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Physics of Electron-Positron Collisions Paul Avery Nov. 17, 2013 Table of Contents 1   Introduction to e+e− Physics Processes ...................................................................................... 1  2   The Physics of e+e− → Hadrons ................................................................................................. 3  

2.1   e+e− → hadrons: cross sections and resonances .................................................................. 3  2.2   e+e− → hadrons cross section and R ................................................................................... 4  2.3   QCD corrections to R ......................................................................................................... 6  2.4   The hadronization process in e+e− quark – antiquark production ....................................... 8  2.5   Jets in e+e− → hadrons ...................................................................................................... 10  2.6   Jets from e+e− → heavy quarks ......................................................................................... 11  

3   Z Physics at e+e− Colliders ....................................................................................................... 12  3.1   Cross sections at the Z peak: interference effects ............................................................. 13  3.2   Forward – backward asymmetries on the Z peak ............................................................. 14  3.3   Z coupling strengths to fermions measured near the Z peak ............................................ 15  3.4   Measuring the number of generations from precision Z measurements ........................... 16  

References ..................................................................................................................................... 18  

1 Introduction to e+e− Physics Processes

Colliding beam electron-positron ( e+e− ) collisions have proven to be an enormously fruitful source of information about fundamental physics since e+e− experiments began taking data in the 1960s . The basic process is shown in the Feynman diagram in Figure 1, in which the elec-tron and positron annihilate to form a fermion – antifermion pair (charged leptons or quarks) via a photon in the intermediate state:

Figure 1: Electron – positron annihilation to a fermion – antifermion pair. The fermions can be ei-ther charged leptons or quarks.

e+

e<

f

fa

2

From our knowledge of Feynman diagrams, we estimate the matrix element to be M Q f e2 / s ,

(product of couplings from each vertex divided by the propagator) where Q f e is the charge of f

and s is the center of mass energy squared. Squaring and using dimensional arguments (cross

sections must have dimensions of GeV–2), the cross section must be σ e+e− → f f( ) ~ Q f

2α 2 / s .

This is close to the correct answer. A relativistic QM calculation gives both the differential and total cross sections (ignoring electron masses, but exact in final state fermion masses):

dσ e+e− → f + f −( )dΩ

=Q f

2α 2

4sβ 1+ cos2θ + 1− β 2( )sin2θ( ) β→1⎯ →⎯⎯

Q f2α 2

4s1+ cos2θ( )

σ e+e− → f + f −( ) = 2πQ f2α 2

3sβ 3− β 2( ) β→1⎯ →⎯⎯

4πQ f2α 2

3s

where β = 1− 4mf

2 / s is the velocity of the final state fermions. For the well-studied process

e+e− → µ+µ− the cross section is (after multiplying the above expression by 2c2 to get correct

units)

σ e+e− → µ+µ−( ) = 86.8

snb

For center of mass energies near 3.5 GeV (the charm region) the e+e− cross section to muons is ~7 nb, while for energies ~10 GeV (bottom region), it drops to 0.8 nb.

The angular distribution of muons in e+e− → µ+µ− is shown in Figure 2. Note that the

1+ cos2θ function that describes the angular distribution is symmetric so that an equal number of muons are emitted in the forward ( cosθ > 0 ) and backward ( cosθ < 0 ) direction. This sym-metry is a general consequence of the fact that the form of the electromagnetic interaction is pre-served under parity (a coordinate transformation in which x →−x ). We will see later that weak interactions violate parity and lead to asymmetric angular distributions.

3

Figure 2: Angular distribution of muons in e+e− → µ+µ−

2 The Physics of e+e− → Hadrons 2.1 e+e− → hadrons: cross sections and resonances

The cross section for e+e− → hadrons is shown in Figure 3 as a function of center of mass energy. The overall 1/s behavior is clearly seen on this log-log graph above ~2 GeV, indicating the valid-ity of the fundamental e

+e− → qq underlying process. However, there are striking resonances as well. The following is a very brief summary.

• The light vector mesons (ρ, ω, φ) dominate the cross section for E < 1, a consequence of the strong coupling of the virtual photon to the mesons. The strong decays of these reso-nances to light mesons account for their notable widths (Γ ~ 4 – 150 MeV).

• The region 3 – 4.5 GeV contains several charmonium ( cc ) resonances, most notably the narrow J/ψ(1S) and ψ(2S), but also some wider ones above 3.77 GeV which decay into charm mesons or charm baryons. Although the charmonium resonances below the DD mass threshold decay strongly, they have very small widths (Γ ~ 100 – 300 keV) because they cannot decay into charm mesons and are forced by their spin and other quantum numbers to decay to three gluons (involving three powers of α s ), with a rate proportional

to α s6 . Charmonium resonances above 3.77 GeV, however, rapidly increase in width be-

cause they can decay directly into charm mesons. A very large amount of charm meson

4

and charm baryon physics has been produced by experiments running in this energy re-gion.

• The narrow bottomonium ( bb ) “Upsilon” resonances in the 10 GeV region can also be clearly seen. The three resonances below BB threshold, like charmonium, have very nar-row widths (Γ ~ 20 – 50 keV) because they cannot decay into bottom mesons and are forced by their quantum numbers to decay to three gluons. Starting with the ϒ(4S), which decays to BB , their decay widths increase rapidly with mass. An enormous amount of B meson data was gathered at experiments such as CLEO, BaBar and Belle which sat on the Υ 4S( )→ BB and measured important phenomena such as CP violation, B − B oscil-lations, very rare processes, etc. and set important limits on physics beyond the Standard Model. New “B factories” are being planned to operate at this resonance but with 50 – 100 times previous luminosities.

• Finally, the Z boson at 91 GeV has an enormous peak. Precision data taken at and above the Z mass region at SLC and LEP have provided some of the most sensitive and strin-gent tests of the Standard Model, including measuring the number of light neutrino spe-cies.

Figure 3: e+e− cross section to hadrons vs CM energy (from PDG 2012 [1]). Note the rich structure of resonances, including the ρ , ω , φ (light quarks) < 2 GeV, charmonium resonances ~3-4 GeV, bot-tomonium resonances ~10 GeV and the Z boson at 90 GeV.

2.2 e+e− → hadrons cross section and R

The e+e− → µ+µ− process has been measured at all available energies and its cross section con-

forms within experimental errors to the prediction 4πα2 / 3s in Section 1. It’s therefore useful to

characterize hadronic production in terms of this cross section to take out the dependence on s and α. Accordingly, we define R as

6 44. Plots of cross sections and related quantities

σ and R in e+e− Collisions

10-8

10-7

10-6

10-5

10-4

10-3

10-2

1 10 102

σ[m

b]

ω

ρ

φ

ρ′

J/ψ

ψ(2S)Υ

Z

10-1

1

10

102

103

1 10 102

Rω

ρ

φ

ρ′

J/ψ ψ(2S)

Υ

Z

√

s [GeV]Figure 44.6: World data on the total cross section of e+e− → hadrons and the ratio R(s) = σ(e+e− → hadrons, s)/σ(e+e− → µ+µ−, s).σ(e+e− → hadrons, s) is the experimental cross section corrected for initial state radiation and electron-positron vertex loops, σ(e+e− →µ+µ−, s) = 4πα2(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” section ofthis Review, Eq. (9.7) or, for more details, K. G. Chetyrkin et al., Nucl. Phys. B586, 56 (2000) (Erratum ibid. B634, 413 (2002)). Breit-Wignerparameterizations of J/ψ, ψ(2S), and Υ(nS), n = 1, 2, 3, 4 are also shown. The full list of references to the original data and the details ofthe R ratio extraction from them can be found in [arXiv:hep-ph/0312114]. Corresponding computer-readable data files are available athttp://pdg.lbl.gov/current/xsect/. (Courtesy of the COMPAS (Protvino) and HEPDATA (Durham) Groups, May 2010.)

5

R =σ e+e− → hadrons( )σ e+e− → µ+µ−( )

For a given CM energy R should depend only on the number of quarks that can be produced at that energy and their charges. Naïvely,

R = Qi

2i∑ , where Qi is the charge in units of e of quark

i and the sum runs over all quarks that can be produced at that energy. But we also have to in-clude the fact that each quark appears in one of three QCD colors, so the prediction is modified to

R = 3 Qi

2i∑ . This simple model predicts the following values for R as a function of CM ener-

gy s :1

53 s <1

2 1< s < 3

103 3< s <10

113 10 < s < 350

5 s > 350

The e+e− R data shown in Figure 4 has many resonances which complicate the experimental pic-ture, but our simple prediction works surprisingly well in the flat regions outside the resonances and shows the expected steps.

Low energy e+e− R data in fact provided the first direct evidence for color, which had been intro-duced partly to explain the symmetry properties of baryons, particularly spin 3/2 baryons with identical quarks such as Δ

++ = uuu( ) , Δ− = ddd( ) and Ω

− = sss( ) . Without color, these wave-functions (functions of space and spin) would be symmetric with respect to interchange of any two quarks, a violation of the spin-statistics theorem of quantum field theory which requires identical fermions (half-integer spin particles) to be antisymmetric under particle interchange. The addition of color to the three quark wavefunction makes it antisymmetric under quark inter-change, as required.2

1 A slight improvement in the model would include quark mass effects near threshold which would lead to a smoother turnon as each quark – antiquark threshold is passed. This is included in the figure. 2 The 3 quark antisymmetric color wavefunction is

rgb− grb+ brg − bgr + gbr − grb( ) / 6 for three colors r, g, b.

6

Figure 4: R from e+e− cross section data compared with a simple quark mode prediction using color.

2.3 QCD corrections to R

Although a simple quark model with color does a reasonable job of explaining the R data, one cannot help notice that the prediction is lower than the data in Figure 4 even in regions outside the resonance regions and doesn’t account for the rise beginning at 40 GeV. The rise is easy to explain as coming from the low tail of the Z boson (Figure 3), which because of the huge size of the Z resonance is measurable even far from the peak. The rest of the discrepancy is explained by taking QCD further into account by including gluon emission and gluon loop corrections to

e+e− → QQ such those shown in Figure 5. The inclusion of these and other QCD diagrams

(which are absent in lepton production) leads to a multiplicative correction of the hadronic cross section by the factor 1+α s s( ) / π in first order, where α s s( ) is the strong coupling constant.

α s s( ) , like α, is a slowly changing (“running”) function of energy. To first order it can be calcu-lated from

α s

−1 s( ) =α s−1 µ2( ) + 33− 2n f

12π

7

where µ is an energy scale where α s has been measured and n f is the number of quark species

with m ff < s . Analysis of e+e− precision data at the Z peak yields

αs mZ

2( ) = 0.1184 ± 0.0007 ,

which can be extrapolated to 10 GeV ( n f = 5 ), 3 GeV (

n f = 4 ) and below 3 GeV (

n f = 3 ) using

this or higher order formulas (Figure 6).3

Figure 5: Examples of diagrams that add QCD modifications to e+e− → QQ .

Figure 6: Calculated value of α s s( ) , using α s mZ

2( ) = 0.1184 ± 0.0007

The comparison of R data to a simple prediction accounting for QCD corrections and the Z peak is shown in Figure 7, showing excellent agreement. Precision analyses of e+e− data to the highest LEP CM energy of 209 GeV show agreement with even fine details of the Z line shape, allowing stringent tests of the Standard Model to be performed.

3 The extrapolation of α s becomes increasingly unreliable at low s because even higher order loop diagrams do not converge rapidly.

a

g

ag

8

Figure 7: R from e+e− cross section data compared with a quark model including a first order con-tribution from QCD corrections and the Z peak.

2.4 The hadronization process in e+e− quark – antiquark production

The fundamental process underlying hadron production in e+e− annihilation is e+e− → QQ ,

where QQ is a quark – antiquark pair of any type, provided that the energy is large enough to produce it. Since quarks cannot exist as free particles a process of hadronization occurs, in which the energy in the increasingly stretched color field connecting Q and Q creates multiple quark-antiquark pairs (from the virtual quantum “sea” of these pairs) that then combine with one anoth-er to form hadrons. The hadronization mechanism is illustrated in Figure 8 for two limiting cas-es. The left diagram outlines e+e− → B+B− in which the e+e− → bb quark process is followed by creation (“popping”) of a single uu pair. This mechanism for producing two heavy mesons begins just above energy threshold for the two mesons (or baryons which requires a diquark –antidiquark pair to be created). At higher energies, hadronization follows the mechanism shown in the right diagram which leads to the production of multiple mesons.4

4 Note that a given q1q2 pair can form a spin 0, spin 1 or even spin 2 meson, though lighter mesons are favored.

9

Energy is required to “pop” qq pairs from the vacuum, so it is no surprise that the hadronization process depends strongly on the quark mass. The mass dependence is quite strong (exponential, in fact), so uu and dd pairs are most commonly created, with ss production about an order of magnitude smaller and cc and bb almost zero. K or K* mesons are thus suppressed relative to π, ρ or ω mesons and Λ and Σ barons are suppressed relative to protons and neutrons. Baryons can be formed during hadronization, but the baryon multiplicity is an order of magnitude smaller than that of mesons because they require the creation of diquark – antidiquark pairs.

Figure 8: Illustration of hadronization. (a) The reaction e+e− → B+B− just above threshold where

the initial e+e− → b+b− process is followed by “popping” a single uu pair from the vacuum to

form the B mesons. (b) The high-energy process e+ e− → hadrons which begins with the fundamen-tal process e

+e− → QQ followed by hadronization in which qq pairs are created from the vacuum from the energy in the color string connecting Q and Q and form multiple mesons and baryons (if diquark – antidiquark pairs are produced).

As the collision energy increases the energy available for hadronization also rises, resulting in a steady increase in average hadron multiplicity (number of hadrons per collision). This is shown in Figure 9, where the average charged particle multiplicity5 is shown for e+e− , ep and pp reac-tions. For each process the multiplicity rises as a low power of log(s).

5 Charged multiplicity is typically measured rather than total multiplicity because it is easy to count charged parti-cles in a tracking chamber. Since pions dominate hadron production and pions of each charge should be produced equally, the total hadronic multiplicity is approximately 50% greater.

(a) (b)

10

Figure 9: Average charged particle multiplicity vs CM energy in e+e−, ep, and pp reactions.

2.5 Jets in e+e− → hadrons

The hadrons that accompany e+e− → QQ production in Figure 8(b) might be expected by mo-

mentum conservation to follow the original directions of the initial quark and antiquark. Hadron-ic “jets”, figuratively shown in Figure 10, are indeed observed and their collimated structure be-comes more obvious at increasing center of mass energies. The angular distribution of the jets follows the predicted 1+ cos2θ quark behavior.

Physicists distinguish the initial hard e+e− → QQ process, which is well-described by perturba-

tive quantum mechanical calculations, from subsequent soft hadronization which can be thought of as arising from emission of multiple virtual gluons (“gluon showering”), splitting of the glu-ons into qq pairs and recombination into hadrons. Virtual gluon radiation takes place preferen-tially at low angles (similar to photon emission from charged particles), so that newly formed hadrons roughly follow the original quark direction. Unlike hard processes, gluon showering and hadronization can only be approximately calculated and predictions rely on simulation computer programs with adjustable parameters that have been fitted to data.

Sometimes the initial hard process is accompanied by the emission of a hard gluon that itself can hadronize, leading to three or even more hadronic jets. The observation in the mid-1970s of 3-jet events at SLAC in e+e− collisions at 3 – 4 GeV center of mass energies provided the first con-

11

vincing evidence of the reality of gluons. Furthermore, the ratio of 3-jet to 2-jet events also pro-vided a measure of the strong coupling constant αs because the amplitude for e

+e− → qqg and

e+e− → qq cross sections differ only by the strong coupling. The ratio was used to measure αs

at this energy for the first time.

Hadronization and jet formation have now been observed in all hard quark processes, whether initiated by e+e− reactions, hadron collisions (Tevatron, LHC), ep reactions (HERA experiments), neutrino interactions or decays of very heavy particles such as t quarks and Z and W bosons. For example, tt creation at the Tevatron and LHC is followed by t → bW → b+ q1q2 decay and re-sults in 6 jets!

Many years of work have led to the development of physics simulation programs (PYTHIA[2], HERWIG[3], SHERPA[4], ARIADNE[5], etc.) that generate many kinds of hard processes and use QCD to reproduce most of the features of hadronization and jet formation. These and other simulation programs (sometimes used in tandem) allow comparisons between experimental re-sults and theory and are essential components of all modern high-energy physics experiments.

Figure 10: Hadronic jets in e+e− → QQ production, illustrating how the final state hadrons are

roughly aligned with the direction of the quark and antiquark produced in the initial hard scatter-ing process.

2.6 Jets from e+e− → heavy quarks

The process e+e− → QQ , where Q is either a charm or bottom quark, provides an interesting

special case of hadronization because of the initial quark’s large mass relative to that of the light quarks popped from the virtual sea. Simple kinematic considerations lead us to expect that the heavy charm or bottom meson should take away a significant fraction of the total momentum of the initial quark, with the effect more pronounced for bottom mesons. The data, shown in the

e+ e<

12

plots in Figure 11, confirm this prediction.[6] The fractional momentum distribution of D0 and D* charm mesons peaks at approximately 60% of the initial quark momentum, while that of bot-tom mesons peaks around 80%.6 The shape of each “fragmentation function” depends only on the quark mass and is almost independent of energy (“scale invariant”) for energies large com-pared to the quark mass. Simple parameterizations of these functions are discussed in the refer-ences within [6].

Figure 11: Distribution of momenta of (a) charm and (b) bottom mesons produced in e+e− collisions to heavy quark – antiquark pairs. The momentum distributions peak at 60% (charm) and 80% (bottom) of the initial quark momentum.

3 Z Physics at e+e− Colliders The Z and W weak bosons were found in 1983 at CERN using pp collisions, leading to two No-bel prizes. However, by far the best and most comprehensive measurements of the Z (and to a lesser extent the W) were made in the 1980s and 1990s by e+e− experiments at CERN’s Large Electron Positron (LEP) (Aleph, Delphi, L3, Opal) and SLD (located at SLAC).

6 The D0 momentum spectrum actually peaks at a slightly lower value than that of the D* because it’s more likely that the D0 is a decay product of the initial charm meson produced by hadronization, which would leave the D0 with slightly less momentum than the original meson. However, higher mass charm mesons are difficult to measure.

18 19. Fragmentation functions in e+

e−, ep and pp collisions

0

2

4

6

8

10

12

14

16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1xp=p/pmax

sB dσ/

dxp

(GeV

2 nb) D0 D*

CLEOBELLE

(a)

xB

1/σ

dσ/

dxB

ALEPH 91 GeVOPAL 91 GeVSLD 91 GeVDELPHI 91 GeV

(b)

0

0.5

1

1.5

2

2.5

3

3.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 19.10: (a) Efficiency-corrected inclusive cross-section measurements forthe production of D0 and D∗+ in e+e− measurements at

√s ≈ 10.6 GeV, excluding

B decay products [193,194]. (b) Measured e+e− fragmentation function of b quarksinto B hadrons at

√s ≈ 91 GeV [206].

June 18, 2012 16:19

18 19. Fragmentation functions in e+

e−, ep and pp collisions

0

2

4

6

8

10

12

14

16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1xp=p/pmax

sB d

σ/dx

p (G

eV2 n

b) D0 D*

CLEOBELLE

(a)

xB

1/σ

dσ/

dxB

ALEPH 91 GeVOPAL 91 GeVSLD 91 GeVDELPHI 91 GeV

(b)

0

0.5

1

1.5

2

2.5

3

3.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 19.10: (a) Efficiency-corrected inclusive cross-section measurements forthe production of D0 and D∗+ in e+e− measurements at

√s ≈ 10.6 GeV, excluding

B decay products [193,194]. (b) Measured e+e− fragmentation function of b quarksinto B hadrons at

√s ≈ 91 GeV [206].

June 18, 2012 16:19

13

Figure 12: R = σ hadrons( ) /σ µ+µ−( ) for e+e− collisions

3.1 Cross sections at the Z peak: interference effects

The matrix element for e+e− → f f (2 fermions, either leptons or quarks) can be written

M = Mγ +MZ , where

Mγ is calculated from photon exchange and MZ is calculated from Z

boson exchange, as shown in the Feynman diagram in Figure 13.

Figure 13: e+ e− collisions to fermion-antifermion pairs in which a photon or Z boson are intermedi-ate particles. f can represent a lepton or quark.

e+

e<

f

fa / Z

14

Squaring the matrix element yields

M

2 = Mγ2+ MZ

2+ 2Re Mγ

∗MZ( )

where the third term is the γ−Z interference term. Thus the differential cross section is the sum of three terms

dσdΩ

=dσγ

dΩ+

dσ ZdΩ

+dσγ Z

dΩ

The first two terms describe the individual cross sections from photon exchange and Z exchange and the last term arises from interference (it can be positive or negative). The PDG gives a nice summary of this physics in the review article on the Standard Model.[7]

3.2 Forward – backward asymmetries on the Z peak A complete calculation of the differential cross section near the Z peak leads to the following formula:

dσdΩ f

= α 2

4sA0

f 1+ cos2θ( ) + A1f cosθ⎡

⎣⎢⎤⎦⎥

The A0f coefficient, which has contributions from the photon plus portions of the Z and interfer-

ence terms, is symmetric in cosθ and by itself predicts an equal number of fermions going in the forward hemisphere ( cosθ > 0 ) as backward ( cosθ < 0 ). But A1

f , which has contributions only from the Z and interference terms, is antisymmetric in cosθ and leads to a forward – backward asymmetry.

We can define the asymmetry more quantitatively using σ + and σ− , the cross sections in the forward and backward hemispheres, respectively:

σ + = dσ

dΩ f2π dcosθ

0

1∫ σ− = dσ

dΩ f2π dcosθ

−1

0∫

Then the forward – backward asymmetry is defined as

AFBf ≡

σ + −σ−σ + +σ−

=A1

f 12 +

12( )

A0f 2+ 2

3( ) =38

A1f

A0f

Thus with knowledge of the Standard Model electromagnetic and Z couplings to the fermions (which differ by species) we can predict the forward – backward asymmetry for each fermion species and compare with data. The latest results (from the PDG article on the Standard Mod-el[7]) are shown in Table 1.

15

Table 1: Forward – backward asymmetries measured at the Z peak

Quantity SM Predicton Measured

AFBe 0.01633 ± 0.00021 0.0145 ± 0.0025

AFBµ 0.01633 ± 0.00021 0.0169 ± 0.0013

AFBτ 0.01633 ± 0.00021 0.0188 ± 0.0017

AFBu

0.0739 ± 0.0005 See discussion

AFBd

0.1035 ± 0.0007 See discussion

AFBs 0.1035 ± 0.0007 0.0976 ± 0.0114

AFBc 0.0739 ± 0.0005 0.0707 ± 0.0035

AFBb 0.1034 ± 0.0007 0.0992 ± 0.0016

No asymmetry measurements are shown for u and d quarks because for a generic hadron event it is impossible to determine whether the initially created quarks were light quarks. The hadrons they produce are no different in momentum or type from other hadrons produced in the event. ss production provides the (somewhat weak) possibility of identification because one can look for relatively high-momentum strange mesons), but this has to be done carefully because of the presence of other strange particles created from ss pairs popped from the sea. The asymmetries in cc and bb production are more straightforward to measure (aside from problems of efficien-cy) because the heavy mesons/baryons they create cannot be produced in hadronization.

3.3 Z coupling strengths to fermions measured near the Z peak

One can also extract the individual Z couplings to fermions from e+e− → f f measurements. For

example, consider the following coupling asymmetry for each fermion species:

A f =

gL2 − gR

2

gR2 + gL

2

Here the Z coupling to the fermion – antifermion pair are gR (right handed) and gL (left hand-ed). For electromagnetic interactions these are equal. Z couplings, however, have different the right and left hand couplings which depend on the fermion species, of which there are 4 catego-ries, as shown in Table 2.

16

Table 2: Left and right Z coupling constants for each fermion class

νν gR = 0 gL = + 12

+− gR = +sin2θW gL = − 12 + sin2θW

uu ,cc ,tt gR = − 23 sin2θW gL = + 1

2 −23 sin2θW

dd ,ss ,bb gR = + 13 sin2θW gL = − 1

2 +13 sin2θW

where θW is known as the Weinberg angle. Its value has been extracted from a number of meas-urements, including neutrino interactions, τ lepton decays, the W and Z masses and measure-ments near the Z peak. The best estimate of sin

2θW (which is the way the quantity is normally

presented) from a number of combined measurements is sin2θW = 0.23116 ± 0.00012 .7

The measurements and SM predictions of the fermion couplings are shown in Table 3, showing excellent agreement between data and prediction for each species.

Table 3: Z coupling asymmetry for each fermion species

Measurement SM Prediction Measured

A0ν l +1.00 (Impossible)

Ae 0.1475 ± 0.0010 0.1514 ± 0.0022 0.1544 ± 0.0060 0.1498 ± 0.0049

Aµ 0.1475 ± 0.0001 0.142 ± 0.015

Aτ 0.1475 ± 0.0001 0.136 ± 0.015 0.144 ± 0.004

AFBs 0.9357 ± 0.0001 0.895 ± 0.091

AFBc 0.6680 ± 0.0004 0.670 ± 0.027

AFBb 0.9348 ± 0.0001 0.923 ± 0.020

3.4 Measuring the number of generations from precision Z measurements The full width of the Z can be written as a sum of contributions over all distinct channels, using the couplings in Section 3.3 for each category:

7 Like α (electromagnetic coupling) and αs (strong coupling), the value of sin

2θW changes slowly (“runs”) with energy. The value shown is evaluated at the Z mass.

17

ΓZ = 3Γ +−( ) + 3Γ νν( ) + 2Γ uu( ) + 3Γ dd( )

where the factor of 3 in front of the charged lepton and neutrino rates accounts for equal produc-tion of each lepton species. Similarly, the factor of 3 in front of the dd rate accounts for charge

− 1

3 quarks ( dd , ss , bb ) while the factor of 2 in front of the uu rate accounts for charge + 2

3 quarks ( uu , cc but not tt ). Table 4 compares Z width data with a precision calculation that in-cludes small mass corrections for the heavier leptons and quarks as well as other effects.[7]

Table 4: Z partial width prediction and measurement for each fermion species

Decay mode Γ (SM prediction) Γ (Experiment)

Z → +− 84.00 ± 0.01 MeV 83.984 ± 0.086 MeV

Z →νν 167.21 ± 0.01 MeV (See discussion)

Z → hadrons 1742.6 ± 1.0 MeV 1744.0 ± 2.0 MeV

Z → invisible 501.69 ± 0.06 MeV 499.0 ± 1.5 MeV

Z → all 2496.1 ± 1.0 MeV 2495.2 ± 2.3 MeV

The decay rate to neutrinos can be inferred experimentally by using the missing contribution to the Z width

Γ Z → invisible( ) = Γ Z → all( )− Γ Z → leptons( )− Γ Z → hadrons( )

and ascribing it to neutrinos. The measured number of neutrino species is then (using all LEP experiments and accounting for small mass and electroweak corrections)[8]

Nν =

Γ Z → invisible( )Γ Z →νν( ) = 499.0 ±1.5

167.21± 0.01→ 2.984 ± 0.008

The calculated number of neutrino species is thus only 0.3% away from 3! The fact that meas-urements of the Z width provide such an accurate determination of the number of neutrino spe-cies is amazing given the number and variety of measurements and precision required.

The number of neutrino species was determined another way in e+e− collisions. The LEP exper-iments collected data for the reaction e

+e− →ννγ (Figure 14), where only the radiated photon is observed. The measurement is in principle simple since the total rate is proportional to the number of neutrino species, but of course a careful correction must be made for events where final state leptons from Z decay escape detection. A large event sample was obtained on and above the Z resonance. A simultaneous analysis of the combined LEP and LEP2 data yielded an independent measure of the light neutrino count, 2.92 ± 0.05.[9]

The value Nν = 3 is consistent with the limits on light neutrinos obtained from Big-Bang nucle-osynthesis and Supernova 1987A. None of these measurements, however, rule out the possibility of additional neutrinos with a heavy mass.

18

Figure 14: The reaction e+e− →ννγ where only the radiated photon is observed.

References 1 PDG cross section plots, http://pdg.lbl.gov/2013/reviews/rpp2012-rev-cross-section-plots.pdf 2 T. Sjostrand et al., Comput. Phys. Commun. 135, 238 (2001) [arXiv:hep-ph/0010017]; T. Sjos-trand, S. Mrenna, and P. Skands, JHEP 0605, 026 (2006) [arXiv:hep-ph/0603175]; http://projects.hepforge.org/pythia6/; T. Sjostrand, S. Mrenna, and P. Skands, Comput. Phys. Commun. 178, 852 (2008) [arXiv:0710.3820 [hep-ph]]; 3 G. Corcella et al., JHEP 0101, 010 (2001) [arXiv:hep-ph/0011363]; http://www.hep.phy.cam.ac.uk/theory/webber/Herwig/; M. Bahr et al., Eur. Phys. J. C58, 639 (2008) [arXiv:0803.0883 [hep-ph]]; http://projects.hepforge.org/herwig/. 4 M.L. Mangano et al., JHEP 0307, 001 (2003) [arXiv:hep-ph/0206293]; http://cern.ch/mlm/alpgen/. 5 Y. Kiyo et al., Nucl. Phys. B823, 269 (2009) [arXiv:0907.2120 [hep-ph]]. 6 Particle Data Group, Fragmentation Functions in e+e−, ep and pp collisions, http://pdg.lbl.gov/2013/reviews/rpp2012-rev-frag-functions.pdf. 7 Electroweak model and constraints on new physics, http://pdg.lbl.gov/2013/reviews/rpp2012-rev-standard-model.pdf 8 Precision electroweak measurements at the Z resonance, http://arxiv.org/abs/hep--ex/0509008. 9 The number of light neutrino types from collider experiments, http://pdg.lbl.gov/2012/reviews/rpp2012-rev-light-neutrino-types.pdf

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