introduction to qcd - eth zpheno/qcdcourse/notes/lecture1.pdf · dis introduction if you assume...
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![Page 1: Introduction to QCD - ETH Zpheno/QCDcourse/notes/lecture1.pdf · DIS introduction If you assume elastic scattering with a constituent carrying a fraction of the proton momentum F](https://reader030.vdocument.in/reader030/viewer/2022040307/5ed0e4cf7f2c081a751d7705/html5/thumbnails/1.jpg)
Introduction to QCDlecture 1: Introduction to color, quarks and gluons
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Quarks in flavour SU(3)
• Why do Hadrons (baryons and mesons) fit the pattern ?
The eightfold way (1961)
discovered as predicted in 1964!
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Quarks in flavour SU(3)1964: Gell-Mann and Zweig propose quarks
u
d
s “Explains” the patternFractional charge!
No free quarks to be seen!
mu ! 3" 9MeV
md ! 1" 5MeV
ms ! 75" 170MeV
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More quarks
e
u ?d s
µ!e !µ
Bjorken and Glashow proposed a fourth quark
to fit the pattern.
GIM mechanism (1970)
1971: discovery at Brookhaven and SLAC
J/!
J/! = (cc)cmc ! 1.1" 1.3GeV
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More quarks
• 1975-1976 naked charm
• 1975: tau discovered at SLAC
• 1977: discovered at Fermilab (E288)
• 1980: naked beauty
• 1995: top quark identified at Tevatron
b
t
! =( bb)
!0b = (udb)
mb ! 4.0" 4.4GeV
mt ! 171GeV
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The spin-statistics issue!++ is a spin 3/2 particle with 3 “identical” up quarks !
St. Pauli’s exclusion principle endangered!
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Color SU(3)Greenberg proposes a new degree of freedom:
Color
u u u There are now 3 kinds of up quarks
u uu
!++
Why 3?
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Adler-Bell-Jackiw anomaly
Z
!
!
Loop diagrams introduce violation of symmetries of the Lagrangian (in this case the chiral symmetry)
example:
The anomaly has to cancel when summing
over fermions.
!
f
e2faf =
12
"!1 + Nc(
49! 1
9)#
b!µ + a!µ!5
aup = 1adown = !1
ae = !1
Nc = 3! anomaly cancelation
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pion decay
!0 = (qq)
!(!0 ! "") = N2c (e2
u " e2d)
2 a2emm3
!
64!3
1f2
!
= 7.63eV (N2
c
3)
7.84± 0.56eVExperimental value:
Nc = 3! pion decay ok.
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Hadron production
!(e+e! ! hadrons)!(e+e! ! µ+µ!)
= Nc
!e2q = Nc
119
q
q
You can therefore measure the number of colors. Experiment yields Nc ! 3.2
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DIS introduction
l = (E,!l)
l! = (E!,!l!)
q = l ! l! = p! ! p
p = (M,!0)
p! = (Ef , !p!)
! = E ! E!
Q2 = !q2 = !(p! p!)2 = !M2 ! p!2 + 2M(M + !)
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DIS introduction
l = (E,!l)
l! = (E!,!l!)
q = l ! l! = p! ! p
p = (M,!0)
p! = (Ef , !p!)
In the elastic scattering case
Q2 = !q2 = !(p! p!)2 = !M2 ! p!2 + 2M(M + !) = 2M!
p!2 = M2 !
xB =Q2
2M!So deviation from elastic scattering
“Bjorken - x”
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DIS introduction
Assuming elastic scattering with a point-like proton
(of spin 1/2)
d!
dQ2=
4"a2
Q4e2q
E
E!
!cos2(#/2) +
Q2
2M2sin2(#/2)
"
l = (E,!l)
l! = (E!,!l!)
q = l ! l! = p! ! p
p = (M,!0)
p! = (Ef , !p!)
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DIS introduction
Assuming elastic scattering with a point-like proton
(of spin 1/2)
l = (E,!l)
l! = (E!,!l!)
q = l ! l! = p! ! p
p = (M,!0)
p! = (Ef , !p!)
d!
dQ2d"=
4#a2
Q4e2q
E
E!
!cos2($/2) +
Q2
2M2sin2($/2)
"%(" ! Q2
2M)
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DIS introduction
Assuming elastic scattering with a point-like proton
(of spin 1/2)
l = (E,!l)
l! = (E!,!l!)
q = l ! l! = p! ! p
p = (M,!0)
p! = (Ef , !p!)
d!
dQ2d"=
4#a2
Q4e2q
E
E!
!cos2($/2) +
Q2
2M2sin2($/2)
"%(" ! Q2
2M)
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DIS introduction
Assuming elastic scattering with a point-like proton
(of spin 1/2)
l = (E,!l)
l! = (E!,!l!)
q = l ! l! = p! ! p
p = (M,!0)
p! = (Ef , !p!)
d!
dQ2d"=
4#a2
Q4
E
E!!W2(Q2, ")cos2($/2) + 2W1(Q2, ")sin2($/2)
"
W2(Q2, !) = e2q"(! !
Q2
2M) W1(Q2, !) = e2
qQ2
4M2"(! ! Q2
2M)
Structure functions
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DIS introduction
If you assume elastic scattering with a
constituent carrying a fraction of the proton
momentumW1(Q2, !) =
!
i
"dxf(xi)e2
iQ2
4xiM2"(! ! Q2
2Mxi) =
!
i
e2i fi(xB)
12M
W2(Q2, !) =!
i
"dxf(xi)e2
i "(! !Q2
2Mxi) =
!
i
e2i fi(xB)
xB
!
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DIS introduction
If you assume elastic scattering with a
constituent carrying a fraction of the proton
momentum
F1(x) = MW1(Q2, !) =12
!
i
e2i fi(x)
F2(x) = !W1(Q2, !) =12
!
i
e2i xfi(x)
Structure functions redefined!
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Bjorken scaling
So, assuming that there are constituents of spin 1/2 and that the scattering is elastic on them, the structure
functions should only depend on Bjorken-x (not on or independently) Q2 !
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Callan-Gross relation
Moreover one expects that
F2(x) = 2xF1(x)F1(x) = MW1(Q2, !) =
12
!
i
e2i fi(x)
F2(x) = !W1(Q2, !) =12
!
i
e2i xfi(x)
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DIS SLAC-MIT experiment
They actually expected rapidly falling structure functions as predicted by the uniform charge distribution assumption (Hofstadter, 1956)
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MIT-SLAC experimentThey found (a) a much milder behavior of the
structure function related part of the
cross section
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MIT-SLAC experiment
...and (b) that both structure functions obey
Bjorken scaling (they only depend on
Bjorken-x)
Friedman’s nobel lecture, RevModPhys.63.615
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Modern DIS data
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DIS with neutrina (charges of quarks)
F eP2 (x) =
!
i
e2i xf(x) = x
"49(u(x) + u(x)) +
19(d(x) + d(x))
#
F eN2 (x) =
!
i
e2i xf(x) = x
"49(d(x) + d(x)) +
19(u(x) + u(x))
#
F eCa2 (x) = x
518
!d(x) + d(x) + u(x) + u(x)
"
F!µCa2 (x) = x
!d(x) + d(x) + u(x) + u(x)
" } charge measurement
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Momentum sum rulesGluons
185
! 1
0dxF eCa
2 (x) =! 1
0dx(u(x) + d(x) + u(x) + d(x)) ! 0.5
The structure functions come from experiment. The sum over all quarks is less than one!
There are other particles inside the proton. Particles that don’t interact
electromagnetically or weakly!
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Scaling violations
Bjorken scaling is only approximate - early calculations showed that in any interacting field theory gross corrections appear to
all orders in perturbation theory.
“however, a mild violation of scaling would be possible in a special class of theories that are asymptotically free-
characterized by effective couplings that approach zero as the renormalization scale increases indefinitely. But, there was no
known example of such a theory at that time.”
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Summary
• Hadrons are composed of quarks
• Quarks are spin 1/2 particles
• They have a color degree of freedom
• The number of different colors is 3
• There is another particle in the hadrons that interacts only strongly