1

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

2

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

3

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。

44

4

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

555

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

66

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

77

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

888

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)

999

• 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. ,

101010

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.

111111

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

121212

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

131313

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.

141414

• We attempt to obtain

for

through search for simultaneous best fit to experimental .

B values( ), ,pp p p p Kp scatterings

tot and ratios

1515

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

16

16

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.

1717

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

181818

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

191919

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

202020

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

212121

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

222222

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

2323

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

2424

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.

2525

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

262626

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)

-

272727

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)

282828

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

292929

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

303030

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

313131

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.

323232

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

333333

• 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

3434

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

353535

• 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

36

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.

37

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

38

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

39

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

40

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