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

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

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?

QED – The Infinities

QED – The Infinities

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

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.

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

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.

Renormalization Constants It implies:

Renormalization Constants Use the relations: Define:

Renormalization Constants Now compare the relations:

Renormalization Constants The coefficient of each term being unity implies:

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)

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.

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

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.

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:

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:

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:

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

Electron Self Energy & Vertex at one Loop

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:

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

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 :

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:

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

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):

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:

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:

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:

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

Running Coupling in QED Thus:

Running Coupling in QED Landau Pole:

Running Coupling in QED Experimental Measurement:

Running Coupling in QCD

Running Coupling in QCD Competition between color and flavor:

Running Coupling in QCD

Running Coupling in QCD Asymptotic Freedom:

Running Coupling in QCD

CONFINEMENT ONE MILLION DOLLARS Confinement in QCD CONFINEMENT ONE MILLION DOLLARS

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?