λ+Λ , BB → c −c which was experimentally observed first in [4] at one energy, and measured in [5] in the mass interval [4.5 5.4] GeV/c2. For this type of reaction the ÷ creation of two light quark pairs is necessary and one could expect some suppression in this channel. However, experimentally the suppression is quite mild, as was discovered in the reaction e+e Λ+Λ in [4, 5]. − → c −c A peak at the Λ+Λ mass around 4.63 GeV/c2 was found in [5], and c −c the nature of this enhancement is still obscure, however different explana- tions were suggested [6, 7]. A discussion of possible mechanisms of similar phenomena in ¯ produced in B meson decays, was given in [8]. Below BB we develop a systematic theory of ¯ production in OZI allowed processes, which is actually a theory of doubBleBstring breaking with ¯ emission, as BB shown in Fig.1. As it will be seen, this theory is a one-step development of the general approach of string breaking, given in [9]. To simplify matter we consider first the case of heavy quarkonium, decaying into heavy-flavor ¯ BB pair. 2 The formalism ¯ ¯ The initial state of our problem is the heavy QQ state, where Q and Q are connected by a string. We are looking for a process, where two light qq¯pairs are created in the field of QQ¯, and hence the basic vertex is the 4q operator 2 in the static QQ¯ confining field. As in the case of one pair qq¯ vertex, it is sufficient to consider the light quark Lagrangian in the field of the static antiquark Q¯ and static quark Q. This situation is shown in Fig.1, where a pair u¯u is created at the point (x,x ) and d¯d at (y,y ) with time growing from left to right. The string 4 4 junction trajectory is shown in Fig. 1 by dotted lines and the string junction positions at each moment of time is defined as the Torricelli points in the ¯ triangles formed by space positions of (cud) and (c¯u¯d). It is important, that points (x,x ) and (y,y ) will be shown to be close 4 4 to each other, and the string junction and anti-string junction are generated at one point in their vicinity, which considerably facilitates the picture of ¯ creation (while the latter is rather complicated ina two-step ¯productioBnB). BB Westartwiththepartitionfunctionofalightquarkinthefieldofexternal current of heavy quarks QQ¯. ¯ Z = DADψDψexp (S +S +S +S +S ) , (1) 0 1 int Q Q¯ − Z h i 1 2 S = d4x Fa , (2) 0 4 µν Z (cid:16) (cid:17) S = i d4xψ¯f(∂ˆ+m)ψf, (3) 1 − Z S = d4xψ¯fgAâtaψf. (4) int − Z Here f is flavor index, S and S refer to action of external quark cur- Q Q¯ ¯ rents, of (possibly high mass) quark Q and antiquark Q. We exploit the background formalism [10] to split gluon field into confining background B and perturbative gluon field a µ µ A = B +a . (5) µ µ µ As in [9] we shall use the simplest contour gauge [11] to express B in µ terms of field strength1 u i B (x) = α (u)F (u)du , α = 1, α = , (6) µ µ iµ i 4 i x ZC(x) i 1 Since the whole construction of S for quark q in the field of antiquark Q¯ is gauge eff invariant, the final result does not depend on gauge [12], and the use of contour gauge is a matter of convenience. 3 and the contour C(x) is going from the point x = (x,x ) to the point (0,x ) 4 4 on the world-line of Q and then along this world-line to x = . Note, 4 −∞ that our final result (11), (12) will be cast in the gauge invariant form, which is the same for all contours, connecting points x,y to the world lines of Q (or Q¯). The independence of the resulting asymptotic expressions from the form of contours is shown in Appendix 3 of [13]. Averaging over fields B ,(F ), one can write µ µν ¯ Z = DψDψexp[ (S +S )], (7) 1 eff − Z Z where S was computed in [12]-[13]. Keeping only quadratic correlators eff and colorelectric fields for simplicity, one obtains (for one flavor) 1 S = d4xd4yψ¯(x)γ [ψ(x)ψ¯(y)]γ ψ(y)J(x,y) (8) eff 4 4 −2 Z ¯ where [ψψ] implies color singlet combination, and J(x,y) is expressed via vacuum correlator of colorelectric fields, g2 x y J(x,y) A (x)A (y) = du dv D(u v). (9) 4 4 i i ≡ N h i − c Z0 Z0 Here D(w) is the np correlator, responsible for confinement [15], g2tr F (u)F (v) = (δ δ δ δ )D(u v)+O(D ) (10) iµ kν ik µν iν µk 1 N h i − − c and we have omitted the (vector) contribution of the correlator D , con- 1 taining perturbative gluon exchange and nonperturbative (np) corrections to it. The properties of the kernel J(x,y) have been studied in [12, 13], here we only mention the general form J(x,y) = xyf(x,y)e(x4 y4)2/4λ2D(0), (11) − x2 where we assumed the Gaussian form for simplicity D(x) = D(0)e−4λ2, and 1 1 x y f(x,y) = ds dte (xˆs yˆt)2, xˆ,yˆ= , , (12) − − 2λ 2λ Z0 Z0 at small xˆ,yˆ, f(0,0) = 1, while asymptotically √π f(x,y) = , cosθ = 1, (13) ∼ max( xˆ , yˆ) | | | | 4 where θ is the angle between x and y. Note also, that D(0) and λ are connected to string tension σ 1 σ = 2πλ2D(0) = D(x)d2x. (14) 2 Z We now turn to the effective action (8), where we write explicitly all flavor and color indices. In the latter case one should carefully restore the gauge invariant combinations, derived in [12], using parallel transporters Φ(u,v) = P exp( vA dz ) and we denote u µ µ R ψ¯ (x)ψ (y) ψ¯ (x)Φ (x,Q¯,y)ψ (y) (15) a¯ a¯ a ab b ≡ with Φ (x,Q¯,y) = Φ (x,x ;0,x )Φ (0,x ;0,y )Φ (0,y ,y,y ), (16) ab ac 4 4 cd 4 4 db 4 4 where 0 is at the position of Q¯. Thus (8) can be rewritten as 1 S = d4xd4yψ¯f(x)γ ψf(x)ψ¯g(y)γ ψg(y)J(x,y). (17) eff −2 a¯ 4 ¯b ¯b 4 a¯ Z We take now into account, that λ 0.1 fm [14], [15] is much smaller, than ≈ all hadron scales, and one can integrate in (17) over d(x y ), using the 4 4 − form (11), yielding x +y S d 4 4 d3xd3y(ψ¯f(x)γ ψf(x))(ψ¯g(y)γ ψg(y))σ(xy)f¯(x,y) eff ≈ − 2 a¯ 4 ¯b ¯b 4 a¯ Z (cid:18) (cid:19) (18) xy where we have used (14) and defined f¯(x,y) = f( , ), so that f¯(x,x) = 1 , 2λ√π ∼ x at large x . | | | | To proceed to the practical calculations with the realistic baryon wave functions, itisconvenient togooverfrombispinorto2 2formalism, asitwas × done in [16] for qq¯vertices, see Appendix 2 of [16]. [Note, that the relativistic formalism for the hadron decay, developed in [9], [17], and adapted for the baryon-antibaryon case in Appendix below, accounts for the full relativistic structure of hadrons, and is exemplified in the factor y¯ , which is the ratio 123 of the vertex Z factors for all hadrons. Below we follow a much simpler n derivation in terms of 2 2 formalism, exploited in [16].] × Wenowtake into account asinAppendix 2 of[16], thateach bispinor ψ of light quark in (18) obeys the Dirac one-body equation (αp+β(m+U))ψ = 5 (ε V)ψ, where U is the scalar confining interaction, U(x) = σ x x , − | − Q¯| and V corresponds to perturbative gluon exchanges; therefore one can write v ψ = , where w ! 1 1 w = (σp)v σpv (19) m+U V +ε → m+ U V +ε − h − i where angular brackets imply averaged value for a given quark in the given hadron, in our case this refers to the average energy and potentials of a light quark in the produced heavy-light baryon , e.g. Λ . We also introduce for c antiquarks bispinors ψc and spinors vc,wc,ψc = (vc,wc). Therefore ψ¯ = C 1ψc = ψc(C 1)T = ψcγ γ ;γ = iβα − − 2 4 i i − and 0 σ v w ψ¯γ ψ = i(vc,wc)β 2 = i(v˜c,w˜c) = i(v˜cw+w˜cv) 4 − σ2 0 ! w ! − v ! − (20) where notation is used, vcσ v˜c, wcσ = w˜c = v˜cσp 1 . Hence 2 ≡ 2 − − ←−m+ U V+ε (18) can be written as (we omit below superscript c in spinorhs v−˜c) i S = dt d3xd3y(v˜fc(x,t )Kvf(x,t ))(v˜g(y,t )Kvg(y,t ))σ(x y)f¯(x,y) eff 4 a¯ 4 ¯b 4 ¯b 4 a¯ 4 · Z (21) where 1 1 σ(p+ p) σP ←− K = σp+σp . ←− m+ U V +ε m+ U V +ε ≡ Ω ≡ Ω h − i h − i (22) We now form the S-wave baryon wave function, which can be written as a product of a symmetric coordinate part and antisymmetric spin-flavor-color factor A ,2 B 1 Ψ = A Ψ(coord)(x ,x ,x ,,t ); A = N e Cfghϕf (i)ϕg (j)ϕh (k) B B B 1 2 3 4 B B √6 abc αβγ aα bβ cγ ijk X (23) 2Weneglectthenonsymmetriccoordinatepartofwavefunction,whichcontributesless than one percent to the nucleon mass, see [18, 19, 20] for more details. See also [20] for estimates of Ω in (22). 6 where abc are color indices, αβγ spinor indices and fgh flavor indices. One can separate the c.m. motion and define the set of bound state wave functions in the c.m. system ψ (x x ,x x ) , h 1 3 2 3 { − − } PR e iEt4 i Ψ(coord)(x ,x ,x ,,t ) = − − Ψ (ξ,η), (24) 1 2 3 4 n √V B 3 where ξ,η are Jacobi coordinates, which can be defined in the relativistic case as [18], (ω = p2 +m2 , ω = ω ) i h i ii + i i q P ω ω ω η = (x x ) 1 2 , ξ = 3 (ω x +ω x (ω +ω )x ) 2− 1 ω (ω +ω ) ω2(ω +ω ) 1 1 2 2− 1 2 3 s + 1 2 s + 1 2 (25) and Ψ (ξ,η) is expanded in the fast converging hyperspherical series, where n the leading term (> 90% in the wave function normalization, see [18], [19] for details) is a function of hyperradius only, Ψ (ξ,η) ψ(ρ), where n ≈ 3 ω ρ2 = i (x R)2 = ξ2 +η2. (26) i ω − i=1 + X In what follows we shall be primarily interested in the charmed baryons, Λ , ,Ξ ,Ω and their orbital (and radial) excitations. As a first example c c c c we consider Λ and take for simplicity only one (leading) component of wave P c function with singlet diquark made of u,d. The explicit forms of A for B p,Λ, ,Ξ are given in Appendix 1. For Λ(Λ ,Λ ) one can write in obvious c b notation P 1 Aα = N e c (i)((ud) (du)) (27) Λc Λc √6 abc aα − jk,bc ijk X where (ud) u (j)d (k)ε , ε = ε = 1. jkbc βb βc β 1 1 As shown in≡Appendix 2, the g2auge−in−v2ariant matrix element in the c.m. system of decaying charmonium state Ψ (r) can be written as n1 G(n P ,n P ,n P ) = (2π)4δ(4)(P P P )J (p) (28) 1 1 2 2 3 3 1 − 2 − 3 nB1Bn2n3 where J (p) is nB1Bn2n3 J (p) = y eiprd3(x u)d3(u v)d3(x y)(Ψ ¯Ψ Ψ ). (29) nB1Bn2n3 123 − − − n1M n2 n3 Z 7 Here r = c(u v), c = ωQ , and Ψ are coordinate space spinor − ωQ+ωu+ωd ni wave functions,while ¯ is defined as M ¯ = σ(xy)f¯(x,y)K K (30) x y M At this point one needs to calculate the matrix element of the operator K K between ¯ wavefunctions, which we write as x y BB (P P ) x y A K K (v˜ σ v ) A = η (v˜ σ v ) (31) h B¯| x y Q¯ i Q | Bi BQ Ω2 Λ¯ i Λ Explicitcalculationyieldscoefficientsη ,giveninAppendix1forΛ , ,p,Ξ. Q c B It is more convenient to go over to momentum space in J (p), and n1n2n3 P using Appendix 2, Eq. (A2.16), one has (we omit the superscript red in (A2.16) here and in what follows) d3p d3p d3q d3q J (p) = y¯ x y x y Ψ+ (cp p p ) nB1Bn2n3 123(2π)3(2π)3(2π)3(2π)3 n1 − x − y × Z Ψ (p ,p )Ψ (p +q ,p +q ) (32) × n2 x y n3 x x y y where q q (v˜ σ v ) y¯ = x y Λ¯ i Λ ˜(q ,q )η (33) 123 2√2N Ω Ω M x y QΛ c x y and ˜(q ,q ) is the Fourier transform of ¯(x,y), Eq. (30), modulo K K , M x y M x y the latter were taken into account in the prefactor of ˜ in (33). Also v˜ M Λ¯ and v are spinors for Λ and Λ+ respectively, while σ refers to the spin of Λ −c c i 1 (QQ¯) state. From (A2.18) one can write −− n ∂ ∂ 1 1 −λ2(qx−qy)2 ˜(q ,q ) = σπ4λ2 dsdt(2π)3δ(3)(tq +sq )e (s+t)2 . M x y −∂q ∂q x y x y Z0 Z0 (34) Insertion of (34) into (33) and (32) yields finally J (p) = y¯ 1 1dsdt d3px d3py d3Q e λ2Q2Ψ+(cp p p )Ψ (p ,p ) nB1Bn2n3 (2π)3(2π)3(2π)3 − 1 − x− y 2 x y × Z0 Z0 Z Ψ (p +sQ,p tQ), (35) 3 x y − 8 where we have differentiated by parts in (q q )M˜(q ,q ), obtaining x y x y 3 21/2π σ y¯= 4 · (v˜ σ v )η . (36) N Ω Ω Λ¯ i Λ QΛ c (cid:18) u d(cid:19) Note the factor 4 in (36), which comes from the accounting for two diagrams in Fig.1, and two diagrams with interchanging u- and d- vertices between points x and y. 3 Baryonic width of heavy quarkonium and the ¯ yield in e+e collisions − BB Using J (p) in (35), one can find the decay probability of the n state n1n2n3 1 of QQ¯ into ¯ in the states n ,n respectively, 2 3 BB d3P d3P dw(1 23) = J (p) 2(2π)4δ(4)( ) 2 3 (37) → | nB1Bn2n3 | P1 −P2 −P3 (2π)6 where bar over J(p) 2 implies averaging over initial and sum over final spin | | projections; in our simple case v˜ σ v 2 = 1. We now turn to a more direct | Λ¯ i Λ| experimental process of ¯ production, namely e+e ¯, which was − BB → BB observed in [4, 5]. The corresponding amplitude can be written as [21] 1 A (p,E) = c (E)T c (E) J (p) n2n3 Xn n nn2n3 ≡ nX,m n Eˆ −E +wˆ(E)!nm mBBn2n3 (38) Here Eˆ and wˆ are matrices in indices n,m, of the QQ¯ system, (Eˆ) = nm E δ , n nm d3p J (p)J (p) w (E) = nn2n3 mn2n3 , (39) nm (2π)3 E E (p) Z nX2n3 − n2n3 where n,m refer to the complete set of charmonium bound states, and J (p) is overlap integral of the n-th charmonium state and n ,n states nn2n3 2 3 of heavy-light mesons. In terms of A the total crossection is n2n3 d3p σ (E) = A (p,E) 2π δ(E E (p)) (40) n2n3 | n2n3 | (2π)3 − n2n3 Z 9 The factor c (E) in (38) accounts for the production of (QQ¯) pair in the n n given process, in case of e+e (QQ¯) one has [21] − n → 4παe √6 Q c = ψ (0), n E2 n and with the definition (38) 6π 12e2 ∆R (E) = · Q ψ (0)T (E) 2dΓ , (41) n2n3 E2 | n nn2n3 | n2n3 X where d3p dΓ = π δ(E E (p)). (42) n2n3 (2π)3 − n2n3 As a result, keeping only one state n in (41) one has for 9e2p(E)ψ2(0) JBB¯ (p) 2 ∆R (E) = Q n | n1n2n3 | (43) n2n3 E E E +w (E) 2 n nn | − | where JBB¯ (p) according to (A2.26) can be written as n1n2n3 JBB¯ = 25/2π1/4 σ λ2β03/2Rn(p)e−cp2R20Υ¯ , (44) n1n2n3 Ω2 λ2 +C¯ 3/2(1+2β2R2)3/2 R2 0 0 0 (cid:16) (cid:17) ¯ ¯ ¯ where parameters β ,R ,Υ,C refer to (QQ) and wave functions and are 0 0 n BB defined numerically in Appendix 2. The polynomial (p) is due to (QQ¯) SHO wave function, and is ob- n n R tained in the way described in Eq. (A.33). It can be approximated as p2 p2 2 (p) = 2.1 1 0.034 0.05 (45) Rn ∼ −  − β02 − β02!    In (44) C¯ and Υ¯ are values of C and Υ, (A2.27),(A2.28) averaged over (s,t) integration region. A rough estimate of ∆R(n) in (43), using (45), near Λ+Λ threshold with ¯ c −c ψ(4S) state for (QQ¯) is BB n p exp( 2.5p2) (4) ∆R ξ − , (46) ¯ ≈ E E E +w (E) 2 BB 4 44 | − | 10