hiukkasfysiikan kesäkoulu - prujut · 2010. 9. 17. · paul hoyer tvärminne 2010 4 neutrino cross...

Paul Hoyer Tvärminne 2010

1

Paul HoyerHelsingin yliopisto

Hiukkasfysiikan kesäkouluTvärminne

24 - 28.05.2010

The Strong Interactions


2

SU(3) x SU(2)L x U(1)

QC

D

Elec

trow

eak

The Standard Model

As simple as 1-2-3?

Not exactly: The strong, electromagnetic and weak interactions manifest themselves very differently in Nature

This suggests that guessing what (if anything) liesBeyond the Standard Model is a challenging task:

LHC is needed!


3

Neutrino vs. Electron in EW

The (unbroken) electroweak theory is fully symmetric under ν ↔ e

The neutrinos and electrons form a doublet of SU(2)EW:(

νee

)

This means that the theory is unchanged if these particles are redefinedby an SU(2) matrix (2 x 2 unitary matrix with determinent = 1)

(ν′ee′

)≡

(a bc d

) (νee

)

We may even choose a = a(t,x), etc: The rotation can be local in space-time.(

a bc d

)= i

(0 11 0

)For we have ν´= i e and e´ = i ν

You might think that the neutrino and electron should have similar properties?Well, think again!


4

Neutrino cross section

The neutrino interacts extremely weakly with matter. Its total cross section on nucleons (= protons and neutrons) is measured to be

σtot(νµN) = (0.667 ± 0.014)Eν

GeV· 10−38 cm2

Where the neutrino energy Eν is expressed in GeV = 109 eV = 1.60 10–10 J

!ne-structure constant α = e2/4 0 c

speed o! light in vacuum cPlanck constant hPlanck constant, reduced h/ 2π

electron charge magnitude econversion constant cconversion constant ( c)2

electron mass meproton mass mp

=

299 792 458 m s− 16.626 068 96(33)× 10− 34 sJ1.054 571 628(53)× 10− 34 sJ= 6.582 118 99(16) × 10− 22 sVeM

1.602 176 487(40)× 10− 19 C = 4.803 204 27(12) × 10− 10 esu 25, 25mfVeM)94(1369623.791

0.389 379 304(19) GeV 2 nrabm

0.510 998 910(13) MeV /c 2 = 9.109 382 15(45) × 10− 31 kg 25, 50938.272 013(23) MeV /c 2 = 1.672 621 637(83) × 10− 27 kg 25, 50= 1.007 276 466 77(10) u = 1836.152 672 47(80) me

l

= 1 / 137.035 999 679(94)

Constants of Nature


5

The cross section σ measures the scattering probability: Typically, there is one neutrino interaction in a length L of matter when there is one nucleon in the volume Lσ.

σ

L

• Nν

In what length L of water does a neutrino with energyEν = 1 MeV typicallyinteract one time?

Compare: 1 MeV electrons in water have L = 5 mm

# nucleonscm3

=1 gcm3

11.67 · 10−24g = 0.6 · 10

24 1cm3

L =cm3

0.6 · 10241

0.67 · 10−41cm2 = 2.5 · 1015 m = 0.25 light-years

Neutrino interaction length


6

Range of weak forceThe neutrino interacts via the exchange of W- and Z-bosons

W, Z

W, Z

W, Z

R


7

Comparison of EM and Weak cross sections

The idea behind the unification of the weak and electromagnetic forces is that their difference is caused by the Higgs mechanism gives large masses to the weak bosons, while leaving the photon massless.

Mγ = 0 (< 2 ⋅ 10–16 eV) MW = 80.4 GeV/c2 MZ = 91.2 GeV/c2

dσ

dQ2(e−µ+ → e−µ+) = 8πα

2

Q4

e– e–

µ+ µ+Qγ

dσ

dQ2(e−µ+ → νeν̄µ) =

8πα2W(Q2 + M2W )2

" 8πα2W

M4W(Q#MW )

e– νe

µ+Qνµ–

W–

α ! 1137

αW ! α

! 8πα2W

Q4(Q"MW )


8Rough estimate of αW

Combining the weak elastic scatteringdσ

dQ2! 8πα

2W

M4W

with the measured total cross section

σtot(νµN) = (0.667 ± 0.014)Eν

GeV· 10−38 cm2

We can determine the effective coupling

α2W = σtot(νN)M4W

16πMNEν

which gives αW ≈ 1/256, not so different from α ≈ 1/137 !This supports the idea that the neutrino cross section is smallbecause MW is large, not because the coupling αW is tiny.


9


dσ

dQ2(e−µ+ → νeν̄µ) =

8πα2W(Q2 + M2W )2

" 8πα2W

Q4(Q#MW )

Today we have also verified experimentally that the electron and neutrino cross sections are similar when MW is small compared to Q:

! dσdQ2

(e−µ+ → e−µ+) = 8πα2

Q4

HERA e(28 GeV) + p(820 GeV) collider at DESY near Hamburg, Germany.

The center-of-mass energy is ca.ECM ≈ 300 GeV >> MW,Z

Hence one can have collisions where

Q >∼ MW,Z


10)

2 (

pb

/Ge

V2

/dQ

!d

-710

-510

-310

-110

10

)2 (GeV2Q

310 410

p CC 03-04 (prel.)+

H1 e

p CC 2005 (prel.)-

H1 e

p CC 2004+

ZEUS e

p CC 04-05 (prel.)-

ZEUS e

p CC (CTEQ6M)+

SM e

p CC (CTEQ6M)-

SM e

p NC 03-04 (prel.)+

H1 e

p NC 2005 (prel.)-

H1 e

p NC 2004+

ZEUS e

p NC 04-05 (prel.)-

ZEUS e

p NC (CTEQ6M)+

SM e

p NC (CTEQ6M)-

SM e

y < 0.9

= 0eP

HERA II

Figure 1. Unpolarised e±p NC and CCsingle differential cross sections, dσ/dQ2.

ZEUS

3

xF

-0.1

0

0.1

0.2

2 = 3000 GeV

2Q

-110 1

-0.1

0

0.1

0.2

2 = 12000 GeV

2Q

2 = 5000 GeV

2Q

-110 1

2 = 20000 GeV

2Q

2 = 8000 GeV

2Q

-110 1

2 = 30000 GeV

2Q

ZEUS NC (prel.)

)-1p (240 pb± e

SM (ZEUS-JETS)

x

Figure 2. The structure function xF 3, as afunction of x in bins of Q2.

where GF is the Fermi constant, MW is the mass of the W± boson and terms in brackets describethe composition of quarks in the proton probed by the W+ or W− boson. Figure 1 shows theunpolarised e±p NC and CC cross sections as a function of Q2. They were measured usingseparate samples of HERA II data with the positive and negative beam polarisation, combinedand corrected for the small residual polarisation[1, 2, 3, 4]. At lower Q2, the NC cross sectionsare identical for e−p and e+p scattering, as they proceed by the exchange of the photon. Athigher Q2, the e−p cross section can be seen to be significantly above the e+p cross section, dueto the Z0 exchange and the contribution from γ−Z0 interference which changes its sign with thelepton charge. At lower Q2, the CC cross section is smaller than the NC cross section becauseof the W± propagator mass term. In the region where Q2 ∼ M2W ,M

2Z the CC and NC cross

sections become similar, which can be considered as a manifestation of electroweak unificationin spacelike scattering. The difference in CC cross section magnitude for e−p and e+p scatteringarises from the fact that the u-quark density in the proton is larger than the d-quark density,additionally suppressed by the helicity factor (1− y)2. Both the shape and the magnitude ofthe NC and CC cross sections are well described by the SM expectation.

In the case of unpolarised beams, the difference in magnitude of the NC cross section fore−p and e+p scattering is described by the interference structure function xF3, sensitive to thevalence quark distribution in the proton. It can be measured from the lepton charge asymmetry:xF3 = Y+/2Y−[σ̃NC(e−p)−σ̃NC(e+p)], where σ̃NC = (xQ4/2πα2Y+)dσNC/dxdQ2 is the reducedcross section. Both H1 and ZEUS have performed measurements of xF3[5, 6]. Figure 2 showsxF3 extracted by the ZEUS experiment using unpolarised e+p HERA-I sample and the full setof polarised e−p HERA-II data, combined to give effective e−p unpolarised sample[6].

3. Polarised CC cross sectionsThe longitudinal polarisation has a particularly strong effect on the CC cross sections, as theyare predicted to be linearly dependent on the polarisation, independently of kinematic variables:

σe±p

CC (Pe) = (1 ± Pe)σe±pCC (Pe = 0). (4)

2

e– → νee+ → νe–

e– → e–

e+ → e+

M2W

M2Z

e±

q qγ,Z0

e±

e±

q qW±

νe

Neutral Current (NC)

Charged Current (CC)


At Q = 1 MeV the ratio is

∼(

100 GeV1 MeV

)4= 1020

Robert Ciesielski, (DESY) J.Phys.Conf.Ser.110:042007,2008.

http://www-library.desy.de/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Ciesielski%2C%20Robert%22http://www-library.desy.de/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Ciesielski%2C%20Robert%22http://www-library.desy.de/spires/find/inst/www?icncp=DESYhttp://www-library.desy.de/spires/find/inst/www?icncp=DESY


11

Spin dependence of Weak cross sections

eP

-1 -0.5 0 0.5 1

(p

b)

CC

!

0

20

40

60

80

100

120

p Scattering±Charged Current e

X" #p -e

X" #p +e

2 > 400 GeV2Q

y < 0.9

MRST 2004

CTEQ6D

H1 2005 (prel.)

H1 98-99ZEUS 04-05 (prel.)

ZEUS 98-99

H1 99-04ZEUS 06-07 (prel.)

ZEUS 99-00

eP

-1 -0.5 0 0.5 1

(p

b)

CC

!

0

20

40

60

80

100

120

Figure 3. Total e±p CC cross sections as afunction of the lepton beam polarisation, Pe.

ZEUS

310

410

-510

-410

-310

-210

)-1

p (78.8 pb-

ZEUS CC (prel.) e

)-1

p (42.7 pb-

ZEUS CC (prel.) e

-210

-110 1

1

10

210

310

= -0.27eSM (ZEUS-JETS) P

= +0.33eSM (ZEUS-JETS) P

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

50

100

150

200

250

)2

(p

b/G

eV2

/dQ

!d

/dx (

pb

)!

d/d

y (

pb

)!

d

)2

(GeV2

Q

x

y

Figure 4. Single differential CC e−p crosssections as a function of Q2 (top), x (middle)and y (bottom) for two lepton polarisations.

Here, Pe is defined as Pe = (NR −NL)/(NR + NL), where NR and NL are the numbers of rightand left-handed leptons in the beam. In the Standard Model only left-handed fermions and right-handed antifermions take part in weak interactions, hence the CC cross sections are expectedto vanish at Pe = 1(−1) for e−p(e+p) scattering. Both H1[4, 7] and ZEUS[8, 9] have measuredthe total CC cross sections at different values of lepton polarisation, in the kinematic rangeQ2 > 200 GeV2 and Q2 > 400 GeV2 and y < 0.9, respectively. Figure 3 shows the total crosssections in the kinematic range of H1 data, together with unpolarised cross sections[11, 12, 13]and SM predictions. The linear dependence on Pe is clearly observed, in agreement with thechiral structure of the SM. The cross sections were fitted with the linear functions and fromthe extrapolation of fits to Pe = 1(−1) for e−p(e+p) scattering the limits on the mass of theright-handed W±R have been set by both experiments[10]. The ZEUS experiment has measuredthe e−p single differential cross sections, dσ/dQ2, dσ/dx and dσ/dy, for two values of electronpolarisation[8], shown in Figure 4. The results agree with the SM prediction and, as expected,the cross section dependence on the polarisation is independent of kinematic variables.

4. Polarised NC cross sectionsSince the contribution from Z0 exchange becomes significant only at higher Q2, the effect of thepolarisation on the NC cross section is expected to depend on Q2. This dependence can be seenby decomposing the generalised structure functions for e±p scattering as follows:

F±2 = Fγ2 − (ve ± Peae)χZF

γ−Z2 + (v

2e + a

2e ± veaePe)χ

2ZF

Z2 (5)

xF±3 = −(ae ± Peve)χZxFγ−Z3 + (2veae ± Pe(v

2e + a

2e))χ

2ZxF

Z3 , (6)

where F γ2 is associated with the pure photon exchange, Fγ−Z2,3 correspond to the γ − Z

0

interference and FZ2,3 describe the pure Z0 contribution. Here, χZ = 1/ sin2(2θW )Q2/(Q2 +M2Z)

3

eP

-1 -0.5 0 0.5 1

(p

b)

CC

!

0

20

40

60

80

100

120


X" #p -e

X" #p +e

2 > 400 GeV2Q

y < 0.9

MRST 2004

CTEQ6D

H1 2005 (prel.)

H1 98-99ZEUS 04-05 (prel.)

ZEUS 98-99

H1 99-04ZEUS 06-07 (prel.)

ZEUS 99-00

eP

-1 -0.5 0 0.5 1

(p

b)

CC

!

0

20

40

60

80

100

120


ZEUS

310

410

-510

-410

-310

-210

)-1

p (78.8 pb-

ZEUS CC (prel.) e

)-1

p (42.7 pb-

ZEUS CC (prel.) e

-210

-110 1

1

10

210

310



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

50

100

150

200

250

)2

(p

b/G

eV2

/dQ

!d

/dx

(p

b)

!d

/dy

(p

b)

!d

)2

(GeV2

Q

x

y





γ−Z2 + (v

2e + a

2e ± veaePe)χ

2ZF

Z2 (5)


2e + a

2e))χ

2ZxF

Z3 , (6)


0


3

eP

-1 -0.5 0 0.5 1

(p

b)

CC

!

0

20

40

60

80

100

120


X" #p -e

X" #p +e

2 > 400 GeV2Q

y < 0.9

MRST 2004

CTEQ6D

H1 2005 (prel.)

H1 98-99ZEUS 04-05 (prel.)

ZEUS 98-99

H1 99-04ZEUS 06-07 (prel.)

ZEUS 99-00

eP

-1 -0.5 0 0.5 1

(p

b)

CC

!

0

20

40

60

80

100

120


ZEUS

310

410

-510

-410

-310

-210

)-1

p (78.8 pb-

ZEUS CC (prel.) e

)-1

p (42.7 pb-

ZEUS CC (prel.) e

-210

-110 1

1

10

210

310



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

50

100

150

200

250

)2

(p

b/G

eV2

/dQ

!d

/dx

(p

b)

!d

/dy

(p

b)

!d

)2

(GeV2

Q

x

y





γ−Z2 + (v

2e + a

2e ± veaePe)χ

2ZF

Z2 (5)


2e + a

2e))χ

2ZxF

Z3 , (6)


0


3

e± polarization

Only left-handed electronsand right-handed positronsinteract via W exchange!

The weak interactions break parity invariance

Charged and neutral current cross sections from HERA

Robert Ciesielski1, on behalf of the H1 and ZEUS Collaborations1 Deutsches Elektronen-Synchrotron, DESY, Notkestr. 85, 22607 Hamburg, Germany

E-mail: [email protected]

Abstract. The cross sections for inclusive neutral and charged current deep inelastic e±pscattering at high Q2 with polarised lepton beams at HERA-II are presented. The electroweakeffects in spacelike scattering are highlighted and compared to the Standard Model prediction.

1. IntroductionThe operation of the HERA ep collider came to an end in June 2007. A good understanding ofQCD and the precise measurements of the proton parton density functions (PDFs) were madepossible by intensive studies of two deep inelastic e±p scattering (DIS) processes: neutral current(NC) interactions, e±p → e±X, and charged current (CC) interactions, e±p → ν̄e(νe)X. TheNC (CC) processes are mediated by the exchange of a photon or Z0 boson (W∓ boson) andthey can be described by three invariant variables: the virtuality of the exchanged boson, Q2,the Bjorken scaling variable, x, and the inelasticity, y (Q2 = sxy). In 2002, the collider wasupgraded (HERA-II) to provide higher instantaneous luminosities and longitudinal polarisationof the lepton beam. This significantly improved the precision in the high-Q2 region where Z0 andW± exchange becomes significant and allowed tests to be made of the electroweak sector of theStandard Model (SM) in the spacelike scattering, complementary to LEP and Tevatron precisionmeasurements. This paper reviews the recent electroweak measurements of the inclusive NC andCC cross sections at high Q2 and high x, performed by H1 and ZEUS collaborations using partof the HERA-II data.

2. Unpolarised NC and CC cross sections and xF3 structure functionThe double differential cross section for e±p NC DIS may be written in terms of proton structurefunctions, F2, xF3 and FL:

dσe±p

dxdQ2=

2πα2

xQ4[Y+F2 ∓ Y−xF3 − y

2FL], (1)

where α is the fine-structure constant and Y± = 1±(1−y)2. The structure functions F2 and xF3contain the sum and the difference of the quark and antiquark PDFs, the longitudinal structurefunction, FL, is sizable only at high y and can be neglected at high Q2 and high x. The doubledifferential cross section for e−p and e+p CC DIS may be written as:

dσe−p

dxdQ2=

G2F2π

M4W(Q2 + M2W )

2[(u + c) + (1 − y)2(d̄ + s̄)] (2)

dσe+p

dxdQ2=

G2F2π

M4W(Q2 + M2W )

2[(ū + c̄) + (1 − y)2(d + s)], (3)

1







dσe±p

dxdQ2=

2πα2


2FL], (1)


dσe−p

dxdQ2=

G2F2π

M4W(Q2 + M2W )

2[(u + c) + (1 − y)2(d̄ + s̄)] (2)

dσe+p

dxdQ2=

G2F2π

M4W(Q2 + M2W )

2[(ū + c̄) + (1 − y)2(d + s)], (3)

1







dσe±p

dxdQ2=

2πα2


2FL], (1)


dσe−p

dxdQ2=

G2F2π

M4W(Q2 + M2W )

2[(u + c) + (1 − y)2(d̄ + s̄)] (2)

dσe+p

dxdQ2=

G2F2π

M4W(Q2 + M2W )

2[(ū + c̄) + (1 − y)2(d + s)], (3)

1

)2

(p

b/G

eV

2/d

Q!

d

-710

-510

-310

-110

10

)2 (GeV2Q

310 410

p CC 03-04 (prel.)+

H1 e

p CC 2005 (prel.)-

H1 e

p CC 2004+

ZEUS e

p CC 04-05 (prel.)-

ZEUS e

p CC (CTEQ6M)+

SM e

p CC (CTEQ6M)-

SM e

p NC 03-04 (prel.)+

H1 e

p NC 2005 (prel.)-

H1 e

p NC 2004+

ZEUS e

p NC 04-05 (prel.)-

ZEUS e

p NC (CTEQ6M)+

SM e

p NC (CTEQ6M)-

SM e

y < 0.9

= 0eP

HERA II


ZEUS

3

xF

-0.1

0

0.1

0.2

2 = 3000 GeV

2Q

-110 1

-0.1

0

0.1

0.2

2 = 12000 GeV

2Q

2 = 5000 GeV

2Q

-110 1

2 = 20000 GeV

2Q

2 = 8000 GeV

2Q

-110 1

2 = 30000 GeV

2Q

ZEUS NC (prel.)

)-1p (240 pb± e

SM (ZEUS-JETS)

x







σe±p

CC (Pe) = (1 ± Pe)σe±pCC (Pe = 0). (4)

Journal of Physics: Conference Series

2

)2

(p

b/G

eV

2/d

Q!

d

-710

-510

-310

-110

10

)2 (GeV2Q

310 410

p CC 03-04 (prel.)+

H1 e

p CC 2005 (prel.)-

H1 e

p CC 2004+

ZEUS e

p CC 04-05 (prel.)-

ZEUS e

p CC (CTEQ6M)+

SM e

p CC (CTEQ6M)-

SM e

p NC 03-04 (prel.)+

H1 e

p NC 2005 (prel.)-

H1 e

p NC 2004+

ZEUS e

p NC 04-05 (prel.)-

ZEUS e

p NC (CTEQ6M)+

SM e

p NC (CTEQ6M)-

SM e

y < 0.9

= 0eP

HERA II


ZEUS

3

xF

-0.1

0

0.1

0.2

2 = 3000 GeV

2Q

-110 1

-0.1

0

0.1

0.2

2 = 12000 GeV

2Q

2 = 5000 GeV

2Q

-110 1

2 = 20000 GeV

2Q

2 = 8000 GeV

2Q

-110 1

2 = 30000 GeV

2Q

ZEUS NC (prel.)

)-1p (240 pb± e

SM (ZEUS-JETS)

x







σe±p

CC (Pe) = (1 ± Pe)σe±pCC (Pe = 0). (4)

Journal of Physics: Conference Series 110 (2008) 042007

2


12

Rutherford’s experiment 1911:

The positive charges in matter arelocated in a tiny nucleus, whose radiusis ~ 10–5 of the atomic size.

Thomson’s atom: Rutherford’s atom:

The Discovery of the Strong Interaction

⇒ There must be a strong, short-rangedforce to counteract the Coulomb repulsion

F =α

r2


13

M. Attisha

Structure of matter after Rutherford


14

The Pion as a carrier of the strong force

π• • •

• • •

• • •

π

π

NR


15

H. Yukawa, PTP, 17, 48 1935

On the Interaction of Elementary Particles

H. Yukawa

(Received 1935)

At the present stage of the quantum theory little is known about the natureof interaction of elementary particles, Heisenberg considered the interaction of“Platzwechsel” between the neutron and the proton to be of importance to thenuclear structure.Recently Fermi treated the problem of β-disintegration on the hypothesis of

“neutrino”. According to this theory, the neutron and the proton can interactby emitting and absorbing a pair of neutrino and electron. Unfortunately theinteraction energy calculated on such assumption is much too small to accountfor the binding energies of neutrons and protons in the nucleus.To remove this defect, it seems natural to modify the theory of Heisenberg

and Fermi in the following way. The transition of a heavy particle from neutronstate to proton state is not always accompanied by the emission of light particles,i.e., a neutrino and an electron, but the energy liberated by the transition istaken up sometimes by another heavy particle, which in turn will be transformedfrom proton state into neutron state. If the probability of occurrence of the latterprocess is much larger than that of the former, the interaction between theneutron and the proton will be much larger than in the case of Fermi, whereasthe probability of emission of light particles is not affected essentially.Now such interaction between the elementary particles can be described by

means of a field of force, just as the interaction between the charged particles isdescribed by the electromagnetic field. The above considerations show that theinteraction of heavy particles with this field is much larger than that of lightparticles with it.In the quantum theory this field should be accompanied by a new sort of

quantum, just as the electromagnetic field is accompanied by the photon.

1

H. Yukawa, PTP, 17, 48 1935


H. Yukawa

(Received 1935)






1

H. Yukawa, PTP, 17, 48 1935


H. Yukawa

(Received 1935)






1

The factor 4πλ2 comes from

∫ ∫ ∫e−λr12

r12dν2 =

4π

λ2.

Hence (17) becomes

4πgg′

λ2

∫ ∫ ∫ν̃(−→r )u(−→r )

∑

k

ψ̃k(−→r )φk(−→r )dν (18)

or by (16)4πgg′

λ2

∫ ∫ ∫ν̃(−→r )u(−→r )

∑

k,l

ψ̃(−→r )δkl′ φ̃′l(−→r )dν, (19)

which is the same as the expression (21) of Fermi, corresponding to the emissionof a neutrino and an electron of positive energy states φ′k(

−→r ) and ψk(−→r ), exceptthat the factor 4πgg

′

λ2 is substituted for Fermi’s g.Thus the result is the same as that of Fermi’s theory, in this approximation,

if we take4πgg′

λ2= 4× 10−50cm3.erg,

from which the constant g′ can be determined. Taking, for example, λ = 5×1012and g = 2× 10−9, we obtain g′ ∼= 4× 10−17, which is about 10−8 times as smallas g.This means that the interaction between the neutrino and the electron is

much smaller than between the neutron and the proton so that the neutrinowill be far more penetrating than the neutron and consequently more difficultto observe. The difference of g and g′ may be due to the difference of masses ofheavy and light particles.

Summary

The interactions of elementary particles are described by considering a hypotheticalquantum which has the elementary charge and the proper mass and which obeysBose’s statistics. The interaction of such a quantum with the heavy particleshould be far greater than that with the light particle in order to account for thelarge interaction of the neutron and the proton as well as the small probabilityof β-disintegration.Such quanta, if they ever exist and approach the matter close enough to

be absorbed, will deliver their charge and energy to the latter. If, then, thequanta with negative charge come out in excess, The matter will be charged toa negative potential.

8

Estimated Mπ = 20 ... 200 MeV based on proton-neutron scattering data

Considered that the pion may also cause β-decay, i.e., have the role ofthe present W boson. “Heavy particle” = proton, neutron

“Light particle” = electron, neutrino

Yukawa’s 1935 paper

Yukawa received the Nobel Prize for his proposal in 1949.


16Pauli’s Neutrino Hypothesis (1930)To explain the continuous energy spectrum in n → p + e– (+ ν)


17Birth of Yang-Mills Theory

In 1954, Yang and Mills generalized this local U(1) gauge symmetry to the SU(2) group of isospin, with the proton and neutron forming an SU(2)

doublet just as in Yukawa’s theory:

Quantum ElectroDynamics (QED) is invariant under local (space and time -dependent) gauge transformations:

ψ(x)→ eieΛ(x)ψ(x)Aµ(x)→ Aµ(x)− ∂µΛ(x)

Electron field:

Photon field:

Λ(x) may be any regular function

(pn

)

This established the structure of non-abelian gauge symmetry. Nature has, however, completely different uses of YM theories.


18Physical Review 96 (1954) 191

ψ(x)→ U(x)ψ(x)Matter field:

Gauge field: Aµ(x)→ U(x)Aµ(x)U†(x)−i

gU(x)∂µU†(x)

For a local gauge transformation defined by an SU(2) matrix U(x), Yang and Mills found that the theory is symmetric provided the fields transform as:

The same rule holds for any group, such as SU(3). In QED,

U(x) = eieΛ(x) Then U1U2 = U2U1, i.e., all group elements commute,hence the U(1) gauge symmetry is said to be abelian.


19Particles found in Experiments

http://fafnir.phyast.pitt.edu/particles/

using cosmic rays

using particle accelerators

Dirac e+Yukawa π±Pauli ν

Einstein γ

http://fafnir.phyast.pitt.edu/particles/http://fafnir.phyast.pitt.edu/particles/


20

Cosmic rays


21Discovery of quarks

Particles of the Standard Model

All strongly interacting particlesfound in experiments (hadrons)have quantum numbers consistentwith the Quark Model

GIM charm


22Quark Model classification of Hadrons

qq̄qqqHadrons may be formed

with any combination of q = u,d,s,c,b,t

p

π–

MesonsBaryons

Are quarks mere mathematical rules, or true particles? (Gell-Mann)Free quarks have never been seen:


23Quarks are for real: Pointlike scattering of electrons

High energy electrons scatter from pointlike quarks inside the proton:e +q → e +q (in analogy to Rutherford’s experiment)The struck quark flies out of the proton and “hadronizes” into a spray (jet) of hadrons (mostly pions).

e + p → e + anything

At relativistic energies quark-antiquark pairs are created to ensure that all quarks end up as constituents of mesons or baryons.

Deep Inelastic Scattering (DIS):

SLAC 1969: Ee = 20 GeV


24Quantum Chromodynamics (1972)

QCD, the quantum field theory describing quark and gluon interactions, is similar to QED, the theory of electrons and photons

Quark-Gluon couplingαs(MZ) =g2

4π= 0.1167± 0.004

Cf. α = e2/4π ≈ 1/137.035 999 679 (94) Electron-Photon coupling

All quarks u,d,s,c,b,t have the same strong coupling g

ψ =

qqq

The SU(3) gauge symmetry is unbrokenand transforms the three colors of the quarks:

ψ → Uψ U ⊂ SU(3)


25Color Confinement

Three quark colors were introduced by O.W. Greenberg in 1964, to make the quark model of the proton compatible with the Pauli exclusion principle:

Baryon wave functions must be antisymmetric under the interchange of quarks. In the Quark Model, the space – spin wf is symmetric. The color wf is antisymmetric, rescuing the Pauli principle.

The antisymmetric color wave function(A,B,C = red, blue, yellow) means thatthe proton is a color singlet (does notchange under gauge transformations).

|p〉 =∑

A,B,C

εABCqAqBqC

U |p〉 = |p〉QCD apparently has Color Confinement: Only color singlet mesons and baryons can propagate over long distances.

Quarks are color triplets and gluons are color octets: Due to color confinement they cannot exist as free particles.

Color confinement is verified numerically in numerical lattice simulations, but the mechanism is still poorly understood.


Scale dependence of the QCD coupling αs

S. Bethke, hep-ex/0606035

〈0|q̄q|0〉 #= 0 and 〈0|FaµνFµνa |0〉 #= 0

F2(x) =∑

q

e2q xfq(x)

x =Q2

Q2 + M 2X

M 2X =Q2(1 − x)

x

αs(Q2) =

12π

(33 − 2nf) log(Q2/Λ2QCD)

The effective coupling

is valid up to correctionsof order 1/log(Q / ΛQCD).

26

ΛQCD ≈ 200 MeV

The large coupling atlow Q may be relatedto color confinement.


27Perspective: The divisibility of matter

Since ancient times we have wondered whether matter can be divided into smaller parts ad infinitum, or whether there is a smallest constituent.

Common sense suggest that these are the two possible alternatives.However, physics requires us to refine our intuition.

Quantum mechanics shows that atoms (or molecules) are the identical smallest constituents of a given substance

– yet they can be taken apart into electrons, protons and neutrons.

Hadron physics gives a new twist to this age-old puzzle: Quarks can be removed from the proton, but cannot be isolated. Relativity – the creation of matter from energy – is the new feature which makes this possible.

We are fortunate to be here to address – and hopefully develop an understanding of – this essentially novel phenomenon!

Democritus, ~ 400 BCVaisheshika school


0 .00 2.00 4.00 6.00 8.00 10.000.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

m2(GeV )

2

Re

(

m)

!2

f (1270)2

4f (2050)

f (2510)6

a (2450)6

a (2020)4

a (1318)2

(2350)"5

(1700)"3(1670)#

3

(770)"

(782)#

Figure 1: Chew-Frautschi plot for the fully exchange-degen erate f , ω, ρ and a2 trajectories.The solid line denotes the trajectory with the parameters obtained in our !t; the dashed lineis the trajectory α(m ) = 0.48 + 0.88m (m in GeV).2 2

P.Desgrolard, M.Giffon, E.Martynov, E.Predazzi, hep-ph/0006244

Spin

Regge trajectory

For unknown reasons, hadron spins are proportional to their mass2

28

Hadron masses are generated by thepotential and kinetic energies of the constituents

Unlike atoms, hadrons have no ionization threshold,where the quark constituentswould be liberated.

Hadron mass spectrum is relativistic


29The parton picture of the proton

Since the constituents are relativistic quark pairs and gluons may be created:

p〉 ≠ uud〉 !

p〉gluonuud

qq̄

The proton state is expressed asas a superposition of “Fock states”, containing any number ofquarks and gluons (partons)

The wave function of each Fock state specifies the momentum distributions of all partons i in a proton:

xi is the fractional longitudinal momentum carried by the partonk⊥i is the relative transverse momentum carried by the parton

The DIS (e + p → e + anything) cross section measures the probability distribution for a quark or gluon to carry momentum fraction x

Partonic picture of the proton


30Quarks move relativistically inside hadrons

Non-relativistic uud stateeq

p

e

γ*

DIS measures thefraction x of theproton energywhich is carriedby the quarks,anti-quarks andgluons.

For non-relativisticinternal motion thex-distribution wouldbe sharply peaked


31

The huge mass ratios

mumt

! 2.5 MeV171 GeV

! 1.5 10−5 ???

In QCD, mass terms can be directly introduced in the lagrangian, but they are not allowed due to EW symmetry. Hence in the SM all masses result from “Yukawa interactions” involving the Higgs field.

Quark masses

The quark masses inferred from experiment are:

mu = 1.5 ... 3.3 MeV

md = 3.5 ... 6.0 MeV

ms = 104 ± 30 MeV

mc = 1.27 ± .10 GeV

mb = 4.20 ± .15 GeV

mt = 171.2 ± 2.1 GeV

remain unexplained:


32Origin of the proton mass

The u, d quarks in the proton have small masses

2mu + mdmp

! 10 MeV938 MeV

! 1%99% of the proton mass is due to interactions!1% is due to Higgs.

Compare this with positronium (e+e–), the lightest QED atom:

2mempos

! 100.00067% Binding energy is tiny wrt mc2

⇒ Ultra-relativistic state

⇒ Nonrelativistic state

The compatibility of the non-relativisticp〉 = uud〉 quark model description of the proton with its ultra-relativistic parton model picture remains a mystery– but a mystery that we can address within QCD.

p〉gluonuud

qq̄

Both are supported by data: =??


33

e+

e–

µ+

µ–

Creating quark pairs using photons

In QED, an electron and a positroncan annihilate via a “virtual” photoninto any charged particle pair allowedby the available energy.

γ*

Since quarks have electric charge,they can be similarly created, at arate given by their charges.

In the total hadroniccross section we sumover the poorlyunderstood processesby which quarks turninto hadrons, whichoccur with probability = 1


〈0|q̄q|0〉 #= 0 and 〈0|FaµνFµνa |0〉 #= 0

F2(x) =∑

q

e2q xfq(x)

x =Q2

Q2 + M 2X

ξ =2x

1 +√

1 + 4m2px2/Q2

y = logE + p‖√m2 + p2⊥

& − log tan(θ/2)

R =σ(e+e− → hadrons)σ(e+e− → µ+µ−)

M 2X =Q2(1− x)

x

34

cc̄ bb̄

= 3∑

q

e2q

(1 +

αsπ

)

ss̄


35

3D2

2900

3100

3300

3500

3700

3900

4100

c(3590)

c(2980)

hc(3525)

(3097)

(3686)

(3770)

(4040)

0(3415)

1(3510)2(3556)

3D1

3D31D2

3P2(~ 3940)3P1(~ 3880)3P0(~ 3800)

(~ 3800)

1 fmC C

Mass [MeV]

DD~ 600 meV-1000

-3000

-5000

-700011S0

13S1

21S0 23S1 2

1P1 23P2 23P1 23P0

031S0 31D2 3

3D2 33D1 33D2

Ionisationsenergie33S1

e+ e-0.1 nm

Binding energy[meV]

8·10-4 eV

10-4 eV

Charmonium Positronium of QCDthe –


36

8 40. Plots of cross sections and related quantities

Annihilation Cross Section Near MZ

Figure 40.8: Combined data from the ALEPH, DELPHI, L3, and OPAL Collaborations for the cross section in e+e− annihilation intohadronic final states as a function of the center-of-mass energy near the Z pole. The curves show the predictions of the Standard Model withtwo, three, and four species of light neutrinos. The asymmetry of the curve is produced by initial-state radiation. Note that the error bars havebeen increased by a factor ten for display purposes. References:

ALEPH: R. Barate et al., Eur. Phys. J. C14, 1 (2000).DELPHI: P. Abreu et al., Eur. Phys. J. C16, 371 (2000).L3: M. Acciarri et al., Eur. Phys. J. C16, 1 (2000).OPAL: G. Abbiendi et al., Eur. Phys. J. C19, 587 (2001).Combination: The ALEPH, DELPHI, L3, OPAL, SLD Collaborations, the LEP Electroweak Working Group,

and the SLD Electroweak and Heavy Flavor Groups, Phys. Rept. 427, 257 (2006) [arXiv:hep-ex/0509008].(Courtesy of M. Grünewald and the LEP Electroweak Working Group, 2007)

LEP determination of neutrino number

e+e− → hadronsat Z peak


23.11.06 5:02 PM

http://l3.web.cern.ch/l3/scan_program/160GeV/events/qqbar2xz_fixed.gif

e+ e– → 2 jets e+ e– → 3 jets

e+

e–Z q

q_

h’s

h’s

h’sg

37

e+

e–Z q

q_

h’s

h’s

Quark and Gluon jets at LEP


Measurement of quarkand gluon color charges in e+ e– annihilations

S. Bethke, hep-ex/0606035 38

quarkcolorcharge

gluon charge


39

Jet production in hadron collisionsarXiv:1002.1708

(log. scalingviolations)

(log. scalingviolations)

(Non-perturbative)

(Non-perturbative)

(Higher order in αs)

(Perturbative)

(Inclusive sum)(or: h + X)


CDF Collab., hep-ex/0701051

pp → jet + X–

ECM = 1.96 TeV

Quarks and gluonsare pointlike down to the best resolutionthat has been reached

Fermilab:

Ex: Estimate the maximumradius of quarks and gluons, given the agreement of QCD with the Fermilab jet data.

Rapidity:

〈0|q̄q|0〉 #= 0 and 〈0|FaµνFµνa |0〉 #= 0

F2(x) =∑

q

e2q xfq(x)

x =Q2

Q2 + M 2X

ξ =2x

1 +√

1 + 4m2px2/Q2

y = logE + p‖√m2 + p2⊥

& − log tan(θ/2)

M 2X =Q2(1− x)

x

αs(Q2) =

12π

(33− 2nf) log(Q2/Λ2QCD)

ECM = 1960 GeV40


41

9. Quantum chromodynamics 7

!"# !"#$ !"#%

&'()*+(

,*-)./0123(45

6.7*)08(-29:;

9((

τ2-(1*?5

@2A0-4B

C)*+D(/4*40./

;

hiukkasfysiikan kesäkoulu - prujut · 2010. 9. 17. · paul hoyer tvärminne 2010 4 neutrino cross...

Documents