К ВОПРОСУ О СИНГУЛЯРНОСТИ ЗАРЯДОВОЙ ПЛОТНОСТИ В

¨¸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’…’“›
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μ²ÊÎ¥´μ 21 ´μÖ¡·Ö 2012 £.