高エネルギーでのハドロン全断面積の...
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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
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Outline of this talk
1. Prediction of σ(pp) at 7TeV, 14TeV (LHC)
2. Test of Universality (BlogBlog22s ) 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
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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 π p、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。
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Test of Universality
• Increase of has been shown to be at most by Froissart (1961) using Analyticity and Unitarity.
• Soft Pomeron fit : Donnachie-Landshoff σtot ~ s0.08 but violates unitarity
• COMPETE collab.(PDG) further assumed
for all hadronic scattering to reduce the number of
adjustable parameters based on the arguments of CGC(Color Glass Condensate) of QCD.
tot 2log s
2
0logtot B s s Z
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Particle Data Group(by COMPETE collab.)
The upper side : σ
The lower side : ρ - ratio
B (Coeff. of 2
0log )s sAssumed to be universal
Theory:Colour Glass Condensate of QCD sug.
Not rigorously proved from QCDTest of univ. of B:necessaryeven empirically.
σ
ρ
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I ncreasing tot.c.s.
• Consider the crossing-even f.s.a.
with
• We assume
at high energies.
2
pp ppf fF
Im4totk
F
'
''2
0 1 22
Im Im Im
log logP
P
P
F R F
c c cM M M M M
M : proton massν, k : incident proton energy, momentum in the laboratory system
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The ratio
• The ratio = the ratio of the real to imaginary part of
F
'
'
Re ReRe
Im ImImP
P
R FF
R FF
'
0.5
1 22 2 log 02
4
0 .
P
tot
c c FM M M M
k
F subtraction const
*
F F
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How to predict σ and ρ for pp at LHC based 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 and in the LHC regions.
tot
(as example)
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• Both and data are fitted through two formulas simultaneously with FESR as a constraint.
• FESR is used as constraint of
and the fitting is done by three parameters:
giving the least .• Therefore, we can determine all the parameters
tot ReF
' ' 0 1 2, ,P P
c c c
2 1 0, ,c c and c
2 ic
'2 1 0, , , , 0P
c c c F
These predict at higher energies including LHC energies. ,
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ISR(=2100GeV)
(a) All region tot
(b)
(c) : High energy region tot
LHCLHC
LHC
Predictions for and
( 62.7 )s GeVThe fit is done for data up to ISR
2100 ( )GeV lab As shown by arrow.
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Summary of Pred. for and at LHC
• Predicted values of 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 of magnitude larger than energy region of the input.
• Now let us test the universality of B.
tot
tot
,tot
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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 the fit
to π p , Kp , , pp ,∑ p ,γ p , 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.
pp
ー
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Particle Data Group(by COMPETE collab.)
The upper side : σ
The lower side : ρ - ratio
B (Coeff. of 2
0log )s sAssumed to be universal
Theory:Colour Glass Condensate of QCD sug.
Not rigorously proved from QCDTest of univ. of B:necessaryeven empirically.
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• We attempt to obtain
for
through search for simultaneous best fit to experimental .
B values( ), ,pp p p p Kp scatterings
tot and ratios
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New Attempt for
• In near future, will be measured at LHC
energy. So, will be determined with good accuracy.
• On the other hand, have been measured only up to k=610 GeV. So, one might doubt to obtain for (as well as ),
with reasonable accuracy.
• We attack this problem in a new light.
,p Kp
pptot
ppB
: 1.8pp s TeVp
tot
B p Kp p KpB B
: 26.4p s GeV
: 24.1Kp s GeV
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Practical Approach for search of BTot. 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 -
40
50
100
Fig.1 pp, pp -
1 10 102 103 104 105 106 log ν(GeV)
●
●
●
●
●
●Tevatron s = 1.8 TeV√
SPS s = 0.9 TeV
√
s = 60 GeV √ISR
σtot (mb)100
50
40Non-Regge comp.(parabola)
• We have good data for large values of log ν compared with LHS, so (ISR, SPS, Tevatron)-data is most important for det. of c2(pp)(or Bpp) with good accuracy.
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Fig.2 πp
• σtot measured only up to
s = 26.4 GeV (cf. with pp, pp).√ -
• So, estimated Bπp
may have large uncertainty.
• 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 ) .
25
30
35
40
45
50
55
60
Fig.2 πp
●
1 10 102 103 logν(GeV)
highest energy s = 26.4 GeV
kL = 610 GeV
√
σtot (mb)
Resonances
↓
50
40
30Non-Regge comp(parabola)
s
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Fitting high-energy data
For pp scatterings We have data in TeV.
σtot = B pp (log s/s0)2 + Z (+ ρ trajectory) in high-energies. parabola of log s
B pp = 0.273(19) mbestimated accurately.Depends the data with the highest energy. (CDF D0)
pp
ー
ISR Ecm<63GeV
SPSEcm<0.9TeV
Tevatron Ecm=1.8TeV
CDFD0
σtot
Fitted energy region
pp
ー
pp- , pp Scattering
cmE
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π p , Kp Scatterings
• No Data in TeV
Estimated Bπ p , B Kp have large uncertainties.
πp π+
p
No Data
K -
p
K +
p
No Data
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Test of Universality of B• Highest energy of Experimetnal data: : Ecm = 0.9TeV SPS; 1.8TeV Tevatron
π- p : Ecm < 26.4GeV
Kp : Ecm < 24.1GeV No data in TeV B : large errors.
B pp = 0.273(19) mb
Bπ p = 0.411(73) mb B pp =? Bπ p =? B Kp ?
B Kp = 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
FESR (1): a kind of P’ sum rule
ppー
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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 , m K , M for π p; Kp;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)* , f K - p (-ν) = f K + p (ν)*
pp ppf f
pp
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Kinematics• Crossing-even amplitudes : F(+)( ー ν)=F(+)(ν)*
average of π- p , π+ p; K - p , K + p; pp ,
Im F(+)asymp(ν) = β P’ /m (ν/m)α’(0)
+(ν/m2)[ c0+c1log ν/m +c2(log ν/m)2] β P’ term : P’trajecctory (f2(1275) ): α P’ (0) ~ 0.5 : Regge Theory
c0,c1,c2 terms : corresponds to Z + B (log s/s0)2
c2 is directly related with B . (s ~ 2M ν)
• Crossing-odd amplitudes : F(-)( ー ν)= ー F(-)(ν)*
Im F(-)asymp(ν) = βV /m (ν/m)αV(0) ρ-trajecctory:αV(0) ~0.5
β P’ , βV is Negligible to σtot( = 4π/k Im F(ν) ) in high energies.
2ap apF f f
2ap apF f f
pp
' 0P
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FESR (1) Duality• Remind that the P’ sum rule in the introd..
• Take two N’s(FESR1)• Taking their difference, we obtain
has open(meson) ch. below ,and div. above th.• If we choose to be fairly larger than we have no difficulty. ( : similar)
No such effects in .
2 2
1 1
2
1 2Im
2
N N
tot asymp
N N
d k k d F
1 2 2 1, ( )N N N N N N
2 20 0
1 2Im
2
N N
asympdk k d Fk
散乱長と結合定数
LHS is estimated from Low-energy exp.data.
RHS is calculable from The low-energy ext. of Im Fasymp.
pp pp
1N m K p
p
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FESR(1) corresponds to n = -1K.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).
Evid.that this sum rule not hold pred. of the traj. with
and the f meson was discovered on the
'P ' 0.5P
'P
2
01
N
tot tot
f Na dk k
M M
0.0015 -0.012 2.22 1
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.
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Average of
in low-energy regions should coincide with
the low-energy extension of the asymptotic
formula
• This relation is used as a constraint between
high-energy parameters:
Very Important Point
Im 1 4 totF k k
Im .asympF
' 2 1 0 ., ,P
c c c
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Choice of N1 for π p Scattering• Many resonances Various values of N1
in π- p & π+ p• The smaller N1 is taken,
the more accurate
c2 (and Bπ p ) obtained.
• We take various N1
corresponding to peak and
dip positions of resonances.
(except for k=N1=0.475GeV)
For each N1,
FESR is derived. Fitting is performed. The results checked.
Δ(1232)N(1520)
N(1650,75,80)
Δ(1700)
Δ(1905,10,20)
-
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N1 dependence of the resultN1(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 151.30 150.51 149.90 148.61 150.39 147.78
• # 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)
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Result of the fit to σtotπ p
c2=(164±29) ・ 10-5 c2=(121±13) ・ 10-5
Bπ p= 0.411±0.073mb Bπ p= 0.304±0.034mb
No FESR FESR used
Fitted region
Fitted region
FE
SR
integralπ- p
π+
pmuch improved
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Result of the fit to σtotKp
c2=(266±95) ・ 10-4 c2=(176±49) ・ 10-4
B Kp= 0.535±0.190mb B Kp= 0.354±0.099mb
large uncertainty much improved
Fitted region
No FESR
Fitted region
FESR
integral
FESR used
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Result of the fit to σtotpp , ppー
c2=(491±34) ・ 10-4 c2=(504±26) ・ 10-4
B pp= 0.273±0.019mb B pp=0.280±0.015mb
Improvement is not remarkable in this case.
No FESR FESR used
Fitted regionlarge
FESR
integral Fitted regionlarge
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Test of the Universal Rise • σtot = B (log s/s0)2 + Z
B (mb)
πp
0.304±0.034
Kp
0.354±0.099
pp
0.280±0.015
B(mb)
0.411±0.073
0.535±0.190
0.273±0.019
FESR used No FESR Bπ p ≠ ? B pp =? B Kp
No definite conclusion in this case.
Bπ p = B pp = B Kp within 1σUniversality suggested.
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Concluding Remarks
• In order to test the universal rise of σtot ,
we have analyzed π±p ; K ±p ; , 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.
pp
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• We obtain Bπ p = B pp = B Kp .
Universality of B
suggested.
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
approx. satisfy ratio predicted by quark model.
Kpπ p pp
, ,p Kp pp pp2:2:3
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Our results
predicts at 7TeV
at 14TeV
at 335TeV
Our Conclusions at 7TeV, 14TeV
will be tested by LHC TOTEM.
• Our Conclusions will be tested by LHC TOTEM.
0.283(15)ppB mb
108.0(1.9)LHCpp mb
96.0 1.4LHCpp mb
176.6 4.5GZKpp mb
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• Finally, let us compare our pred. at 14TeV with other pred.
ref. Ishida-Igi (this work)
Igi-Ishida (2005) Block-Halzen (2005) COMPETE (2002) Landshoff (2007)• Pred. in various models have a wide range.• The LHC(TOTEM) will select correct one.
pptot mb
108.0 1.9106.3 5.1 2.4sys stat 107.3 1.2115.5 1.2 4.1
125 25
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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.
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Predictions for up to ultra-high energies including GZK
Fitted energy region
LHCEcm=7, 14TeV
GZKν=6×1010GeV
From Resonances
Non-Reggecomp.(parabola)
SPSTevatron
Cosmic rays
pp
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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。 の値の予言のためにも、 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。
GZKpp
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Appendices:FESR(1)
Define
We obtain
0( ) ( )' 0 0PF F R F
'
( )20
0.5
2 20 0
2 ImRe
1 2Im 1
2
N NP
tot
P FF M d
k
PBorn dk k d R FESR
k M M
+項
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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)
'
2
0 0
0 0
1Im
4
Im Im (2)
M N
tot
N N
P
F d k k dk
R d F d FESR