Introduction to Regge theory and soft Pomeron

1
Introduction to Regge theory and soft Pomeron

• K. Itakura (KEK)
• 1st NIC (New Interaction Code) meeting
• May 31th _at_ ICRR, Kashiwa

2
????Pomeron??????????

• R.Engel, Very High Energy Cosmic Rays and Their
  Interactions
• Nucl. Phys. B(Proc. Suppl.) 151 (2006)
  437.??3?????????????
• ????
• An analysis of the same data set with different
  hadronic interaction models can lead to a wide
  range of different results. () The largest
  uncertainty in EAS simulation stems from the
  unknown characteristics of hadronic multiparticle
  production.
• Three energy regions
• 1. At very low energy (lt a few GeV)
  production and decay of resonances
• 2. Intermediate energy (lt103 GeV)
  fragmentation of two color strings
• 3. High energy (gt103 GeV) minijet
  production and multiple parton-
• parton interaction becomes important
• Central assumptions that characterize a models
  high energy extrapolation
• 1. Size and energy dependence of the QCD
  minijet cross section
• 2. Distribution of partons in transverse
  space (profile function)
• 3. Scaling of leading particle
  distributions or scaling violation
• 4. Treatment of nuclear effects
  (semi-superposition model, Gribov-Glauber
• approximation, increased parton
  shadowing, etc)

3
????Pomeron??????????

• ???????QGSJET vs SIBYLL???2?????????????
• 2. Parton distributions
• QGSJET01 pre-HERA parton distribution
• SIBYLL 2.1 post-HERA parton distribution
• 1. Minijet transverse momentum cutoff
  (???pQCD???)
• QGSJET01 energy independent constant cutoff
  1.5 GeV
• SIBYLL energy-dependent cutoff
• 1.5GeV at low energy, 8
  GeV at 1020 GeV
• gluon saturation
  ???????????
• (cutoff???????????????
  ??????)
• --------------------------------------------------
  --------------------------------------------------
  -
• QGSJET II QGSJET? post-HERA data
  ?????????????
• ?????cutoff ?energy independent??????
• ?????????nonperturbative???????
• soft Pomeron?????????????????
• BBL SIBYLL2.1???saturation???(Black-dis
  k limit)????????
• ? CGC(nonlinear hard Pomeron)??? (soft
  ?????)

dN/dkt2
1/kt2
kt2
4
????????

• ???????? Regge??
• ?????????? ??????????????????

??????????
?????????????
a
c
s(papb)2
s
????????
t(pa - pc)2
d
b
s gtgt t
t

• W2 (pq)2 g-p????????????
• Q2 -q2 ??????
• W2 gtgt Q2
• ???? x Q2 /(W2 Q2 )? 0

?????????
Cf) Bjorken limit xQ2/2pq ????? Q2 ?8,
2pm qm W2 Q2 ?8
5
???????

• ????????????????µ??QCD???
• ?QCD QCD???????????????? 200-400MeV
• ?) 1 loop
• ???(???) µ ?QCD
• ???(????) µ lt ?QCD

• ?)
• ??????(?????? ?????? t0 ???)
• DIS???????????????
• Q2 ?? ????????(????????????????)
• Q2 ? ? ??(or vector meson)???????????????
• ???????????????????????????????????????

6
?????

• QCD??
• 1943 Heisenberg S????????
• 1958 Mandelstam ????S?????(Mandelstam?????)
• 1959 Regge Regge????(????)
• 1961 Chew-Frautschi ?????Regge?????? ? soft
  Pomeron
• ? ????dual resonance model, Veneziano
  amp, string theory
• QCD??
• 1970?? QCD???
• 1976-78 BFKL QCD???????????(LO-BFKL???)???
• ? hard Pomeron (?
  1993????HERA???)
• 2000 NLO-BFKL??? ? ??????????
• 1983 GLR(Gribov-Levin-Ryskin)???saturation???.
  BFKL???. CGC???
• 1986 Mueller-Qiu DLA(DGLAP?small-x
  limit)?????????
• 1994 McLerran-Venugopalan?? ?????????????
• 2000 Iancu, McLerran, etc. GLR?
  MV???????????????????
• ? JIMWLK???, BK???(LO) ????????
• ? Color
  Glass Condensate (2001 Geometric scaling at HERA)

? ?
? ? ?
7
Regge??

• ????????????S??????????
• S??????????????????????????(??????????)???????
• QCD???????????QCD?????????
• ??????????????????????????????????????????????

Kinematics (2???)
Mandelstam?? ? Lorentz??
??????2???
s(p1p2)2
1
3
t(p1 p3)2
s
4
u(p1 p4)2
2
t
???????? mi
????? s ? t ???????? A(s,t)
8
S???????

• ??
• (?)Outstate b,outgt? instate?????????
  ltb,outa,ingtltb,in S a,ingt????? S ???????
• S??????3????
• I. S???Lorentz???????? 2????? S(s, t)
• II. Cutkosky? ? ????
• III. S???simple pole cut ?????????

S???????????? a,ingt ?? ?? b,outgt????????
Sba ltb, outa,
ingt ?????????????a,ingt (b,outgt)? t?-infinity
(t?infinity)?? ???????????????????????

1. S???Lorentz?????
2. S??????????? S S S S 1 (?????)
3. S?????????Lorentz????????????
   ???????????????????

9
??II???

• ???? A(s,t)
• Cutkosky rule (SS1??)
• Optical theorem (Cutkosky rule? ab???)

????????????? ???? c ?????????
????????????? (c) ?????????

2 Im

.
a ? b
???? t 0
?? a ???????????? ? ????
p1?COM??????????
10
??III???

• Cutkosky?????????n????????
• 1????
• 2????
• n????

S

2 Im

a ? b
??m????? ? simple pole
s gt (2m)2 ??????? s 4m2 ?branch point???Cut???
s gt (nm)2 ???????s (nm)2 ?branch point???Cut???
s
?????
9m2
m2
4m2
???????????
11
????????????

• ?????
• ???????? (Sommerfeld-Watson??)

??????????????? f(q ) ?????????????
????? Legendre?? (k ????????)
?????(t channel ???????? s ? t ???? q ??????)
l
??a(l,t)?????????????????????????????????? ?
signature h,- ???
??????()/?(-)??????????
12
Regge?

• ??????

l
?????? Re l gt 0 ???pole??????
Regge pole
??????signature???
Regge pole ??????????????????????? s ???? Pa(
zt ) ? zt12s/t ???????? Regge limit s/t ?
infinity ??????????????? Regge pole
????????? Re a ????????? ????t-channel
?????? spin a ???????????
Reggeon
13
Regge??

• Regge??

t channel ??????????Reggeon???????? ????????????
??? J ?? M ??????????? a(t)
??????????????
??????????pole ??? t ???? pole???????????a ? t
?????? ?????????
???????? J ??? M ???????
Regge trajectory a(0) 0.55 lt 1
? ?? a 0.86 GeV 2 ? ??
14
Pomeron

• ????
• Reggeon????intercept ?1????????????? s ? infty
  ???
• Pomeranchuk??? (1956)
• ??????????????? s ? infty ???????
• Foldy-Peierls (1963)
• s ? infty ??????????????????????????????
• (isospin 0, charge conjugation
  even)?????????????
• ????????????????????????????????
• ? Reggeon ( a(0) lt 1 ) ?????trajectory?????!
• ????Pomeron ???? ( a(0)
  gt 1 )

15
Pomeron vs exp. data (standard picture)

• ????????????

Donnachie-Landshoff, 1992 Pomeron Reggeon
???? aP(0)1.08 gt 1, aR(0)0.55 lt
1 ???????????pp, ppbar? ????? Reggeon?aR(0)0.55?
Regge ?? ??????? pp, p-p, g p?????????
Fit region
aP(0)1.08 ???exchange? soft Pomeron???
pp, ppbar?????????? aP 0.25 GeV-2
16
Pomeron????

• Pomeron trajectory
• ????????????? (Reggeon????????)
• Spin 2 , M1.9GeV ??? f2(1950) ,
  JPC2 ??
• ????????glueball?
• (??????????????????)

17
Beyond single Pomeron exchange

• Froissart??
• 1 Pomeron exchange????????????????
• ??????????????????????????????
• Froissart
  1961, Martin 1966
• 
  (s0????????????????????)
• ????1 Pomeron ?????????????????????
• ??????????????????
• ? ??Pomeron???Pomeron???? ???
• (??1) s0 ????????????????????????????
• ????????????????????s0 1GeV2 ??????
• ???????????
• (??2) ??????? ??????????????????

18
Froissart ????????

• Heisenberg (1952) ?????????????????????????shock
  wave??????????(!)

Total energy
Saturation is implicit
???1/4??????!
19
???????

• ??????????
• ln s, ln2 s (Froissart bound), or sl (l0.08)
  (Pomeron) ???
• ??? PDG ? ln2 s ????????????COMPETE Collab.???
• B??????????
• 
  Fit??cosmic ray pp data of AKENO Flys eye
• ???

S1/2 10 102 103 104
GeV
p/mp2 62 mb
?????B?Froissart?????????????? ?????? log2
behavior ??????????????????? ???CGC?
dipole-proton ?????????COMPETE????????
LO BFKL B 2.09 8.68 mb (aS0.1 0.2)
rNLO BFKL B 0.446 mb (aS0.1 )
20
Pomeron????

• Single diffractive event

???????triple Reggeon vertex?????? ?
?????????????Pomeron???? Pomeron???????????????
? ???1Pomeron??????????? (Reggeon Field
Theory?????????)
21
???a

• ???????????????S???????????????????Pomeron???Regge
  on???????????????????????????????????????????????P
  omeron??????????????????????????????
• 1Pomeron?????????????????????????????????????(Froi
  ssart??)? ??Pomeron???Pomeron?????????????
• ??????????????????????interplay. ?????????forward
  cross section????????????soft Pomeron??????hard,
  semi-hard??????????????

Hard (pQCD)
cross section
Semi-hard (CGC)
Soft (soft Pomeron, string)
energy