1. Adnan Bashir, UMSNH, Mexico
QCD: A BRIDGE BETWEEN
PARTONS & HADRONS
Adnan Bashir, UMSNH, Mexico
November 2017
CINVESTAV

2. Hadron Physics & QCD Part 2: From Quarks to QCD: QCD as a gauge theory
Feynman rules
Higher orders and infinities
Regularization
Renormalization schemes
One loop renormalization
Anomalous dimensions
Running coupling
Beta function
Asymptotic freedom

3. Contents QED – The Bare Lagrangian Renormalization Constants
Electromagnetic Coupling
The Photon Propagator
Photon Propagator at One Loop
RGE and Anomalous Dimensions
Electron Propagator at One Loop
The Vertex and the Charge Renormalization
QED Running Coupling
QCD Running Coupling
What Next?

4. QED – The Infinities

5. QED – The Infinities

6. QED – The Bare Lagrangian
Let us start with the QED Lagrangian:

7. Quantum Electrodynamics
Employing the QED Lagrangian, when we compute physical
observables beyond tree level, they come out infinite.
However, QED is a renormalizable theory. We can add
counter terms of the same form as present in the original
Lagrangian to come up with a new Lagrangian whose
predictions are consistent with experimental results.
As the counter terms are of the same form, it is straight
forward to add them to the original Lagrangian to get the
modified Lagrangian whose coefficients are infinitely large,
namely the bare Lagrangian LB.

8. Quantum Electrodynamics
Thus we can write the bare QED Lagrangian as:
Explicitly, bare QED Lagrangian is:

9. Renormalization Constants
Quantities in this bare Lagrangian are connected to the
renormalized quantities of the original Lagrangian through
infinite multiplicative renormalization constants for each
term:
This implies the certain relations between bare and the
renormalized quantities.

10. Renormalization Constants
It implies:

11. Renormalization Constants
Use the relations:
Define:

12. Renormalization Constants
Now compare the relations:

13. Renormalization Constants
The coefficient of each term being unity implies:

14. Electromagnetic Coupling
Renormalization constants have the structure:
We must define the renormalized coupling in such a way
that it remains dimensionless in dimension d.
The action is dimensionless, because it appears in the
exponent in the Feynman path integral. The action is an
integral of L over d-dimensional space-time.
Therefore, the dimensionality of the Lagrangian L is [L]=d
(in mass units).
Show that: (d=4-2)

15. Electromagnetic Coupling
In order to define a dimensionless coupling, we have to
introduce a parameter  with the dimensionality of mass
(called the renormalization scale).
Thus in the
MS scheme:
where  is the Euler constant.
Use:
Thus inversely:
Thus a physical quantity is first expressed in terms of the
bare coupling and then expressed in terms of the
renormalized coupling.

16. QED Feynman Rules Massless QED: Warning!!! Notation
Additional Feynman rules (loops):
Traceology:
A (-1) for every fermion loop.
A trace for the closed
fermion loop.
Integration over undetermined
loop momentum ddk/(2)d

17. The Photon Propagator The photon propagator has the structure:
where the photon self-energy i(p) (denoted by a shaded
blob) is the sum of all one-particle-irreducible diagrams
(diagrams which cannot be cut into two disconnected pieces
by cutting a single photon line), not including the external
photon propagators.

18. The Photon Propagator The series can also be rewritten as:
Thus the inverse of the photon propagator is:
The Ward identity reads:
Thus the general form of the photon self energy is:
Therefore:

19. The Photon Propagator
Thus the longitudinal part of the full propagator gets no
corrections, to all orders of perturbation theory.
As the photon propagator involves the product of two
photon field vectors, full bare propagator is related to the
renormalized one by (watch out for notation!):
Thus Ward identity implies:

20. Photon Propagator at One Loop
Let us start with one
loop photon propagator:
Hence the self energy
can be written as:
Contract with g, take
trace and simplify:

21. Photon Propagator at One Loop
Hence:
Therefore,
bare one loop
photon propagator is:
Thus:
It requires:
rendering:

22. Electron Self Energy & Vertex at one Loop

23. The  Function of QED
Recall the relation between renormalized and bare charge
in the Msbar scheme:
Keeping the –dependent quantities on one-side (the right
hand side), we can rearrange the above expression as:
Taking the log of both sides:

24. The  Function of QED
Taking the derivative with respect to log , we have:
The -function of QED is defined as:
We thus have:

25. The  Function of QED The last equation can be written as:
We can rearrange this equation as follows:
We are interested in it till order :

26. The  Function of QED
The knowledge of Z to one-loop allows us to write:
With the appropriate differentiation:
This enables us to evaluate the 1-loop -function:

27. The  Function of QED Hence the –function is given by:
Therefore the QED –function to 1-loop is:

28. The Running Coupling in QED
Let us start again from the equation:
And work in the limit   0:
It can be re-written as (inserting expansion of -function):

29. The Running Coupling in QED
The last equation can be simplified as follows:
Or further rearranged as:
The solution to this equation can be written as:

30. The Running Coupling in QED
The last equation can be simplified as follows:
The inverse of this equation is:
The running coupling of QED is:

31. The Running Coupling in QED
The running coupling could also be written as:
Inserting the calculated value of 0 (-4/3):
In another set
of variables:

32. The Running Coupling in QED
One loop running
coupling in QED
Recall the
expansion:

33. Running Coupling in QED
Thus:

34. Running Coupling in QED
Landau Pole:

35. Running Coupling in QED
Experimental Measurement:

36. Running Coupling in QCD

37. Running Coupling in QCD
Competition between color and flavor:

38. Running Coupling in QCD

39. Running Coupling in QCD
Asymptotic Freedom:

40. Running Coupling in QCD

41. CONFINEMENT ONE MILLION DOLLARS
Confinement in QCD
CONFINEMENT
ONE MILLION DOLLARS

42. What Next? How can we study hadron physics starting from QCD?
What are the fundamental equations of QCD which can
study all momentum regimes of the coupling?
How can we make connections with the experimental
results of the existing hadron physics facilities?
What is the current status of these studies?