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# Quantum Chromodynamics and Strong Interaction

Quantum Chromodynamics and Strong

J.P. Ma, Institute for Theoretical , Academia Sinica, Beijing

2013.06. 14. 中国科技大学 Content:

1. Elementary Particles and

2. as Theory of Strong interaction

3. Predictions of QCD and Structure

4. Outlook

1. Elementary Particles and Interactions

Matter Structure: • Molecular: + Atoms • : Nuclei + • Nucleus: +

General Feature: • Non-relativistic bound state • Bound state with fixed constituents. • .

What are elementary constituents or elementary particles?? Answer: Elementary particles before 04. July 2012

Now Higgs 125GeV Size: Four fundamental interactions, (now fifth…….?) There are 4 types of interactions:

u Gravitation ( We don’t know its quantum-- theory version)

uWeak interaction ( Combined with QED, the )

u Electromagnetic interaction ( , QED)

uStrong interaction (A Part of this talk)

Interactions: The Standard Model(SM)

The main interactions are fixed by gauge , the gauge group is SU(3) SU(2) U(1) ⌦ ⌦

The gauge group for The product of two gauge strong interaction groups for . After symmetry-breaking, one gauge symmetry remains…. 2. Quantum Chromodynamics as Theory of Strong interaction

Quark field: it has three colors The gauge fields: 8 q 1 µ a,µ a q = q G = G T , 0 2 1 q3 With mass m @ A q The eight ½ Gellman The field strength tensor: matrices for SU(3)

Gµ = µG G + ig Gµ,G s  Massless. Nonabliean gauge field Yang-Mills theory

1 = Gµ G + qi¯ ⇥µ + ig Gµ q m qq¯ L 4 µ µ s q q X ✓ ◆ q = u, d, s, c, b, t gs The constant

Gauge Symmetry q(x) U(x)q(x),U(x):SU(3) phase, ! µ µ 1 µ 1 G (x) U(x)G (x)U (x)+ig U(x) U (x) ! s The Lagrangian is invariant under the transformations.

The phase can also depend on gauge fields…….

Total: 7 parameters……?? The masses: m m 5MeV,m 120MeV u, d, s: u ⇠ d ⇠ s ⇠

Heavy quarks: c, b and t, m 1.5GeV,m 4.8GeV,m 175GeV c ⇠ b ⇠ t ⇠

Proton mass: Mp = 938MeV

In high energy scattering involving light : E > 2 GeV one can neglect heavy quarks and masses of light quarks.

In this special case, QCD is of three massless quarks + massless

QCD has no dimensional parameter, but 2 dimensionless parameters.

Dimensional transmutation The two parameters:

The number of colors: Nc Assumed to be 3 g2 The coupling: (Q)= s s 4⇥ 1 Q: Energy scale Q ⇠ r The depends on the energy scale or distance.

Scale Dependence of the Strong Coupling

 D. Gross, D. Politzer, and F. Wilczek

found in 1973 that the momentum dependence of the strong coupling due to the vacuum effect is just opposite to that of the e & m coupling (β-function is negative!)

The strong interaction coupling gets smaller as the distance between quarks gets closer à ! For QCD one can define the beta-function: µ = Q ⇤ (µ) µ s = ⇥ ( (µ)) ⇤µ s 1 2 = ↵2b + (↵3),b= 11 N . s 0 O s 0 4⇡ 3 F 

NF is the number of quark flavors, it is 3--6. Additional fluctuation: g⇤ g⇤ + g⇤ g⇤ ! ! The beta –function is negative if NF is not large: 4⇡ µ ↵ (µ) 0 ↵s(µ)= 2 2 s b0 ln(µ /⇤ ) !1 !

An intrinsic scale Asymptotic Freedom vs Experiment:

↵ (M )=0.1184 0.0007 s Z ± Three Classes of Problems in QCD:

A: Only short-distance effect, totally perturbative ( fewer) E.g., R-Ratio at large Energy,…

B: Short-distance effect combined long-distance effect, perturbative theory can be used if one can prove factorization theorems. E.g., Scattering with large momentum transfer(s)……

C: Only long-distance effect, perturbative theory is useless. E.g., mass………………..

E.g., one problem of the 3. class: 牛顿第零定律 M(3樱桃)=3×M(樱桃)

With Gell-Man’s classification: Proton: 2 u-”quarks” + 1 d-”quark”, (uud)

m m 5MeV,m 120MeV u ⇠ d ⇠ s ⇠ Mp = 938MeV

In fact, this law is no correct for proton or hadrons!!

With all these experiments QCD as a theory of strong interaction is well tested! Further, we learn:

Proton, and all hadrons are bound states of partons(gluons, quarks) and:

• All partons move relativistic! （quark models??)

• The number of partons is not fixed because . (Gellman’s classification is only by keeping minimum numbers of partons, by quantum numbers.)

• No free parton was found! “Quark confinement”??

In comparison with quark models ……..

3. Predictions of QCD

A detailed question: How to calculate amplitudes of hadron scattering or cross-section from perturbative QCD

Perturbation theory of quantum field theories

In general, it is divergent because of summation of all quantum fluctuation.

Ultraviolet divergence: The large momentum of fluctuation,

QCD is a perfect theory: Large momentum è Asymptotic freedom

 How to deal with these divergences when only hadrons in initial- or final state??

 We do not know the structure of hadrons. How to calculate their scattering??

Physical interpretation of parton distributions:

fa/H (x, µ),a= q, G

The probability of finding in the hadron H a parton with the momentum fraction x.

It contains bound state effects, nonperturbative effects.

It is well-defined in QCD with QCD operators.

It is universal……

One can extract pdf’s from experimental data…

One can prove the factorization theorem for DIS: F (x, Q2)=xC f + xC f + , ⇠ 2 q ⌦ q/P g ⌦ g/P ··· The functions C’s can be calculated without any soft divergence.

But we still do not know the structure function or the cross-section because unknown pdf’s

What is predicted from QCD ??? if QCD is right, one can extract pdf’s—information about the structure of the hadron from measured structure functions !!

The true predictions: The Q-dependence or mu-dependence:

F2 depends on Q:

Ca(⇠,Q,µ),fa/P (z,µ),a= q, g, ... Q2 ln Taking µ = Q to eliminate log’s µ2 F2 does not depend on µ Famous DGLAP equation: @f (x, µ) ↵ q/P = s P f + P f , @ ln µ2 2⇡ qq ⌦ q/P gq ⌦ g/P @fg/P (x, µ) ↵ = s P f + P f , @ ln µ2 2⇡ gg ⌦ g/P qg ⌦ q/P Once we know pdf’s at one scale, we also know pdf’s at another scale.

Q-Dependence: µ = Q F (x, Q2)=xC (⇠,Q,Q) f (z,Q)+ 2 q ⌦ q/P ··· The Q-dependence is determined by DGLAP!! It can be tested. There is a Q-dependence even at tree-level:

F2(x, Q)=xfq/P (x, Q) In the early days

Momentum sum rule:

dxxfa/H (x)+ dxxfg/H(x)=1 a=q,q¯ X Z Z dxxf (x) 0.5 Exp.: a/H ⇡ a=q,q¯ X Z

Evidence of the existence of gluons

Q2 -dependence of the structure function HERA experiment VS theory: Modern days of QCD E.g., One-dimensional structure of proton For polarized case, one can also establish a collinear factorization and extract from experiment the polarized pdf’s.

There are interesting and unsolved problems….

The “crisis” of proton :

A new research area: Spin physics The effects of orbital angular momenta of partons? Various factorization theorems can be established or can not be. For processes involving large momentum transfers one may expect factorizations. Well studied cases: + hA + hB ` + ` + X + ! e + e h + X ! h + h A + X A B ! hA + hB + X ……… ! Pdf’s are universal…….

Given the fact that QCD is a correct theory, hadron structure??

How to extract information of hadron structure??

Collinear factorization: one-dimensional distribution.

Physics of Semi-Inclusive DIS

k'

k q Ph X

P • momentum q is in the Bjorken limit. • Final state hadron h can be characterized by fraction of parton momentum z and transverse

momentum Ph┴ Three cases for measured Ph┴

A. Ph ┴ ~ Q : Ph┴ generated from QCD hard scattering, factorization theorem exists. (Standard collinear factorization)

B. Q >> Ph┴ >>ΛQCD : Still perturbative, but resummation is needed. It is important for many processes.

C. Ph┴ ~ΛQCD Nonperturbative! Ph┴ is generated from partons inside of hadrons. It gives a possible way to learn 3-dimensional structure of hadrons!!!!!

A factorization theorem is needed for the case Ph┴ ~ΛQCD ! The TMD factorization for SIDIS: (Unpolarized case) d⇥ F (x, z, Q, q ) d2⌅k d2p⌅ d2⌅⇤ ⇠ ? ⇠ ? ? ? Z 2 q(x, p )ˆq(z,p )S(⇤ )H(Q) (z⌅k + p⌅ ⌅⇤ P⌅h ) ? ? ? ? ? ? ?

TMD parton distribution, 3-dimensional

TMD parton fragmentation function

A soft-factor

The perturbative part

With the help of the TMD factorization, one is able to extract from experiment the 3-dim. structure of a hadron. H = proton: (uncompleted list)

Nucleon Unpol. Long. Trans. Quark

Unpol. q(x, k┴) qT(x, k┴)

Long. ΔqL(x, k┴) ΔqT(x, k┴)

δqT(x, k┴) Trans. δq(x, k┴) δqL(x, k┴) δqT'(x, k┴)

They are intensively studied in experiments Lattice QCD:

Path integral: µ i d4x = [dq][dq¯][dG ]exp LQCD ZQCD Z R A functional integral, complex number

Formulate it in Euclidean space by x ix 0 ! 0 µ d4x = [dq][dq¯][dG ]exp LQCD ZQCD Z R Discretizing the space-time as a lattice, one can numerically calculate the path integral, hence solve QCD Still a long way to go…… Summary:

Only a small part of QCD is touched here!

“QCD is our most perfect physical theory” and its scope is wide: 1. 2. Accelerator physics 3. Cosmology 4. Extreme astrophysics 5. and nature philosophy 6. Condense physics of QCD …………………………………