¨¸Ó³ ¢ —Ÿ. 2013. ’. 10, º 7(184). ‘. 998Ä1001 ”ˆ‡ˆŠ ‹…Œ…’›• —‘’ˆ– ˆ ’Œƒ Ÿ„. ’…ˆŸ Š ‚‘“ ‘ˆƒ“‹Ÿ‘’ˆ ‡Ÿ„‚‰ ‹’‘’ˆ ‚ –…’… π-Œ…‡ . . ‚ ²² ,1 , ˆ. . ¥·¥¢ ²μ¢ ,2 , Œ. ‚. μ²Ö±μ¢ ¡,3 , . Š. ‘μ±μ²Ó´¨±μ¢ ,4 ¡ ”ƒ“ ‚ ˆƒ“, ˆ·±Êɸ±, μ¸¸¨Ö ˆ´¸É¨ÉÊÉ É¥μ·¥É¨Î¥¸±μ° ˨§¨±¨ II, Ê·¸±¨° Ê´¨¢¥·¸¨É¥É μÌʳ , μÌʳ, ƒ¥·³ ´¨Ö ¡μ¡Ð ¥É¸Ö ¨ ´ ²¨§¨·Ê¥É¸Ö ¢Ò· ¦¥´¨¥ ¤²Ö ±¢ ·±μ¢μ° ¶²μÉ´μ¸É¨ π-³¥§μ´ ¸ ÊÎ¥Éμ³ ±¢ ´Éμ¢μ° ¶·¨·μ¤Ò ¶·¨Í¥²Ó´μ£μ ¶ · ³¥É· ¢ μ¡² ¸É¨ ³ ²ÒÌ Ë §μ¢ÒÌ μ¡Ñ¥³μ¢. ‘²¥¤¸É¢¨¥³ ÔÉμ£μ Ö¢²Ö¥É¸Ö μɸÊɸɸɢ¨¥ ´¥Ë¨§¨Î¥¸±μ° ¸¨´£Ê²Ö·´μ¸É¨ ¢ Í¥´É· ²Ó´μ° μ¡² ¸É¨ ¶¨-³¥§μ´ . We generalize and analyze the expression for the quark density of the pion with an allowance for quantum nature of the impact parameter in the small phase space volume region. As a result, there is no singularity in the center of the pion. PACS: 11.55.-m; 03.65.Nk; 11.30.-j ‚ · ¡μÉ Ì [1, 2] ¶μ¤·μ¡´μ · ¸¸³μÉ·¥´ ¢μ¶·μ¸ μ ¶μ¢¥¤¥´¨¨ § ·Ö¤μ¢μ° ¶²μÉ´μ¸É¨ ¢ Í¥´É·¥ ¶¨μ´ . ·¨ ÔÉμ³ § ·Ö¤μ¢ Ö ¶²μÉ´μ¸ÉÓ μ¶·¥¤¥²Ö¥É¸Ö ± ± ¤¢Ê³¥·´μ¥ ¶·¥μ¡· §μ¢ ´¨¥ ”Ê·Ó¥ Ô²¥±É·μ³ £´¨É´μ£μ Ëμ·³Ë ±Éμ· Fπ ¶μ ¶·¨Í¥²Ó´μ³Ê ¶ · ³¥É·Ê b: dq⊥ Fπ (Q2 = q2⊥ ) e−iq⊥ b . (1) ρ(b) = (2π)2 ‘ ¤·Ê£μ° ¸Éμ·μ´Ò, ÔÉ ¶²μÉ´μ¸ÉÓ ¶μ²ÊÎ ¥É¸Ö ± ± ·¥§Ê²ÓÉ É ¶·¥μ¡· §μ¢ ´¨Ö ¤μ´ ¢¥·μÖÉ´μ¸É´μ£μ · ¸¶·¥¤¥²¥´¨Ö ±¢ ·±μ¢ ¢ ¤·μ´¥ (ËÊ´±Í¨¨ ‚¨£´¥· ) [3, 4]: 1 (2) ρ(b2 ) = ρ̆(b2 ) = W (x, q)δ (2) b − 2 [q · [x · q]] dx dq, q £¤¥ W (x, q) Å ËÊ´±Í¨Ö ‚¨£´¥· . ‡¤¥¸Ó ρ̆(b2 ) μ§´ Î ¥É ¶·¥μ¡· §μ¢ ´¨¥ ¤μ´ , ¢ ¸μμÉ¢¥É¸É¢¨¨ ¸ μ¡μ§´ Î¥´¨¥³, ¶·¨´ÖÉÒ³ ¢ · ¡μÉ¥ [3]. 1 E-mail: anvall@mail.ru IrenAdler1@rambler.ru 3 E-mail: maxim.polyakov@tp2.ruhr-uni-bochum.de 4 E-mail: sokolnikova.aleksandra@gmail.com 2 E-mail: Š ¢μ¶·μ¸Ê μ ¸¨´£Ê²Ö·´μ¸É¨ § ·Ö¤μ¢μ° ¶²μÉ´μ¸É¨ ¢ Í¥´É·¥ π-³¥§μ´ 999 ‘μ£² ¸μ¢ ´¨¥ ¸μμÉ´μÏ¥´¨° (1), (2) Å ÔÉμ ¸μ£² ¸μ¢ ´¨¥ Ô±¸¶¥·¨³¥´É ²Ó´ÒÌ ¤ ´´ÒÌ ¶μ Ô²¥±É·μ³ £´¨É´μ³Ê Ëμ·³Ë ±Éμ·Ê Fπ , ¸ μ¤´μ° ¸Éμ·μ´Ò, ¨ ³μ¤¥²Ö³¨ ¶ ·Éμ´´ÒÌ · ¸¶·¥¤¥²¥´¨° ¨§ ¤ ´´ÒÌ ¶μ £²Ê¡μ±μ´¥Ê¶·Ê£μ³Ê · ¸¸¥Ö´¨Õ ´ π-³¥§μ´¥ Å ¸ ¤·Ê£μ°. ŒÒ ¶·¥¤² £ ¥³ ¥¸É¥¸É¢¥´´μ¥ μ¡μ¡Ð¥´¨¥ ¶·¥¤¸É ¢²¥´¨Ö (1) ¢ ¢¨¤¥ · §²μ¦¥´¨Ö ¶μ ¶²μ¸±¨³ ¢μ²´ ³ ´ £·Ê¶¶¥ SO(2, 1) (ËÊ´±Í¨¨ ˜ ¶¨·μ ξ(q⊥ , μ)): 1 (3) ξ̄(q⊥ , μ)Fπ (q⊥ ) dΩq , ρ(μ) = (2π)2 1 £¤¥ dΩq = dq⊥ , dq⊥ = q⊥ dq⊥ dϕ. 2 2 q q − q⊥ ¸´μ¢ ´¨¥³ ± ÔÉμ³Ê ¸²Ê¦¨É Éμ, ÎÉμ ¢ ÔÉμ³ ¸²ÊÎ ¥ ¨´É¥£·¨·μ¢ ´¨¥ ¢ ¨³¶Ê²Ó¸´μ³ ¶·μ¸É· ´¸É¢¥ ¨¤¥É ¶μ ˨§¨Î¥¸±μ° μ¡² ¸É¨ ¨§³¥´¥´¨° q⊥ , É ±¦¥ ÉμÉ Ë ±É, ÎÉμ ´ ¶μ¢¥·Ì´μ¸É¨, § ¤ ¢ ¥³μ° ·£Ê³¥´Éμ³ δ-ËÊ´±Í¨¨ ¢ ¸μμÉ´μÏ¥´¨¨ (2), ËÊ´±Í¨¨ ˜ ¶¨·μ μ¡· §ÊÕÉ ¶μ²´ÊÕ μ·Éμ´μ·³¨·μ¢ ´´ÊÕ ¸¨¸É¥³Ê [5]. ‚ μ¡² ¸É¨ ¡μ²ÓÏ¨Ì ¶·¨Í¥²Ó´ÒÌ ¶ · ³¥É·μ¢ b ¶·¥¤¸É ¢²¥´¨Ö (1) ¨ (3) ¸μ¢¶ ¤ ÕÉ. ‘²¥¤¸É¢¨¥³ ±¢ ´Éμ¢ ´¨Ö ±μ³¶μ´¥´É ¶·¨Í¥²Ó´μ£μ ¶ · ³¥É· bi Ö¢²Ö¥É¸Ö ¶μÖ¢²¥´¨¥ ³¨´¨³ ²Ó´μ£μ ¢μ§³μ¦´μ£μ §´ Î¥´¨Ö ¶ · ³¥É· b2 = b2min = 2 /4q 2 , ÎÉμ ¸μμÉ¢¥É¸É¢Ê¥É §´ Î¥´¨Õ μ = 0. ¸¸³μÉ·¨³ · §²μ¦¥´¨¥ ¶²μÉ´μ¸É¨ ρ(μ) ¢ μ±·¥¸É´μ¸É¨ μ 0. ‚ ¸μμÉ´μÏ¥´¨¨ (3) ¢μ§Ó³¥³ ¨´É¥£· ² ¶μ ¶μ²Ö·´μ³Ê Ê£²Ê ϕ, ÊΨÉÒ¢ Ö, ÎÉμ Ëμ·³Ë ±Éμ· Fπ μÉ ´¥£μ ´¥ § ¢¨¸¨É. ’죤 ¶μ²ÊΨ³ 1 q q⊥ dq⊥ ρ(μ) = (4) Fπ (q⊥ ) . 2 P−1/2+iμ 2 2π q 2 − q⊥ q 2 − q⊥ „²Ö Ëμ·³Ë ±Éμ· Fπ ¢μ§Ó³¥³ ³μ¤¥²Ó, Ìμ·μÏμ ¸μ£² ¸ÊÕÐÊÕ¸Ö ¸ Ô±¸¶¥·¨³¥´É ²Ó´Ò³¨ ¤ ´´Ò³¨ [6]: 2 Fπ (Q2 = q⊥ )= 1 1 2 . 2 2 + 2 q⊥ Rm Rd2 q⊥ 1+ 1+ 62 122 (5) ”¨É¨·μ¢ ´¨¥ Ô±¸¶¥·¨³¥´É ²Ó´ÒÌ ¤ ´´ÒÌ ¢ ¨´É¥·¢ ²¥ Q2 μÉ 0,60 ¤μ 2,45 ƒÔ‚2 ¶·¨¢μ¤¨É ± §´ Î¥´¨Ö³ 2 = 0,431 ˳2 , Rd2 = 0,411 ˳2 . Rm ¸¸³μÉ·¨³ μɤ¥²Ó´μ ³μ´μ¶μ²Ó´μ¥ ¸² £ ¥³μ¥ Ëμ·³Ë ±Éμ· : Fπ 2 mon (q⊥ ) = 1 62 , = 2 q 2 (1 − 1/u2 ) 62 + Rm R2 q 2 0 1 + m 2⊥ 6 2 , q2 = q2 + q2 . £¤¥ u0 = q/ q 2 − q⊥ 3 ⊥ ’죤 ¤²Ö ¶²μÉ´μ¸É¨ · ¸¶·¥¤¥²¥´¨Ö ρmon (μ) ¶μ²ÊΨ³ [7, 8]: 62 1 Rm q Rm q π − + P , ρmon (μ) = P −1/2+iμ −1/2+iμ 2 ch πμ 2π 2ζm ζm ζm (6) 1000 ‚ ²² . . ¨ ¤·. 2 2 2 £¤¥ ¢¢¥¤¥´μ μ¡μ§´ Î¥´¨¥ ζm = 62 + Rm q , ¶·¨Î¥³ −1 < Rm q < 1. ζm ɸդ ¶·¨ μ = 0 ¶μ²ÊΨ³ 32 1 1 Rm q Rm q ρm (0) = 2 K +K , 1+ 1− πζm 2 ζm 2 ζm (7) £¤¥ K(z) Å ¶μ²´Ò° Ô²²¨¶É¨Î¥¸±¨° ¨´É¥£· ² ¶¥·¢μ£μ ·μ¤ . ’¥¶¥·Ó · ¸¸³μÉ·¨³ ¤¨¶μ²Ó´μ¥ ¸² £ ¥³μ¥ ¢ Ëμ·³Ë ±Éμ·¥ (5): Fπ 2 dip (q⊥ ) 1 (122 )2 = 2 = 2 . 2 2 (122 + Rd2 q 2 (1 − 1/u20 )) Rd q⊥ 1+ 122 ‚ ÔÉμ³ ¸²ÊÎ ¥ ¶²μÉ´μ¸ÉÓ · ¸¶·¥¤¥²¥´¨Ö ρdip (μ): (122 )2 ρdip (μ) = 2π ∞ P−1/2+iμ (u0 ) 1 ζd2 2 u30 du0 2, (ζd2 u20 − Rd2 q 2 ) (8) Rd2 q 2 . = 12 + ´ ²μ£¨Î´Ò¥ ¶·¥¤Ò¤ÊШ³ ¢ÒΨ¸²¥´¨Ö ¶·¨¢μ¤ÖÉ ¤²Ö ρdip (μ) ± ¸²¥¤ÊÕÐ¥³Ê ¢Ò· ¦¥´¨Õ: Rd2 q 2 (1/2 + iμ) 364 Rd q Rd q 1+ ρdip (μ) = 4 + P− 12 +iμ + P− 12 +iμ − ζd ch (πμ) 242 ζd ζd Rd q Rd q 32 Rd q(1/2 + iμ) − P 12 +iμ . (9) P 12 +iμ − + 2ζd3 ch (πμ) ζd ζd £¤¥ ·¨´¨³ Ö ¢μ ¢´¨³ ´¨¥, ÎÉμ 1−x 1 3 P 12 (x) = F − , ; 1; , 2 2 2 μ±μ´Î É¥²Ó´μ ¤²Ö ¶²μÉ´μ¸É¨ ρ(0) ¶μ²ÊΨ³ ρ(0) = ρmon (0) + ρdip (0) = 32 Rm q Rm q 1 1 +K + = K 1+ 1− 2 πζm 2 ζm 2 ζm 1 1 724 Rd2 q 2 Rd q Rd q + 1+ K +K + 1+ 1− πζd4 482 2 ζd 2 ζd 1 Rd q 1 Rd q 32 Rd q 1 3 1 3 + F − , ; 1; + − F − , ; 1; − . (10) 4ζd3 2 2 2 2ζd 2 2 2 2ζd Š ¢μ¶·μ¸Ê μ ¸¨´£Ê²Ö·´μ¸É¨ § ·Ö¤μ¢μ° ¶²μÉ´μ¸É¨ ¢ Í¥´É·¥ π-³¥§μ´ 1001 ‡Š‹—…ˆ… ’ ±¨³ μ¡· §μ³, ±¢ ·±μ¢ Ö ¶²μÉ´μ¸ÉÓ π-³¥§μ´ , § ¤ ¢ ¥³ Ö ¸μμÉ´μÏ¥´¨¥³ (3), ´¥ ¨³¥¥É ¸¨´£Ê²Ö·´μ¸É¨ ¢ ˨§¨Î¥¸±μ° μ¡ ¸É¨ ¨§³¥´¥´¨Ö ¶·¨Í¥²Ó´μ£μ ¶ · ³¥É· 2 /4q 2 b2 < ∞. É μ¡² ¸ÉÓ ¸²¥¤Ê¥É ¨§ ±¢ ´Éμ¢μ-³¥Ì ´¨Î¥¸±μ£μ μ¡μ¡Ð¥´¨Ö ¶·¨Í¥²Ó´μ£μ ¶ · ³¥É· [5]. ‚Ò· ¦ ¥³ £²Ê¡μ±ÊÕ ¡² £μ¤ ·´μ¸ÉÓ ¤μ±Éμ·Ê ˨§.-³ É. ´ ʱ ²¥±¸ ´¤·Ê ‹¥μ´¨¤μ¢¨ÎÊ ² ´¤¨´Ê ¨ ¤μ±Éμ·Ê ˨§.-³ É. ´ ʱ ·¨Õ ¤μ²ÓËμ¢¨ÎÊ Œ ·±μ¢Ê § μ¡¸Ê¦¤¥´¨¥ · ¡μÉÒ ¨ ±·¨É¨Î¥¸±¨¥ § ³¥Î ´¨Ö. ¡μÉ ¢Ò¶μ²´¥´ ¢ · ³± Ì ·μ£· ³³Ò ¸É· É¥£¨Î¥¸±μ£μ · §¢¨É¨Ö ˆƒ“ ´ 2012Ä 2016 ££., ¶·¨ ˨´ ´¸μ¢μ° ¶μ¤¤¥·¦±¥ ”– ® Êδҥ ¨ ´ Êδμ-¶¥¤ £μ£¨Î¥¸±¨¥ ± ¤·Ò ¨´´μ¢ Í¨μ´´μ° μ¸¸¨¨¯ ´ 2009Ä2013 ££., ‘μ£² Ï¥´¨¥ º 14.B37.21.0910, ƒŠ º 16.740.11.0154, º 1197. ‘ˆ‘Š ‹ˆ’…’“› 1. Miller G. A. Singular Charge Density at the Center of the Pion? // Phys. Rev. C. 2009. V. 79. P. 055204; arXiv:0901.1117v1 [nucl-th]. 2. Hoyer P. Measuring Transverse Size with Virtual Photons // Nuovo Cim. C. 2012. V. 035N2. P. 277; arXiv:1110.3393v1 [hep-ph]. 3. ƒ¥²ÓË ´¤ ˆ. Œ., ƒ· ¥¢ Œ. ˆ., ‚¨²¥´±¨´ . Ÿ. ·¥μ¡· §μ¢ ´¨¥ ¤μ´ μ¸´μ¢´ÒÌ ¨ μ¡μ¡Ð¥´´ÒÌ ËÊ´±Í¨° ¢ ¢¥Ð¥¸É¢¥´´μ³ ˨´´μ³ ¶·μ¸É· ´¸É¢¥ // ƒ¥²ÓË ´¤ ˆ. Œ., ƒ· ¥¢ Œ. ˆ., ‚¨²¥´±¨´ . Ÿ. ˆ´É¥£· ²Ó´ Ö £¥μ³¥É·¨Ö ¨ ¸¢Ö§ ´´Ò¥ ¸ ´¥° ¢μ¶·μ¸Ò É¥μ·¨¨ ¶·¥¤¸É ¢²¥´¨°. Œ.: ƒμ¸. ¨§¤-¢μ ˨§.-³ É. ²¨É., 1962. C. 15Ä110. 4. Belitsky A. V., Radyushkin A. V. Unraveling Hadron Structure with Generalized Parton Distributions // Phys. Rep. 2005. V. 418. P. 1Ä387. 5. Vall A. N. et al. Spatial Description of the Particle Production Region in Elastic and Quasi-Elastic Processes on the SO(2.1) Group // Phys. Part. Nucl. 2009. V. 40, No. 7. P. 1030Ä1058. 6. Huber G. M. et al. Charged Pion Form Factor between Q2 = 0.60 and 2.45 GeV2 . II. Determination of, and Results for, the Pion Form Factor // Phys. Rev. C. 2008. V. 78. P. 045203; arXiv:0809.3052v1 [nucl-ex]. 7. ¥°É³ ´ ƒ., ·¤¥°¨ . ‚ҸϨ¥ É· ´¸Í¥´¤¥´É´Ò¥ ËÊ´±Í¨¨. Œ.: ʱ , 1965. 296 ¸. 8. ƒ· ¤ÏÉ¥°´ ˆ. ‘., Ò¦¨± ˆ. Œ. ’ ¡²¨ÍÒ ¨´É¥£· ²μ¢, ¸Ê³³, ·Ö¤μ¢ ¨ ¶·μ¨§¢¥¤¥´¨°. Œ.: ”¨§³ É£¨§, 1962. 1100 ¸. μ²ÊÎ¥´μ 21 ´μÖ¡·Ö 2012 £.