1. QED and QCD, an introduction.
Bilal Masud
Centre for High Energy Physics
Punjab University.
8th LHC School, August 2019, NCP, Islamabad.

2. Outline Nuclear force and the exchanged mesons.
Spins half and one: QED and its Feynman diagrams.
Renormalization of QED: electric charge is variable.
QED’s explanation of electron proton inelastic scattering: Bjorken scaling.
QCD correction to above: Feynman diagram with quark-quark-gluon vertex
Colour charge and its bigger variations; Feynman diagrams become useless.
Non-perturbative QCD: lattice gauge theory.
Potential Model, string model and the constituent quark mass.
Integral equations of QCD & continuum models also give quark mass. Predictions for hadron masses and decays, etc.
Slide no

3. Nuclear Force
Nuclear force (also known as nuclear interaction or strong force) is the force that binds protons and neutrons.
Powerfully attractive at distances of about 1 femtometre (fm, or 1.0×10−15 meters).
Rapidly decreases at distances beyond about 2.5 fm.
At less than 0.7 fm, the nuclear force becomes repulsive.
Maltman and Isgur, Phys. Rev. D29 (1984) 952
Ruprecht Machleidt (2014), Scholarpedia, 9(1):30710.

4. Fourier Transform; Propagator
𝑉 𝑥 = 𝐺 𝑥,𝑦 𝑠(𝑦)𝑑𝑦
The propagators are Green's functions for the Klein-Gordan equation : functions G(x, y) satisfying
□ 𝑧 + 𝑚 2 𝐺 𝑥,𝑦 =−𝛿 𝑥−𝑦
Kinetic energy and rest mass energy operators.
Natural units:
𝑝=ħ𝑘 written as 𝑝=𝑘, etc.
−𝑝 2 +𝑚 2 = −𝐸 2 + 𝒑 2 +𝑚 2

5. Exchanged Particles
Each integrated momentum 𝒌 or 𝐩 is of a plane wave 𝑁 𝑒 𝑖(𝒌.𝒙−ω𝑡) .
In the Yukawa case, it is a meson with ∆ 𝑭 𝒌 being the weightage of the exchanged meson with momentum 𝒌.

6. The field and the exchanged photon between electric charges
No 𝑘 2 − 𝑚 2 . Why?

7. Quantizations of currents: spin half.
Before quantization: the sources 𝑠 𝜇 or currents 𝑗 𝜇 are a collection of ordinary charge and current densities.
First quantization: charges are integer multiples of electron’s which obey relativistic quantum Dirac equation.
𝑗 𝜇 =𝑖𝑒 𝜓 𝛾 𝜇 𝜓,
Second Quantization: Fourier coefficients c 𝒑 and 𝑑 † 𝒑 in the Fourier series for the Dirac wave function
become lowering operator for electron number and raising for positron. In predictions, momentum dependence in spinors 𝑢 𝒑 and 𝑣 𝒑 .
𝑗 0 =𝜌=constant 𝜓 ∗ 𝜓
𝜓 𝑥 = 𝑝 𝑐 𝐩 𝑢 𝐩 𝑒 −𝑖𝑝.𝑥 + 𝑑 † 𝐩 𝑣 𝐩 𝑒 𝑖𝑝.𝑥

8. Quantizations of Electromagnetic field: Spin 1 photons
No quantization, e.m. energy 𝐸=𝑝𝑐. Why not 𝐸 2 = 𝑝 2 𝑐 2 − 𝑚 2 𝑐 4 ?
First quantization: operators 𝑖 ħ 𝜕 𝜕𝑡 and −𝑖 ħ𝜵 : 𝜕 2 𝜕𝑡 2 𝑬= 𝜵 𝟐 𝑬. What is this equation and where is wave function?
Second quantization: Coefficients 𝑎 𝒌 in the Fourier series for the related scalar and vector potentials 𝐴 0 𝑥 =𝜑(𝑥) and 𝑨 𝑥 i.e.
𝐴 𝜇 𝑥 = 𝑘 … 𝜀 𝜇 𝒌 [𝑎 𝒌 𝑒 −𝑖𝑘.𝑥 + 𝑎 † 𝒌 𝑒 𝑖𝑘.𝑥 ]
become raising and lowering operators for the number of spin 1 photons. In predictions the momentum dependence is in the polarization vectors 𝜀 𝜇 𝒌 .

9. Think/Assignment: Raising and Lower Operators
How many (and of which kind) raising and lowering operators would model
Compton scattering by electron, Compton scattering by positron, pair annihilation, pair creation electron-electron scattering, positron-positron scattering, electron-positron scattering electron and positron self energies, photon self energy?

10. QED Hamiltonian and Lagrangian
Ket in the middle side
𝑗 𝜇 =𝑖𝑒 𝜓 𝛾 𝜇 𝜓

11. If “Theory of Everything”, amplitude for electron-electron scattering is infinite!
𝑞+𝑝 𝑞
= lim 𝑀→∞ 0 𝑀 𝑑 4 𝑞 𝑓(𝑞)
𝑒 𝑅 = 𝑒 𝑅 𝑒, 𝑝 2 ,𝑀 , with an infinity in this expression.
Comprise: Renormalization:
Take 𝑒 𝑅 from one experiment by relating it to, say, 𝑒 𝑒 scattering
Solve for 𝑒 𝑅 : the measured cross section =𝑓 𝑒 𝑅 .
Then predict for other quantities.

12. Prediction: Running Electric Charge
Horizontal?

13. Quark Model
Starting 1965: Static model of mesons and baryons (like protons and neutrons)
Include
With
explained as

14. 3 same-flavor Fermions not identical due to color charge (Greenberg)
A quark of any flavor (up, down, strange, charm, …) can have three kinds of charges, called red, blue, green only because these three charges themselves can form color neutral (singlet) combinations, like color-anticolour.

15. Modern Perception of Nuclear Force
The nuclear force is a residual effect of the more fundamental strong force, or strong interaction that binds quarks together.
Analogous to the short-range forces between neutral atoms or molecules: London forces or instantaneous dipole–induced dipole forces.
Similarly, nucleons are "color neutral“. But some combinations of quarks and gluons leak away; short-range nuclear force Quark Interchange:
Maltman and Isgur, Phys. Rev. D29 (1984) 952

16. Explanation of electron-proton scattering by QED: Parton model.

17. Educational Background: Electron Muon Scattering
𝑆 𝑓𝑖 = 𝛿 𝑓𝑖 + 2𝜋 4 𝛿 (4) 𝑝 𝑓 ′ − 𝑝 𝑖 𝑖 𝑉 𝐸 𝑖 1/2 𝑓 𝑉 𝐸 𝑓 ′ 1/2 𝑙 (2 𝑚 𝑙 ) 1/2 ℳ
𝑑𝜎= 𝑙𝑖𝑚 𝑇→∞ 𝑆 𝑓𝑖 2 𝑇 𝑉 𝜐 𝑟𝑒𝑙 𝑓 𝑉 𝑑 3 𝑝 𝑓 ′ 2𝜋 3
= 1 4𝑀𝐸 𝑑 3 𝑘 ′ (2𝜋) 3 2 𝐸 ′ 𝑑 3 𝑃 ′ 2𝜋 3 2 𝑃 0 ′ 𝑒 4 𝑞 𝐿 𝑒 ) 𝜇𝜈 𝐿 𝜇𝜈 𝑚𝑢𝑜𝑛 (2𝜋) 4 𝛿 (4) ( 𝑃+𝑞− 𝑃 ′ ).
6 Integrations, two Dirac deltas. Net: two integrations.
𝑑𝜎 𝑑𝐸 ′ 𝑑Ω 𝑒𝜇→𝑒𝜇 = 4 𝛼 2 𝐸 ′ 2 𝑞 𝐶𝑜𝑠 2 𝜃 2 − 𝑞 2 2 𝑚 2 𝑆𝑖𝑛 2 𝜃 2 𝛿 𝜐+ 𝑞 2 2𝑚 ,
with 𝜈= 𝑃.𝑞 𝑀 (= 𝑃 0 𝑞 0 𝑀 = 𝑘′ 0 − 𝑘 0 =E’-E in the rest frame of target muon).

18. Inelastic electron-proton scattering
𝑑𝜎 𝑑𝐸 ′ 𝑑Ω 𝑒𝑝→𝑒𝑋 = 4 𝛼 2 𝐸 ′ 2 𝑞 4 𝑊 2 𝜈, 𝑞 2 𝑐𝑜𝑠 2 𝜃 2 +2 𝑊 1 𝜈, 𝑞 2 𝑠𝑖𝑛 2 𝜃 2
No 𝛿 4 𝑝 2 − 𝑚 constraint.
Inelastic form factors W’s are functions of two scalar variables that can be constructed from p and q. Chosen to be 𝑞 2 and 𝜈= 𝑝.𝑞 𝑀 .

19. The Parton Model Limit: If the struck particle has a rest mass…
If the scattered particle (other than electron) has some rest mass m that is apparently unknown,
2 𝑊 1 𝑝𝑜𝑖𝑛𝑡 (ν, 𝑄 2 )= 𝑄 2 2 𝑚 2 δ 𝜈− 𝑄 2 2𝑚 = 1 𝑀 1 ω 𝑥 2 δ 1− 1 𝑥ω
𝑊 2 𝑝𝑜𝑖𝑛𝑡 (ν, 𝑄 2 )=δ 𝜈− 𝑄 2 2𝑚 = 1 𝜈 δ 1− 1 𝑥ω
𝑄 2 ≡− 𝑞 2 , ω≡ 2𝑞.𝑝 𝑄 2 = 2𝑀𝜈 𝑄 2 and (𝑑𝑒𝑓𝑖𝑛𝑡𝑖𝑜𝑛 x≡ 𝑚 𝑀 ).
But then this constraint 𝑖𝑡𝑠𝑒𝑙𝑓 would tell an experimentalist
the value of mass 𝑚 through the equation
𝑚 𝑀 =𝑥= 1 𝜔 = 𝑄 2 2𝑀𝜈
because 𝑀 is for proton and 𝑞= 𝑘 ′ −𝑘 and 𝜈=E’−E are experimentally knowable through electron.

20. Structure Functions and Momentum Distributions
For Bjorken scaling 𝜈𝑊 2 𝑝𝑜𝑖𝑛𝑡 (ν, 𝑄 2 )≡ 𝐹 2 𝑥 ≡ 𝑖 𝑒 𝑖 2 𝑥 𝑓 𝑖 𝑥
and M 𝑊 1 𝑝𝑜𝑖𝑛𝑡 (ν, 𝑄 2 )≡ 𝐹 1 𝑥 .
𝐹 1 𝑥 and 𝐹 2 𝑥 are dimensionless structure functions of proton.
𝑓 𝑖 𝑥 are parton momentum distributions.
𝑓 𝑖 𝑥 is the probability density 𝑑 𝑃 𝑖 𝑑𝑥 that the struck parton i carries a fraction x of the proton’s four momentum p.
𝑖 ′ 𝑑𝑥 𝑥 𝑓 𝑖 ′ 𝑥 =1.

21. Only QED!

22. Bjorken Scaling (a Parton model, QED result) is verified but only approximately

23. Writing in terms of “cross-sections”
𝜎 𝑡𝑜𝑡 𝛾𝑝→𝑋 = 4 𝜋 2 𝛼 𝐾 𝜀 𝜇∗ 𝜀 ν 𝑊 𝜇ν
with 𝑊 𝜇ν being the hadronic tensor in 𝑑𝜎∼ 𝐿 𝜇𝜈 𝑒 𝑊 𝜇𝜈 . For real photons, sum over two transverse polarizations of the incident photon.
𝜎 𝜆 𝑡𝑜𝑡 = 4 𝜋 2 𝛼 𝐾 𝜀 𝜆 𝜇∗ 𝜀 𝜆 𝜈 𝑊 𝜇ν for virtual photons as well.
𝜎 𝑇 = 𝜎 + 𝑡𝑜𝑡 + 𝜎 − 𝑡𝑜𝑡 = 4 𝜋 2 𝛼 𝐾 𝑊 1 𝜈, 𝑄 2

24. Modification by QCD
Changing
to “Compton Scattering”

25. Modification to Virtual (Total) Cross-Sections by Gluon Scattering
𝐹 2 (𝑥, 𝑄 2 ) 𝑥 = 𝜎 𝑇 (𝑥, 𝑄 2 ) 𝜎 0 = 𝑖 𝑒 𝑖 2 𝑥 1 𝑑𝑦 𝑦 𝑓 𝑖 𝑦 𝛿 1− 𝑥 𝑦 + 𝛼 𝑠 2𝜋 𝑃 𝑞𝑞 𝑥 𝑦 log 𝑄 2 𝜇 2 . This makes the structure function 𝐹 2 (𝑥, 𝑄 2 ) 𝑥 changing with 𝑄 2 even for a given 𝑥. Bjorken scaling violated by QCD effects.

26. Only QED!
Here, we need QCD!

27. QCD was used in above… 𝛼 𝛼 𝑠 4 3 To replace 𝛼 2 by
𝛼 𝛼 𝑠 4 3
To replace
𝛼 2 by
This means: a possibly different value of charge and different number of charges.

28. Electric Field and Color (Chromo) Fields.
The electric field spreads.
The color field glues and hence
forms flux tubes.
Each term in the Fourier Series
for the electric field is an
(exchanged or mediating) photon.
Similarly, the color field is
composed of gluons.
The string model

29. QCD Lagrangian; AAA and AAAA terms

30. Some QCD Feynman Diagrams
𝑝 1
𝑝 2
k = 𝒑 𝟐 − 𝒑 𝟏
𝑝 3
𝑝 4

31. There are also particles (gluons) interchanged between quarks inside a nucleon.
Small and large mass values: current and consitutent masses
Blue dots in the second figure are the valence
quarks that give a hadron its quantum numbers.
Note: Loops give infinity results managed by
renormalization, as in QED.

32. QCD; problems with the Perturbative expansion
The color charge g and its square increases for lower momentum transfers eventually making theory unsolvable for the usual hadronic physics of spectroscopy and decays. 𝛼 𝑟

33. Summing all terms? The perturbative series: can be summed up as: If
Exponential and its derivative infinite series….differentiate and it is related this way to itself. Not needed in actual QED. But to explore…. If

34. The (infinite) sum is a function of itself.
Expanding the last blob, we can draw:
This series can be written in terms of itself:
Make this series can be written TILL THE END as one second animation
com;pre with a differetial the 3rd
Compare with a differential equation: 𝑦=𝑓(𝑥) is written in terms of its derivative: e.g.
𝑦= 𝑑𝑦 𝑑𝑥

35. Integral Equations of QCD: A Schwinger-Dyson Equation
q=p-k p p k
𝑆 −1 𝑝 =𝑖𝛾.𝑝+ 𝑚 0 − 𝑑𝑘 𝜋 4 𝑔 0 2 𝐷 𝜇𝜈 (𝑞) 𝜆 𝑎 2 𝛾 𝜇 𝑆(𝑘) Γ 𝜈 𝑎 (𝑘,𝑝)
Make a QED eq pics
The simplest new
Compare: Mathematical definition of an integral equation. An example
𝑦 𝑥 = 𝑦 𝑑𝑥 𝑦(𝑥)
The simple integral equation can be solved analytically (what is Solution here?), but Schwinger-Dyson equation is not that easy.

36. Schwinger-Dyson Equations (QED)

37. Truncation needed; Constraints on it
Two point one-particle irreducible (1PI) Green's functions (propagators) are related to three point 1PI Green's functions (vertices) and three point Green's functions to the four-point Green's functions scattering kernel and so on….
Truncation by introducing model(s) of some Green’s functions that incorporate, as far as possible, the effects of all the higher point Green's functions.
The models should not contradict perturbative QED for high momentum transfers, etc.
Truncation new

38. How to find the mass function M 𝑝 2 ?
Taking the trace of
𝑆 −1 𝑝 =𝑖𝛾.𝑝+ 𝑚 0 − 𝑑𝑘 𝜋 4 𝑔 0 2 𝐷 𝜇𝜈 (𝑞) 𝜆 𝑎 2 𝛾 𝜇 𝑆(𝑘) Γ 𝜈 𝑎 (𝑘,𝑝)
and noting that gamma matrices 𝛾 𝜇 have zero trace, we get, with
some appropriate truncation,
If w new
For m=0, this is, after discretization, a Column=Matrix*Column
𝑀 𝑝 = 𝑘 𝑊 𝑝𝑘 𝑀 𝑘 or 𝑘 (𝐼−𝑊) 𝑝𝑘 𝑀 𝑘 =0 or (I−W)M=0.
If W was independent of k, this can have a non−trivial solution if det(I−W)=0. But above is a non-linear problem, with W a function of k.

39. The Quark Mass 𝑘 (𝐼−𝑊) 𝑝𝑘 𝑀 𝑘 =0
𝑘 (𝐼−𝑊) 𝑝𝑘 𝑀 𝑘 =0
is a set of as many equations as are values of p. Simultaneous equations can be solved.
Now, 𝑝 2 = rest mass square.
Changing 𝑚 0 to this intersection is the dynamical mass generation.

40. Yellow in 1st animation, rest in the seccond

41. Path Integrals..
Derivation:
Using

42. Classical and Quantum Paths

43. Statistical Mechanics
Path Integrals
Statistical Mechanics
𝑍= 𝑖 𝑒 −𝛽 𝐸 𝑖 .
Path integral 𝑘 𝑑 𝑥 𝑘 𝑒 𝑖𝑆 ,
< 𝑥 𝑖 𝑥 𝑗 > = 1 𝑍 𝑘 𝑑 𝑥 𝑘 𝑥 𝑖 𝑥 𝑗 𝑒 𝑖𝑆 .
correlation <𝐴𝐵> = 𝑖𝑗 𝐴 𝑖 𝐵 𝑗 𝑃 𝑖𝑗 = 𝑖𝑗 𝐴 𝑖 𝐵 𝑗 𝑒 −𝛽 𝐸 𝑖𝑗
In scalar F.T. n-Point Green Function; 2 Point is propagator, 3 Point is vertex. Vacuum expectation value= 𝑛 𝑎 𝑛 𝑃 𝑛 .
In QCD, scalar field is replaced by quark field and the gluonic field

44. Dressed/Full/Interacting Propagator
replacing the free action 𝑆 0 by full interacting action S gives
Ω 𝑇 𝜓( 𝑥 1 ), 𝜓 ( 𝑥 2 ) Ω ≡
Expectation value in the ground state of full Hamiltonian H, not only free part 𝐻 0

45. Lattice Gauge Theory: Path Integral over group elements
In lattice QCD, a pure gluonic Euclidean (imaginary time) path integral (using adiabatic approximation) with is an expectation value. Imaginary time; Wick’s rotation (of integration limits.
The strong coupling expansion in inverse coupling 𝛽,through bare.
to get an agreement with an integral of 𝐹 𝜇ν 𝐹 𝜇ν .

46. Confinement “derived” in the strong coupling limit.
Quantum Mechanics (“Quarks, Gluons and Lattices” by Creutz)
If the operator is taken to be U’s on the perimeter of a rectangle of width as separation 𝑅 and height as time 𝑇, the expectation value is Wilson Loop
In the strong coupling limit, Energy is proportional to the separation.

47. But in the continuum limit of zero lattice spacing , bare inverse coupling 𝛽 is not small; we need a calculational procedure valid for all 𝛽.
Lattice numerical evaluation of the path integral
with importance sampling of this sharply peaked integrand, can be performed for all values of g and energies:
(A set of) random number(s)> 𝑒 −𝑆 is rejected, meaning that probability of acceptance (and processing) is 𝑒 −𝑆 .
Then take a simple average 𝑖 𝐴 𝑖 𝐶 𝑖 and not 𝑖𝑗 𝐴 𝑖 𝐵 𝑗 𝑒 −𝛽 𝐸 𝑖𝑗 .

48. Numerical Simulations of QCD
Energy of the ground state color field vs 𝑞 𝑞 separation
Gluonic ground state energy (black curve) fitted by optimum values of the quark potential model parameters , b and σ.
Quark potential model extension to hybrids (mesons with excited gluonic field).
Change bracketed a with alpha
Energy of the excited state
color field vs 𝑞 𝑞 separation
Bilal Masud (CHEP, PU)

49. Quarks and gluons remain inside hadrons: