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Relativistic description of the double P -wave charmonium production in e + e - annihilation
A.P. Martynenko1,2 , A.M. Trunin2

2

Samara State University Samara State Aerospace University

1


Double charmonium production
1. K. Abe et al., Phys. Rev. D 70, 071102 (2004) 2. B. Aubert et al., Phys. Rev. D 72, 031101 (2005) 3. K.-Y. Liu, Z.-G. He and K.-T. Chao, Phys. Lett. B 557, 45 (2003) 4. V.V. Braguta, A.K. Likhoded and A.V. Luchinsky, Phys. Lett. B 635, 299 (2006) 5. Y.-J. Zhang, Y.-J. Gao and K.-T. Chao, Phys. Rev. Lett. 96, 092001 (2006) 6. E. Braaten and J. Lee, Phys. Rev. D 67, 054007 (2003); ibid. 72, 099901(E) (2005) 7. A.E. Bondar and V.L. Chernyak, Phys. Lett. B 612, 215 (2005) 8. D. Ebert and A.P. Martynenko, Phys. Rev. D 74, 054008 (2006) 9. A.V. Berezhnoy, Phys. Atom. Nucl. 71, 1803 (2007) 10. G.T. Bodwin, J. Lee and Ch.Yu, Phys. Rev. D 77, 094018 (2008) 11. D. Ebert, R.N. Faustov, V.O. Galkin and A.P. Martynenko, Phys. Lett. B 672, 264 (2009) 12. E.N. Elekina, A.P. Martynenko, Phys. Rev. D 81, 054006 (2010) 13. N. Brambilla, S. Eidelman, B.K. Heltsley et al. EPJ C 71, 1534 (2011)
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General formalism
Two sources of the enhancement of the nonrelativistic cross section for the double charmonium production are revealed to the present: the radiative corrections of order O (s ) and relative motion of c -quarks forming the bound states. In the following we investigate second relative order relativistic corrections to the double P -wave charmonium production amplitudes in LO s as well as to the quarks bound state wave functions. Two stages of the pro duction process: 1. The virtual photon produces two heavy c -quarks and two heavy ¯ c -antiquarks. Pertubative QCD. 1 p1,2 = P ± p , 2 (p · P ) = 0, 1 q1,2 = Q ± q , 2 (q · Q ) = 0, (1)

P, Q the total four-momenta, p = LP (0, p), q = LQ (0, q) the relative four-momenta.
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General formalism
2. In the second nonpertubative stage quarks and antiquarks form the final P -wave mesons hc and cJ , J = 0, 1, 2. Quasipotential approach to the relativistic quark model. The color-singlet mechanism is considered as a basic one for the pair charmonium production. The production amplitude [8]: M(p- , p+ , P , Q ) = 8 2 s (4m2 )Qc ¯ v (p+ ) u (p- ) 3s dp (2 )3 dq â (2 )3

¯ ¯ â Tr hc (p , P )1 (p , q , P , Q )c (q , Q ) + ¯ ¯ + c (q , Q )2 (p , q , P , Q )hc (p , P )

, (2)

Qc
2

c -quark electric charge, vertex functions.
4 / 16

s = l = (p+ + p- )2 , 1
,2
[8] D. Eb ert and A.P. Martynenko, Phys. Rev. D 74, 054008 (2006)


Production diagrams
The leading order in s : e
-

hc


e

-

h
J

c



c



e

+

e+ + final states permutations. (^ - q1 + m) l^ D (l - q1 )2 - m2 ^ (q2 - ^ + m) l µ D = (l - q2 )2 - m2
µ µ

cJ

1 = 2

µ

^ (p1 - ^ + m) l µ D (l - p1 )2 - m2 (^ - p2 + m) l^ (k1 ) + µ D (l - p2 )2 - m2 (k2 ) +


µ

(k2 ), (3) (k1 ),

µ

k1,2 = p1,2 + q1,2 gluon four-momenta, l = p+ + p- = P + Q .
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Bound quarks wave functions
The relativistic wave functions was transformed from the rest frame (CM) to the moving one with four-momenta P , Q : ¯ hc (p , P ) = ¯ hc (p) 0
(p ) ( (p )+m) m 2m

^ ^ v1 - 1 p2 p ^ + v1 - â 2 2m( (p ) + m) 2m

^ â5 (1 + v1 ) ¯ c (q , Q ) =

^ ^ p2 p v1 + 1 ^ + v1 + , 2 2m( (p ) + m) 2m (4) ¯ c (q) ^ ^ v2 - 1 q2 q 0 ^ + v2 + â (q ) ( (q )+m) 2 2m( (q ) + m) 2m
m 2m

^ â c (Q , Sz )(1 + v2 ) ^ v1 =
P Mh

^ ^ v2 + 1 q2 q ^ + v2 - , 2 2m( (q ) + m) 2m

, v2 =

Q M

,

(p ) = m2 + p2 , c (Q , Sz ) the polarization vector of the spin-triplet state cJ .
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Expansions of quark and gluon propagators
1
2 k1, 2

=

4 1 s

4(lp + lq ) 4(p + q )2 16(lp + lq )2 - + + ... , s s s2 1±2 lp p 2 4(lp )2 - + + ... , w w w2 (5)

1 1 = (l - p1,2 )2 - m2 w

1 1 lq q 2 4(lq )2 = 1±2 - + + ... , (l - q1,2 )2 - m2 x x x x2 w= s s 1 2 2 2Mc - Mhc - 4m2 , + J 24 2 s s 1 2 2 x= + 2Mhc - Mc - 4m2 . J 24 2

(6)

Relativistic corrections was preserved up to the second relative order in |p|/s , |q|/s .
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The angular integration


dq i q (q) = - (Q , Lz ) q3 RP (q)d q, 3 (2 ) 6 0 i dq P q q qµ 0 (q) µ (Q , Lz )P + = (2 )3 5 6
P 0

+ (Q , Lz )P µ + (Q , Lz )P P = (g


µ 0

q5 RP (q)d q, (7)

- v v ).

S

Summing over Sz and Lz with Clebsch-Gordon coefficients: 1 (g - v v ), J = 0, 2 2 3 i 1, Lz ; 1, Sz |J , Jz (Q , Lz ) (Q , Sz ) = e v (Q , J ), J = 1, z 2 2 z ,Lz (Q , Jz ), J = 2, (8)
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Effective relativistic Hamiltonian
H = H0 + U1 + U2 + U3 , CF s ~ H 0 = 2 p2 + m 2 - 2m - + Ar + B , r 2 CF s U1 (r ) = - 20 ln( r ) + a1 + 2E 0 , 4 r r(rp)p CF s 3CF s p2 + U2 (r ) = - 2 + (SL)- 2 2m r r 2m 2 r 3 2 CF s S2 (Sr)2 CA CF s - -3 5 - , 2m2 r3 r 2mr 2 U3 (r ) = fV A 2m 2 r 42 22 (Sr)2 S - 1 + 3(SL) - S -3 2 3 3 r A -(1 - fV ) 2 SL, 2m r s (m2 ) 0.242. ~ - (12) (9) (10)

(11)

s (m2 ) 0.314,

(13)
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Effective relativistic Hamiltonian
A = 0.18 GeV 2 , Kinetic energy operator: T =2 p2 + m 2 = 2 ~ m= p2 + m
2 2

B = -0.16 GeV,

m = 1.55 GeV. p2 2m2 , + ~ ~ m E

(14)

p2 + m



(15) (16)

~ E 1 = 2 2

p2 + m2 . eff

Table I: The parameters of the effective relativistic Hamiltonian and masses of P -wave charmonium states. ¯ (c c ) c0 c1 c2 hc n
2S +1

L

J

13

P0 3P 11 13 P2 11 P1

J PC 0++ 1++ 2++ 1+ -

p

eff

2

, GeV 0.54 0.54 0.54 0.54

2

~ m, GeV 0.857 0.857 0.857 0.857

M

exp

, GeV 3.415 3.511 3.556 3.526

M

theor

, GeV 3.418 3.493 3.557 3.499
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The structure of the production amplitudes
M(p- , p+ , P , Q ) =
3

16 s (4m2 )Q r 45
6 M0 2
5 2

6
5 2

u (1 - u )
3

¯ v (p+ ) u (p- )â (q) + m (q) d q, 2 (q) (17)

â
0

(p) + m p Rhc (p) dp 2 (p)
0

K (p, q) q R



cJ

K (hc + c0 ) = A1 eµ v1 v2 µ , hc

K (hc + c1 ) = B1 v1 +B4 c h

c1



c1



hc

+ B2 v2

c1

h

c





c1



+ B3 v1



c1

v2 v2 ,

h

c

v1 +

v1 + B5 v1

v2

hc

v2 + B6 c h +



c1

K (hc + c2 ) = C1 eµ + C2 +C5

c2

µ c2 v1 v2 hc

(18) + +



+ C3

c2 v1 v1

+ C4
hc

c2 v2 v1 h
c

e

µ µ v1 v2 hc

µ c2 v1 v1 eµ v1 v2 hc

+ C6

+ C7 v2 .

v1



µ c2 eµ v1 v2

+C8

µ c2 v1 eµ v1 hc

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The structure of the production amplitudes
Ai = Ai (p, q; , u ; r ), Bi = Bi (p, q; , u ; r ), = McJ m , u= , M0 M0 M0 = Mhc + McJ , r2 = Ci = Ci (p, q; , u ; r )

(19)

2 M0 (20) . s Dependence on p, q can be described in terms of the following functions:

cij (p, q) =

m - (p) i m - (q) m + (p) m + (q) i = 0 . . . 2, j = 0 . . . 2.

j

,

(21)

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The structure of the cross sections
2 22 s (4m2 )Q2 r 6 1 - r 2 1 - r 2 (2u - 1)2 c (hc + cJ ) = â 94 u 11 (1 - u )11 ~ ~ |Rhc (0)|2 |Rc (0)|2 7 (J ) J Fi (r 2 )i , â s (McJ + Mhc )10
i =0

(22)

Jn =
0

q 3 RP (q )

(q ) + m 2 (q )

m - (q ) m + (q ) 2 J0 , J2 (hc ) , J0 (hc )
2 4

n

dq , (23)

1 ~ RP (0) = 3 0 = 1, 1 = J1 (hc ) , J0 (hc ) J2 (cJ 5 = J0 (cJ 2 = ) , )

2 3 = 1 ,

J1 (cJ ) 4 = , J0 (cJ )

(24) 7 = 1 4 .
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6 =

,


Numerical results

Table I I: Numerical values of the charmonium wave functions integrals ¯ (c c ) c0 c1 c2 hc n
2S +1

L

J

13 13 13 11

P0 P1 P2 P1

J PC 0++ 1++ 2++ 1+ -

~ RP (0), GeV 0.33 0.20 0.13 0.17

5 2

1 or 4 -0.28 -0.18 -0.08 -0.14

2 or 5 0.13 0.07 0.01 0.04

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Numerical results
Table I I I: Comparison of the obtained results with previous theoretical predictions and exp State Belle â B aB ar â [3], [4], [5], LO [6], H1 H2 âB>2 [1], fb âB>2 [2], fb fb fb fb fb 17.9 +1.4 J / + c0 6.4 ± 1.7 ± 1.0 10.3 ± 2.5-1.8 6.7 14.4 2.4 ± 1.02 (6.35) J / + c1 1.1 0.38 ± 0.12 J / + c2 1.6 0.69 ± 0.13 c + hc 1.6 0.308 ± 0.017 hc + c0 0.053 ± 0.019 hc + c1 0.258 ± 0.064 hc + c2 0.017 ± 0.002 erimental data Our result (22), fb 14.47 ± 5.64 1.78 0.44 0.25 0.075 0.132 0.004 ± ± ± ± ± ± 0 0 0 0 0 0 .69 .17 .10 .029 .051 .002

Table IV: The role of the relativistic corrections (r.c.) to (fb) State H1 H2 hc + c hc + c hc + c w/o r.c.
0 1 2

r.c. to ampl. only 0.10 0.83 0.11

r.c. to w.f. only 0.097 0.15 0.0039

r.c. to w.f. and ampl. 0.13 0.37 0.015

0.14 0.60 0.035

r.c. to w.f. and ampl. 1 u = 2, = 1 4 0.075 0.13 0.0036

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Conclusion
We presented a systematic treatment of relativistic effects in the P -wave double charmonium production in e + e - annihilation. We separated two different types of relativistic contributions to the production cross sections: v /c corrections to the wave functions and p / s corrections appearing from the expansion of the quark and gluon propagators in amplitude. Relativistic corrections to the quark bound state wave functions in the rest frame was considered by means of the Breit-like potential. Mass spectrum of P -wave charmonium was obtained. It turns out that the examined effects change essentially the nonrelativistic results of the cross section for the reaction e + e - hc + cJ at the center-of-mass energy s = 10.6 GeV, e.g. relativistic corrections to the amplitude increase (hc + c1 ) and (hc + c2 ) in 1.38 and 3.14 times respectively. Nevertheless, all types of corrections taken together significantly decrease double P -wave production cross sections.
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