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A QCD analysis of and

Where is the pion in the proton ?

B. Pire, L. Szymanowski

[0.5cm]
CPhT, École Polytechnique, F-91128 Palaiseau, France^{1}^{1}1
Unité mixte C7644 du CNRS.

[0.2cm] Sołtan Institute for Nuclear Studies, Hoża 69,

00-681 Warsaw, Poland

[0.2cm] Université de Liège, B4000 Liège, Belgium

[1.0cm]

We study the scaling regime of nucleon -
anti-nucleon annihilation into a deeply virtual photon and a photon or meson,
, , in the forward direction.
The leading twist amplitude factorizes into
an antiproton distribution amplitude,
a short-distance matrix element and a long-distance dominated
transition distribution amplitude (TDA) which describes the nucleon to
meson or photon transition.
The impact representation of this TDA maps out the transverse locations of
the small size core and the meson or photon cloud inside the proton.

## 1 A new factorization

The understanding of the hadronic structure needs appropriate tools to be manufactured[1]. It recently appeared that a fruitful approach could be accessed through exclusive hard quasi forward scattering, the prototype reaction being deep virtual Compton scattering in the forward region. We have generalized [2, 3] this analysis to the reactions

which will be accessible at future intense antiproton facilities[4]. Our arguments for the factorization of the short distance hard subprocess from the usual distribution amplitude and a new transition distribution amplitude, defined below, are a succession of logical steps generalizing the factorization proof [5] of deep exclusive meson electroproduction on a meson in the forward direction, to its time reversed [6] , to meson-meson annihilation with the meson-photon analogy proven by the studies of the photon structure functions. The ultimate generalization from the meson case to the baryon case, implying three quark exchanges is advocated to be safe on the basis of the QCD analysis of baryon form factors.

We thus propose to write the amplitude as

(1) |

where is the antiproton distribution amplitude, the hard scattering amplitude, calculated in the colinear approximation and the new TDAs.

## 2 Transition Distribution Amplitudes

To define the transition distribution amplitudes from a nucleon to a pseudoscalar meson, we introduce light-cone coordinates and transverse components for any four-vector . The skewedness variable with and describes the loss of plus-momentum of the incident hadron in the proton meson transition. We parametrize the quark momenta as shown on Fig. 1. The fractions of + momenta are labelled , and , and their supports are within . Momentum conservation implies : The fields with positive momentum fractions, , describe creation of quarks, whereas those with negative momentum fractions, , the absorption of antiquarks. The eight leading twist TDAs for the (which can be expressed in terms of eight independent helicity amplitudes for transition) then reads :

(2) | |||

where , is the charge conjugation matrix and the nucleon spinor. is the pion decay constant ( MeV) and is the constant which determines the value of the nucleon wave function at the origin, and which has been estimated through QCD sum rules to be of order GeV . Each TDA is then Fourier transformed to get the usual representation in terms of the momentum fractions, through the relation

(3) |

where stands for and . The first three terms in (2) are the only ones surviving the forward limit . The constants in front of these three terms have been chosen in reference to the soft pion () limit results :

(4) | |||

where is the standard leading twist DA.

## 3 Impact Parameter Picture

As in the case of generalized parton distributions [7] and distribution amplitudes [8] the simultaneous presence of two transverse scales and , allows through a Fourier transform to map the impact parameter dependence of the scattering amplitude. In the case under study, the dependence of the transition distribution amplitude allows in its ERBL region (namely, when all ) a transverse scan of the location of the small sized (of the order of ) hard core made of three quarks when a pion carries the rest of the momentum of the nucleon. This may be phrased alternatively as detecting the transverse mean position of a pion inside the proton, when the proton state is of the ”next to leading Fock ” order, namely . This is shown on Fig. 2. The other regions have slightly different interpretations.

The study of different TDAs such as the ones for and , related to the and reactions, may shed light on the versus components of the proton.

## 4 Conclusion

The formalism developed for the proton antiproton exclusive annihilation may as well be used for related channels such as backward virtual Compton scattering or backward electroproduction of a meson, where data exist for moderate values of . These spacelike analogs of the processes discussed here share the same virtues and their studies should allow a first look at the internal structure of the states inside the nucleon. Mesonic channels may also be studied as , or in the near forward region. The TDAs are then not much different from the mesonic GPDs.

Work of L.Sz. is supported by the Polish Grant 1 P03B 028 28. He is a Visiting Fellow of the FNRS (Belgium).

## References

- [1] B. Pire, Annales Henri Poincare 4 (2003) S243 [arXiv:hep-ph/0211093].
- [2] B. Pire and L. Szymanowski, Phys. Lett. B 622 (2005) 83 [arXiv:hep-ph/0504255].
- [3] B. Pire and L. Szymanowski, Phys. Rev. D 71 (2005) 111501 [arXiv:hep-ph/0411387].
- [4] “An Int. Accelerator Facility for Beams of Ions and Antiprotons”, GSI Conceptual Design Report, Nov. 2001. V. Barone et al. [PAX Collaboration], arXiv:hep-ex/0505054.
- [5] J. C. Collins et al., Phys. Rev. D 56, 2982 (1997).
- [6] E. R. Berger, M. Diehl and B. Pire, Eur. Phys. J. C 23 (2002) 675 and Phys. Lett. B 523, 265 (2001).
- [7] M. Burkardt, Phys. Rev. D 62 (2000) 071503 [Erratum-ibid. D 66 (2002) 119903]; J. P. Ralston and B. Pire, Phys. Rev. D 66 (2002) 111501; M. Diehl, Eur. Phys. J. C 25 (2002) 223 [Erratum-ibid. C 31 (2003) 277].
- [8] B. Pire and L. Szymanowski, Phys. Lett. B 556 (2003) 129 [arXiv:hep-ph/0212296].