Document

高エネルギーでのハドロン全断面積の
普遍的増加とＬＨＣでのｐｐ全断面積の予言
理研仁科センター
猪木慶治
２０１１年３月９日京都大学基研研究会
素粒子物理学の進展２０１１
Phys.Lett.B670(2009)395
Phys.Rev.D79(2009)096003
+α
In collaboration with Muneyuki Ishida
1
Outline of this talk
1. Prediction of σ(pp) at 7TeV, 14TeV (LHC)
2. Test of Universality (Blog2s )
Is B universal between 2 body had. scatt.?
Answer :Yes
3. σ（＋）：parabola as function of logν
4. 高エネルギーのデータを使って B を決める
のが普通、parabola左側の共鳴領域のデータ
をも使うことができ、Ｂの決定の精度大。
Universality
2
5. それをつかって、ＬＨＣにおいて、7TeV,
14TeV での予言。さらに、他のグループで
はできないπｐ、Ｋｐの予言。
6. 最後にＧＺＫエネルギー（Ｅ＝３３５ＴｅＶ）を
含む超高エネルギーでの予言
Auger 観測所との比較が楽しみ。
3
Test of Universality
• Increase of  tot has been shown to be at most log2 s
by Froissart (1961) using Analyticity and Unitarity.
• Soft Pomeron fit : Donnachie-Landshoff
σtot ~ s0.08 but violates unitarity
• COMPETE collab.(PDG) further assumed
 tot  B  log s s0   Z
2
for all hadronic scattering to reduce the number of
adjustable parameters based on the arguments of
CGC(Color Glass Condensate) of QCD.
4
4
Particle Data Group
(by COMPETE collab.)
The upper side：σ
The lower side：ρ－ratio
 log s s0 
σ
2
)
B (Coeff. of
Assumed to be universal
Theory:Colour Glass
Condensate of QCD sug.
Not rigorously proved from QCD
Test of univ. of B:necessary
even empirically.
ρ
5
Ｉncreasing tot.c.s.
• Consider the crossing-even f.s.a.
pp
pp
f


f




 
F   
2

k

with Im F       tot  
4
• We assume

 
Im F       Im R    Im FP'  


  P   
2
c

c
log

c
log
0
1
2




2
M 
M
M  M M 
'
P'
at high energies.
M : proton mass
ν, k : incident proton energy, momentum in the laboratory system
6
F
The 
 
 
     F   
*
ratio
 
• The  ratio = the ratio of the real to
 
imaginary part of F  


  
Re F     
Im F

 

Re R    Re FP'  
Im R    Im FP'  
 
  P   
c  2c2 log



2  1
'
M  M M 

k tot
4
F     0   subtraction const .

2M 
0.5
 F   0 
7
How to predict σ and ρ for pp at LHC based
(as example)
on FESR duality?
• We searched for simultaneous best fit of σ
and ρ up to some energy(e.g.,ISR) in terms
of high-energy parameters constrained by
FESR.
 
 

• We then predicted tot and in the LHC
regions.
8
• Both tot and Re F   data are fitted through two
formulas simultaneously with FESR as a constraint.
• FESR is used as constraint of  P   P  c0 , c1 , c2 
and the fitting is done by three parameters:
 

'
'
c2 , c1 , and c0
giving the least  2 .
   ci  
• Therefore, we can determine all the parameters
c2 , c1 , c0 ,  P' , F
 
0
These predict  ,  at higher energies including LHC energies.
9
LHC
ISR(=2100GeV)
(a)
 tot
All region
LHC
(c)    : High energy region
tot
Predictions for
 and 
The fit is done for data up to ISR
( s  62.7 GeV )
LHC
(b)
  
  2100 GeV (lab)
As shown by arrow.
10
Summary of Pred. for   and   at
LHC
• Predicted values of tot agree with pp
exptl. data at cosmic-ray regions within
errors.
• It is very important to notice that energy
 

range of predicted
is several orders
tot
of magnitude larger than energy region of
the  tot  ,    input.
• Now let us test the universality of B.
11
Main Topic:Universal Rise of σtot ?
• B (coeff. of (log s/s0)2) :
Universal for all hadronic scatterings ?
• Phenomenologically B is taken to be universal in
ー
the fit to πｐ,Ｋｐ, ｐｐ ,ｐｐ,∑ｐ,γｐ, γγ forward scatt.
COMPETE collab. (adopted in Particle Data Group)
• Theoretically Colour Glass Condensate of QCD
suggests the B universality.
Ferreiro,Iancu,Itakura(KEK),McLerran(Head of TH group,RIKEN-BNL)’02
Not rigourously proved yet only from QCD.
 Test of Universality of B is Necessary even
empirically.
12
Particle Data Group
(by COMPETE collab.)
The upper side：σ
The lower side：ρ－ratio
 log s s0 
2
)
B (Coeff. of
Assumed to be universal
Theory:Colour Glass
Condensate of QCD sug.
Not rigorously proved from QCD
Test of univ. of B:necessary
even empirically.
13
• We attempt to obtain B values
for pp( p p), p, Kp scatterings
through search for simultaneous best fit to
experimental  tot and  ratios .
14
New Attempt for  p, Kp
• In near future,  totpp will be measured at LHC
energy. So, Bpp will be determined with
good accuracy.
pp : s  1.8TeV
p

• On the other hand, tot have been measured
only up to k=610 GeV. So, one might doubt
to obtain B for  p (as well as Kp), B p  BKp 
with reasonable accuracy.  p : s  26.4GeV
Kp : s  24.1GeV
• We attack this problem in a new light.
15
Practical Approach for search of B
Tot. cross sec.= Non-Reggeon comp. + Reggeon(P’) comp.
• Non-Reg. comp. shows shape of parabola as a fn. of logν with a min.
• Inf. of low-energy res. gives inf. on P’ term. Subtracting this P’ term
from σtot(+), we can obtain the dash-dotted line(parabola).
Fig.1 pp, pp
σtot (mb)
Fig.1 pp, pp
100
100
• We have good data for large
values of log ν compared with
LHS, so (ISR, SPS, Tevatron)data is most important for det.
50
of c2(pp)(or Bpp) with good
40
accuracy.
Non-Regge
Tevatron
√s = 1.8 TeV
●
SPS
●
●
●
√ s = 0.9 TeV
ISR
50
●
40

comp.(parabola)
1
●
√ s = 60 GeV
16
10
102
103
104
105
106
log ν(GeV)
16
σtot (mb)
60
Fig.2 πp
Fig.2 πp
55
5050
• σtot measured only up to
√ s = 26.4 GeV (cf. with pp, -pp).
4040
• So, estimated Bπp
Resonances
45
↓
s
35
highest energy
√ s = 26.4 GeV
●
3030
Non-Regge 
25
comp(parabola)
kL = 610 GeV
may have large uncertainty.
1
10
102
103
logν(GeV)
• The πp has many res. at low energies, however.
So, inf. on LHS of parabola obtained by subtracting P’ term from σtot(+)
is very helpful to obtain accurate value of B(πp).
(res. with k < 10GeV).
•( Kp : similar to πp )
.
17
Fitting high-energy data
-pp , pp Scattering
σtot
Ecm
Tevatron
σtot = Bｐｐ (log s/s0)2 + Z
(+ ρ trajectory)
in high-energies.
parabola of log s
Ecm=1.8TeV
SPS
ー
ｐｐ
ISR
Forー
pp scatterings
We have data in TeV.
Ecm<0.9TeV
Ecm<63GeV
CDF
D0
Bｐｐ = 0.273(19) mb
estimated accurately.
ｐｐ
Fitted energy region
Depends the data with
the highest energy. (CDF
18
 D0)
πｐ , Ｋｐ Scatterings
No
Data
πｐ
π+ｐ
No
Data
Ｋ-ｐ
Ｋ+ｐ
• No Data in TeV
 Estimated Bπｐ , BＫｐ have large uncertainties.
19
Test of Universality of B
• Highest energy of Experimetnal data:
ー
ｐｐ
: Ecm = 0.9TeV SPS; 1.8TeV Tevatron
π-ｐ： Ecm < 26.4GeV
Ｋｐ： Ecm < 24.1GeV No data in TeV B : large errors.
Bｐｐ = 0.273(19) mb
Bπｐ = 0.411(73) mb  Bｐｐ =? Bπｐ =? BＫｐ ?
BＫｐ = 0.535(190) mb
No definite conclusion
• It is impossible to test of Universality of B only by
using data in high-energy regions.
• We attack this problem using duality constraint from
20
ＦＥＳＲ(1): a kind of P’ sum rule
Kinematics
• ν:
Laboratory energy of the incident particle
s =Ecm2 = 2Mν+M2+m2 ~ 2Mν
M : proton mass of the target. Crossing transf. ν  ー ν
m : mass of the incident particle
m=mπ , mＫ , M for πｐ；Ｋｐ；ｐｐ; pp
k = (ν2 – m2)1/2 : Laboratory momentum ~ ν
• Forward scattering amplitudes
fap(ν): a = ｐ,π+,Ｋ+
Im fap (ν) = (k / 4 π) σtotap : optical theorem
• Crossing relation for forward amplitudes:
f π-ｐ(-ν) = fπ+ｐ(ν)* , fＫ-ｐ(-ν) = fＫ+ｐ(ν)*

pp
pp
f     f  
21
Kinematics
• Crossing-even amplitudes : F(+)(ーν)=F(+)(ν)*


F       f ap    f ap   2
average of π-ｐ, π+ｐ；Ｋ-ｐ, Ｋ+ｐ；ｐｐ,
pp
Im F(+)asymp(ν) = βＰ’ /m (ν/m)α (0)
+(ν/m2)[ c0+c1log ν/m +c2(log ν/m)2]
’
βＰ’ term : P’trajecctory (f2(1275) ): α (0) ~ 0.5
c0,c1,c2 terms : corresponds to Z + B (log s/s0)2
Ｐ’
 P  0
'
: Regge Theory
c2 is directly related with B . (s～2M ν)
• Crossing-odd amplitudes : F(-)(ーν)= ーF(-)(ν)*
F      f ap    f ap   2


Im F(-)asymp(ν) = βV /m (ν/m)αV(0) ρ-trajecctory:αV(0) ~0.5
βＰ’ , βV is Negligible to σtot( = 4π/k Im F(ν) ) in high energies.22
ＦＥＳＲ(1) Ｄｕａｌｉｔｙ
• Remind that the P’ sum rule in the introd..
1
2

 
 
dk

k

d

Im
F

 散乱長と結合定数




asymp
2 
2

2 0
0
k
N
N
N  N1 , N  N2 ( N2  N1 )
• Take two N’s(FESR1)
• Taking their difference, we obtain
N
N
1 2
2 2

d k  tot  k    d Im Fasymp  
2 
2 N1
 N1
LHS is estimated from
Low-energy exp.data.
RHS is calculable from
The low-energy ext. of Im Fasymp.
pp has open(meson) ch. below pp ,and div. above th.
• If we choose N1 to be fairly larger than m we have no difficulty. ( K  p : similar)
No such effects in  p .
23
FESR(1) corresponds to n = -1
K.IGI.,PRL,9(1962)76
• The following sum rule has to hold under the
assumption that there is no sing.with
vac.q.n. except for Pomeron(P).


1 
 M

N
f2

N
  




a



dk

k







tot
tot




0
M


0.0015
-0.012
2.22
 1 
Evid.that this sum rule not hold  pred. of
 P'  0.5
the P' traj. with
and the f meson was discovered on the P'
VIP: The first paper which predicts high-energy from low-energy ( FESR1）
Moment sum rule において、n=-1 とおくと P’ sum rule に reduce.
24


Average of Im F    1 4  k tot  k 
in low-energy regions should coincide with
the low-energy extension of the asymptotic
 
formula Im Fasymp
 .
 

• This relation is used as a constraint between
high-energy parameters: P' , c2 , c1 c0 .
Very Important Point
25
Choice of N1 for πｐ Scattering
• Many resonances
in π-ｐ & π+ｐ
• The smaller N1 is taken,
the more accurate
c2 (and Bπｐ) obtained.
Various values of N1
Δ(1232)
N(1520)
N(1650,75,80)
• We take various N1
corresponding to peak and
dip positions of resonances.
(except for k=N1=0.475GeV)
Δ(1905,10,20)
For each N1,
Δ(1700)
FESR is derived. Fitting is performed. The results checked.
26
N1 dependence of the result
N1(GeV)
10
7
5
4
3.02
2.035
1.476
c2(10-5)
142(21)
136(19)
132(18)
129(17)
124(16)
117(15)
116(14)
χtot2
149.05
149.35
149.65
149.93
150.44
151.25
151.38
N1(GeV)
0.9958
0.818
0.723
(0.475)
0.281
No SR
c2(10-5)
116(14)
121(13)
126(13) (140(13))
121(12)
164(29)
χtot2
151.30
150.51
149.90
150.39
147.78
148.61
• # of Data points : 162.
• best-fitted c2 : very stable.
• We choose N1=0.818GeV
as a representative.
• Compared with the fit by
6 param fit with
No use of FESR(No SR)
27
Result of the fit to σtotπｐ
No FESR
Fitted region
FESR integral
Fitted region
π-ｐ
FESR used
π+ｐ
much improved
c2=(164±29)・10-5
Bπｐ＝0.411±0.073mb
c2=(121±13)・10-5
Bπｐ＝0.304±0.034mb
28
Result of the fit to σtot
No FESR
FESR used
Fitted region
FESR integral
Fitted region
c2=(266±95)・10-4
BＫｐ＝0.535±0.190mb
large uncertainty
Ｋｐ
c2=(176±49)・10-4
BＫｐ＝0.354±0.099mb
much improved
29
ー
ｐｐ,ｐｐ
Result of the fit to σtot
No FESR
FESR used
FESR integral
Fitted region
large
Fitted region
large
c2=(491±34)・10-4
c2=(504±26)・10-4
Bｐｐ＝0.273±0.019mb
Bｐｐ＝0.280±0.015mb
Improvement is not remarkable in this case. 30
Test of the Universal Rise
• σtot = B (log s/s0)2 + Z
B (mb)
πｐ
B(mb)
0.304±0.034
0.411±0.073
Ｋｐ 0.354±0.099
0.535±0.190
ｐｐ
0.273±0.019
0.280±0.015
FESR used
Bπｐ= Bｐｐ= BＫｐ within 1σ
Universality suggested.
No FESR
Bπｐ ≠? Bｐｐ =? BＫｐ
No definite conclusion in this
case.
31
Concluding Remarks
• In order to test the universal rise of σtot ,
we have analyzed π±ｐ;Ｋ±ｐ; pp,ｐｐ
independently.
• Rich information of low-energy scattering data
constrain, through FESR(1), the high-energy
parameters B to fit experimental σtot and ρ
ratios.
• The values of B are estimated individually for
three processes.
32
• We obtain Bπｐ= Bｐｐ= BＫｐ.
Universality of B
suggested.
Ｋｐ
πｐ
ｐｐ
Use of FESR is essential
to lead to this conclusion.
• Universality of B suggests
gluon scatt. gives dominant cont. at very high
energies( flav. ind. ).
• It is also interesting to note that Z for  p, Kp, pp  pp 
approx. satisfy ratio 2:2:3 predicted by quark model.
33
Our results Bpp  0.283(15) mb
predicts  ppLHC  96.0 1.4 mb at 7TeV
LHC
 pp  108.0(1.9) mb at 14TeV
GZK
 pp  176.6  4.5 mb at 335TeV
Our Conclusions at 7TeV, 14TeV
will be tested by LHC TOTEM.
34
• Finally, let us compare our pred. at 14TeV
with other pred.
pp

ref.
tot  mb 
Ishida-Igi (this work) 108.0  1.9
106.3  5.1sys  2.4stat
Igi-Ishida (2005)
107.3  1.2
Block-Halzen (2005)
115.5  1.2  4.1
COMPETE (2002)
125  25
Landshoff (2007)
• Pred. in various models have a wide range.
• The LHC(TOTEM) will select correct one.
35
Very Important Point
Ishida-Igi’approach gives predictions
not only for pp
but also for πp, Kp scatterings,
although experiments are not so easy
in the very near future.
36
Predictions for  pp up to
ultra-high energies including GZK
From Resonances
Cosmic
rays
GZK
ν=6×1010GeV
Tevatron
SPS
LHC
Ecm=7, 14TeV
Non-Regge
comp.
(parabola)
Fitted energy region
37
Concluding Remarks の続き
• この図からわかるように、入射陽子が宇宙背景
輻射の光子と衝突してエネルギーを失うＧＺＫエ
ネルギー（335TeV)でも、我々の予言の誤差は驚
くほど小さくなっている。
• Bの値は最高エネルギーのデータポイントの値に
比較的強く依存する。  GZK
の値の予言のため
pp
にも、 LHCでの測定は大変重要。
• また、LHCの測定で、CDF,D0のどちらが正しい
かも決まる。
38
Appendices:
FESR(1)
  0   0  0
Define F    F    R    F  
We obtain
()
( )
P'
Re F
()
M  
 Born項+
2P

1
2
2


0

N
0
 Im F
k2
d

dk tot
k  
2P


N
0
d
 
 P   0.5 
Im R   
  
2 
k 
M  M  
'
FESR 1
39
FESR(2)

M
 Im F
0

1
  d 
4

N
0
k  tot  k  dk
2

   Im R   d    Im FP'   d
N
0
N
0
FESR(2)
• Dolen-Horn-Schmid : (nth)-moment sum
rule において、
• n=1とおくと、FESR(2)
• n= -1とおくと、FESR(1)=P’ sum rule(1962)
40

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