ρ ω The R-ratio: comparison to data - becher.itp.unibe.ch · R √ s [GeV] Andrea Banﬁ Lecture 2 The R-ratio: comparison to data QCD Hadron Collider Summer School ’08 G.Zanderighi

6 40. Plots of cross sections and related quantities

! and R in e+e! Collisions

10-8

10-7

10-6

10-5

10-4

10-3

10-2

1 10 102

![m

b]

!

"

#

"!

J/$

$(2S)!

Z

10-1

1

10

10 2

10 3

1 10 102

R !

"

#

"!

J/$ $(2S)

!

Z

!

s [GeV]Figure 40.6: World data on the total cross section of e+e! ! hadrons and the ratio R(s) = !(e+e! ! hadrons, s)/!(e+e! ! µ+µ!, s).!(e+e! ! hadrons, s) is the experimental cross section corrected for initial state radiation and electron-positron vertex loops, !(e+e! !µ+µ!, s) = 4"#2(s)/3s. Data errors are total below 2 GeV and statistical above 2 GeV. The curves are an educative guide: the broken one(green) is a naive quark-parton model prediction, and the solid one (red) is 3-loop pQCD prediction (see “Quantum Chromodynamics” sectionof this Review, Eq. (9.12) or, for more details, K. G. Chetyrkin et al., Nucl. Phys. B586, 56 (2000) (Erratum ibid. B634, 413 (2002)).Breit-Wigner parameterizations of J/$, $(2S), and % (nS), n = 1, 2, 3, 4 are also shown. The full list of references to the original data and thedetails of the R ratio extraction from them can be found in [arXiv:hep-ph/0312114]. Corresponding computer-readable data files are availableat http://pdg.lbl.gov/current/xsect/. (Courtesy of the COMPAS (Protvino) and HEPDATA (Durham) Groups, August 2007. Correctionsby P. Janot (CERN) and M. Schmitt (Northwestern U.)) See full-color version on color pages at end of book.

The R-ratio: comparison to data

QCD Hadron Collider Summer School ’08 G.Zanderighi

Renormalisation group

QCD beta function

Short-distance observables

Comparison of R̂ to experimental data

10-1

1

10

10 2

0.5 1 1.5 2 2.5 3

Sum of exclusivemeasurements

Inclusivemeasurements

3 loop pQCD

Naive quark model

u, d, s

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9.5 10 10.5 11

MD-1ARGUS CLEO CUSB DHHM

Crystal Ball CLEO II DASP LENA

!(1S)!(2S)

!(3S)

!(4S)

b

R

!

s [GeV]

Andrea Banfi Lecture 2




QCD beta function



10-1

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!(1S)!(2S)

!(3S)

!(4S)

b

R

!

s [GeV]


e+e− to hadrons: cross section (PDG)

6 40. Plots of cross sections and related quantities

! and R in e+e! Collisions

10-8

10-7

10-6

10-5

10-4

10-3

10-2

1 10 102

![m

b]

!

"

#

"!

J/$

$(2S)!

Z

10-1

1

10

10 2

10 3

1 10 102

R !

"

#

"!

J/$ $(2S)

!

Z

!

s [GeV]Figure 40.6: World data on the total cross section of e+e! ! hadrons and the ratio R(s) = !(e+e! ! hadrons, s)/!(e+e! ! µ+µ!, s).!(e+e! ! hadrons, s) is the experimental cross section corrected for initial state radiation and electron-positron vertex loops, !(e+e! !µ+µ!, s) = 4"#2(s)/3s. Data errors are total below 2 GeV and statistical above 2 GeV. The curves are an educative guide: the broken one(green) is a naive quark-parton model prediction, and the solid one (red) is 3-loop pQCD prediction (see “Quantum Chromodynamics” sectionof this Review, Eq. (9.12) or, for more details, K. G. Chetyrkin et al., Nucl. Phys. B586, 56 (2000) (Erratum ibid. B634, 413 (2002)).Breit-Wigner parameterizations of J/$, $(2S), and % (nS), n = 1, 2, 3, 4 are also shown. The full list of references to the original data and thedetails of the R ratio extraction from them can be found in [arXiv:hep-ph/0312114]. Corresponding computer-readable data files are availableat http://pdg.lbl.gov/current/xsect/. (Courtesy of the COMPAS (Protvino) and HEPDATA (Durham) Groups, August 2007. Correctionsby P. Janot (CERN) and M. Schmitt (Northwestern U.)) See full-color version on color pages at end of book.

e+e− to hadrons: R-ratio (PDG)

First experimental evidence for colour

I. Existance of !++ particle: particle with three up quarks of the same spin and with symmetric spacial wave function. Without an additional quantum number Pauli’s principle would be violated ⇒ color

II.R-ratio: ratio of (e+e- ! hadrons)/(e+e- ! µ+µ-)

U †U = UU † = 1 det(U) = 1

!

i

!!i !i !

!

ijk

U!ij!jUik!k =

!

k

!!k!k

!

ijk

!i!j!k !!

ijki!j!k!U!

ii!Ujj!Ukk!!i!!j!!k! =!

i!j!k!!i!!j!!k!

nq " nq̄ = n · 3 with n integer

R # e+e" ! hadrons

e+e" ! µ+µ" $ Nc

!

f

Q2f

1

Data compatible with Nc=3 (r, g, b). Will come back to R later.

charge 2/3mass=

upfew MeV

charm~1.25 GeV

top~172 GeV

charge -1/3mass =

downfew MeV

strange~100 MeV

bottom~5 GeV





QCD beta function



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QCD beta function



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!(1S)!(2S)

!(3S)

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s [GeV]


charm quark region .√

s � 2mc

40. Plots of cross sections and related quantities 7

R in Light-Flavor, Charm, and Beauty Threshold Regions

10-1

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3 loop pQCD

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Mark-IMark-I + LGWMark-IIPLUTODASPCrystal BallBES

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MD-1 ARGUS CLEO CUSB DHHM


!(1S)!(2S)

!(3S)

!(4S)

b

R

!s [GeV]

Figure 40.7: R in the light-flavor, charm, and beauty threshold regions. Data errors are total below 2 GeV and statistical above 2 GeV.The curves are the same as in Fig. 40.6. Note: CLEO data above ! (4S) were not fully corrected for radiative e!ects, and we retainthem on the plot only for illustrative purposes with a normalization factor of 0.8. The full list of references to the original data andthe details of the R ratio extraction from them can be found in [arXiv:hep-ph/0312114]. The computer-readable data are available athttp://pdg.lbl.gov/current/xsect/. (Courtesy of the COMPAS (Protvino) and HEPDATA (Durham) Groups, August 2007)




QCD beta function



10-1

1

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3 loop pQCD

Naive quark model

u, d, s

!

"

#

!!

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3 3.5 4 4.5 5

Mark-I

Mark-I + LGW

Mark-II

PLUTO

DASP

Crystal Ball

BES

J/$ $(2S)

$3770

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!(1S)!(2S)

!(3S)

!(4S)

b

R

!

s [GeV]


bottom quark region . √

s � 2mb

40. Plots of cross sections and related quantities 7

R in Light-Flavor, Charm, and Beauty Threshold Regions

10-1

1

10

10 2

0.5 1 1.5 2 2.5 3



3 loop pQCD

Naive quark model

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Mark-IMark-I + LGWMark-IIPLUTODASPCrystal BallBES

J/$ $(2S)

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9.5 10 10.5 11

MD-1 ARGUS CLEO CUSB DHHM


!(1S)!(2S)

!(3S)

!(4S)

b

R

!s [GeV]

Figure 40.7: R in the light-flavor, charm, and beauty threshold regions. Data errors are total below 2 GeV and statistical above 2 GeV.The curves are the same as in Fig. 40.6. Note: CLEO data above ! (4S) were not fully corrected for radiative e!ects, and we retainthem on the plot only for illustrative purposes with a normalization factor of 0.8. The full list of references to the original data andthe details of the R ratio extraction from them can be found in [arXiv:hep-ph/0312114]. The computer-readable data are available athttp://pdg.lbl.gov/current/xsect/. (Courtesy of the COMPAS (Protvino) and HEPDATA (Durham) Groups, August 2007)

R = K Nc

�

q

e2q and K̄ = 1−K

34th International Conference on High Energy Physics, Philadelphia, 2008

HPQCD + Karlsruhe 08

Kuehn, Steinhauser, Sturm 07

Buchmueller, Flaecher 05

Hoang, Manohar 05

Hoang, Jamin 04

deDivitiis et al. 03

Rolf, Sint 02

Becirevic, Lubicz, Martinelli 02

Kuehn, Steinhauser 01

QWG 2004

PDG 2006

mc(3 GeV)

lattice + pQCD

low-moment sum rules, NNNLO

B decays !s2"0

B decays !s2"0

NNLO moments

lattice quenched

lattice (ALPHA) quenched

lattice quenched

low-moment sum rules, NNLO

0.8 0.9 1 1.1 1.2 1.3 1.4

Kuehn, Steinhauser, Sturm 07Pineda, Signer 06Della Morte et al. 06Buchmueller, Flaecher 05Mc Neile, Michael, Thompson 04deDivitiis et al. 03Penin, Steinhauser 02Pineda 01Kuehn, Steinhauser 01Hoang 00QWG 2004PDG 2006

mb(mb)

low-moment sum rules, NNNLO

# sum rules, NNLL (not complete)

lattice (ALPHA) quenched

B decays !s2"0

lattice (UKCD)

lattice quenched

#(1S), NNNLO

#(1S), NNLO

low-moment sum rules, NNLO

# sum rules, NNLO

4.1 4.2 4.3 4.4 4.5 4.6 4.7

Figure 1: Comparison of recent determinations of mc(3GeV) and mb(mb).

sensitive to non-perturbative contributions from condensates, to the Coulombic higher order e!ects, the variation of

µ and the parametric !s dependence. For n = 1:

mc(3 GeV) = 0.986(13) GeV . (3)

The moment with n = 2 is less sensitive to data for R(s) from the continuum region above 5 GeV, where experimental

results are scarce and the aforementioned theory uncertainties are still relatively small. The agreement between

n = 1 and n = 2 (mc(3 GeV) = 0.976(16)GeV), together with the nice convergence with increasing order in !s can

be considered as additional confirmation of this approach.

Instead of measuring the moments Mexpn in e+e! annihilation they can also be determined in lattice simulations.

This approach has recently been pioneered in [10] using the Highly Improved Staggered Quarks (HISQ) discretization

of the quark action in combination with four-loop perturbative results [3, 4, 5, 9, 11]. The final result, mc(3 GeV) =

0.986(10) GeV corresponds to a scale-invariant mass mc(mc) = 1.268(9)GeV and is in excellent agreement with the

determinations based on e+e! data.

The approach based on e+e! data is also applicable to the determination of mb. The three results based on n = 1, 2

and 3 are of comparable precision. The relative size of the contributions from the threshold and the continuum region

decreases for the moments n = 2 and 3. On the other hand, the theory uncertainty is still small. Therefore the

result from n = 2 was taken as the final answer [5], despite the fact that C̄2 was not yet known. The result,

mb(10 GeV) = 3.609(25) GeV, corresponds to mb(mb) = 4.164(25) GeV. The recent evaluation [9] of C̄2 has lead to

a decrease of the central value by 2 MeV and a reduction of the error from 25 MeV to 19 MeV. A comparison of a

few selected mb- and mc-determinations is shown in Fig. 1.

2. The strong coupling constant

One of the most precise and theoretically safe determination of !s is based on measurements of the cross section

for electron-positron annihilation into hadrons [12]. These have been performed in the low-energy region between

2 GeV and 10 GeV and, in particular, at and around the Z resonance at 91.2 GeV. Conceptually closely related is

the measurement of the semileptonic decay rate of the " -lepton, leading to a determination of !s at a scale below

2 GeV [13]. The perturbative expansion for the ratio R(s) ! #(e+e! " hadrons)/#(e+e! " µ+µ!) in numerical

form is given by

R = 1 + as + (1.9857 # 0.1152 nf) a2s + (#6.63694# 1.20013nf # 0.00518n2

f) a3s

+ (#156.61 + 18.77 nf # 0.7974 n2f + 0.0215 n3

f) a4s . (4)

Here as ! !s/$ and the normalization scale µ2 = s. The a4s corrections are conveniently classified according to their

power of nf , with nf denoting the number of light quarks. The a4sn

3f term is part of the “renormalon chain”, the

2

Documents

ρ ω The R-ratio: comparison to data - becher.itp.unibe.ch · R √ s [GeV] Andrea Banﬁ Lecture 2 The R-ratio: comparison to data QCD Hadron Collider Summer School ’08 G.Zanderighi