Basics of Quantum Chromodynamics (Two Lectures)

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Basics of Quantum Chromodynamics (Two lectures)

Faisal Akram

5th School on LHC Physics, 15-26 August 2016 NCP Islamabad 2015 Outlines

• A brief introduction of QCD Classical QCD Lagrangian Quantization Green functions of QCD and SDE’s

• Perturbative QCD Perturbative calculation of QCD Green functions Feynman Rules of QCD Renormalization Running of QCD coupling (Asymptotic freedom)

• Non-Perturbative QCD Confinement QCD phase transition Dynamical breaking of chiral symmetry Elementary Particle Physics Today

Elementary particles: Quarks Leptons Gauge Bosons Higgs Boson (can interact through (cannot interact through (mediate interactions) (impart mass to the elementary strong interaction) strong interaction) particles) 푢 푐 푡 Meson: 푠 = 0,1,2 … Quarks: Hadrons: 1 3 푑 푠 푏 Baryons: 푠 = , , … 2 2

푣푒 푣휇 푣휏 Meson: Baryon Leptons: 푒 휇 휏

Gauge Bosons: 훾, 푊±, 푍0, and 8 gluons

399 Mesons, 574 Baryons have been discovered Higgs Bosons: 퐻 (God/Mother particle)

• Strong interaction (Quantum Chromodynamics) • Electro-weak interaction (Quantum electro-flavor dynamics) The Standard Model

Explaining the properties of the Hadrons in terms of QCD’s fundamental degrees of freedom is the Problem laying at the forefront of Hadronic physics. How these elementary particles and the SM is discovered?

1. Scattering Experiments Cross sections, Decay Constants, 2. Decay Processes Masses, Spin , Couplings, Form factors, etc. 3. Study of bound states Models

Particle Accelerators Models and detectors

Experimental Particle Physics Theoretical Particle Particle Physicist Phenomenologist Physicist

Measured values of Calculated values of physical observables ≈ physical observables

It doesn’t matter how beautifull your theory is, It is more important to have beauty It doesn’t matter how smart you are. If it doesn’t in one's equations than to have agree with experiment, it’s wrong… R.P. Feynmann them fit experiment..... P.A.M Dirac

Decades of observation and calculations show that the Standard Model of particle physics can describe almost every thing which we have observed in the labs of high energy physics. A Quick review of QFT’s • The standard model is a quantum field theory.

• In field theories, fields (defined by field functions) act as fundamental dof of the system.

• To every different kind of a particle we associate a field. (e.g., electron field, proton field, photon field etc)

• Particles appear as quanta of field.

Quantization Quanta of field has particle properties. 휙(푥)

• Equation of motion of the fields (푠 = 0) (푠 = 1) (푠 = 1) Free Scalar Field: Free abelian Vector Field: Fee Non-abelian Vector Field: 휇 휈 휈 휇 휇 2 휕 휕 퐴 + 휕 (휕 퐴 ) = 0 1 휇휈 (휕휇휕 + 푚 )휙(푥) = 0 휇 휇 ℒ = − 퐹 퐹 1 0 휇휈푎 푎 ℒ = 휕 휙†휕휇휙 − 푚2휙†휙 휇휈 4 0 휇 ℒ0 = − 퐹휇휈퐹 휇휈 휇 휈 휈 휇 휇 휈 4 퐹푎 = 휕 퐴푎 − 휕 퐴푎 + 푔0푓푎푏푐퐴푏퐴푐 (푠 = 1/2) 퐹휇휈 = 휕휇퐴휈 − 휕휈퐴휇 Free Dirac Field: 휇 푖훾 휕휇 − 푚 휓(푥) = 0 휇 ℒ0 = 푖휓 훾 휕휇휓 − 푚휓 휓 A Quick review of QFT’s

• Interaction is introduced by the coupling of fields .

휇 For example: ℒ퐼 = 푒휓 훾 휓퐴휇 for QED

휓 훾

휓

• Only Lorentz invariant couplings are allowed.

휇 휇 5 5 휇 ℒ퐼 = 휓 훾 휓퐴휇; 휓 훾 훾 휓퐴휇; 휓 휓휙; 휓 훾 휓휙; 휓 훾 휓휓 훾휇휓; . . . .

• In gauge theories coupling are further constrained by the gauge symmetries.

i) 푆푈 2 퐿 ⊗ 푈(1)푌 for electroweak interaction. ii) 푆푈(3)푐 for strong interaction. Quantum Chromo Dynamics (QCD) (Foundations)

• QCD is the theory of strong interaction of quarks, which is based on SU(3)c color symmetry.

• It assumes each flavor of quark comes in three different colors.

• The color states of quarks are SU(3) triplets.

푓 = 1,2,3 … 푁 푞 푞 푓 푓 푁 = 6 푞 Quark Fields: 휓퐶 퐶 = 푅, 퐺, 퐵 푓 푅 퐺 퐵 Free Lagrangian of the Quarks:

푁푓 푓 휇 푓 푓 ℒ0 = 휓퐶 푖훾 휕휇 − 푚0 휓퐶 푓=1 퐶=푅,퐺,퐵

The color states of quarks are SU(3)c triplets. 푁 휓푓 푓 푅 푓 휇 푓 푓 푓 푓 ℒ0 = 휓 푖훾 휕휇 − 푚0 휓 휓 = 휓퐺 푓 f iata f 푓=1 휓퐵    e  This Lagrangian is invariant under Where 푡푎 are Gell-Mann matrices. global SU(3)c gauge transformation.

[ta ,tb ]  ifabctc Free and Classical QCD Lagrangian Quantum Chromo Dynamics (QCD) (Foundations)

• Extending the global gauge symmetry to local symmetry. Local gauge transformation: f f f ia (x)ta f f    U      e   U     Global gauge transformation f f f    U    i( a )taU Local gauge transformation Free Lagrangian does not possess local symmetry:   D    ig t A (x) 푓 휇 푓 푓 푓 푓    0 a a ℒ0 = 휓 푖훾 휕휇휓 + 푚0 휓 휓 f f Such that D  U D 

1 i 1  ta Aa  Uta AaU   U U g0

푓 휇 푓 푓 푓 푓 ℒ = 휓 푖훾 퐷휇휓 + 푚0 휓 휓

푓 휇 푓 푓 푓 푓 푓 휇 푓 ℒ = 휓 푖훾 휕휇휓 + 푚0 휓 휓 + 푔0휓 훾 푡푎휓 퐴푎휇

[ta ,tb ]  ifabctc Including kinetic energy term of gauge fields 1 ℒ = 휓 푓푖훾휇휕 휓푓 + 푚푓휓 푓휓푓 + 푔 휓 푓훾휇푡 휓푓퐴 − 퐹 퐹휇휈 휇 0 0 푎 푎휇 4 푎휇휈 푎

휇휈 휇 휈 휈 휇 휇 휈 where, 퐹푎 = 휕 퐴푎 − 휕 퐴푎 + 푔0푓푎푏푐퐴푏퐴푐 , (푎 = 1, 2, . . . , 8) 26 fields = (6 flavors x 3 Color)+(8 gauge bosons); 7 parameter = 6 masses + 1 coupling

This complete Lagrangian is invariant under local SU(3)c gauge transformation. Quantum Chromo Dynamics (QCD) (Foundations) Interacting Classical QCD Lagrangian: 1 ℒ = 휓 푓푖훾휇휕 휓푓 + 푚푓휓 푓휓푓 + 푔 휓 푓훾휇푡 휓푓퐴 − 퐹 퐹휇휈 휇 0 0 푎 푎휇 4 휇휈푎 푎

휇휈 휇 휈 휈 휇 휇 휈 where, 퐹푎 = 휕 퐴푎 − 휕 퐴푎 + 푔0푓푎푏푐퐴푏퐴푐 , (푎 = 1, 2, . . . , 8)

1 1 1  F F    ( A   A )(  A   A )  g f ( A   A )A A 4 a a 4  a  a a a 2 0 abc  a  a b c g 2  0 f f A A A A . 4 abc ab'c' b c b' c'

• Quark-gluon interaction • 3-point gluon interaction • 4-point gluon interaction Quantum Chromo Dynamics (QCD) (Quantization)

Green Functions of QFT 퐺 푥1, 푥2, … , 푥푛 ≡ Ω 푇 휙 푥1 휙 푥2 … 휙 푥푛 Ω

1 −푖훿 −푖훿 −푖훿 퐺 푥1, 푥2, … , 푥푛 = . . . . 푍(퐽) 퐽=0 Cross sections 푍(0) 훿퐽(푥1) 훿퐽(푥2) 훿퐽 푥푛 Decay constants 푇 Bound State masses 푖 푑푡 ℒ+퐽(푥)휙(푥) Couplings Generating function: 푍(퐽) = lim 푑휙 푒 −푇 푡→∞(1−푖휖) Form factors

Generating function of QCD:

i dx4 (L  f  f  f f  J  A ) Z( ,, J )  d[A ]e  QCD a a   

The path integration over gauge fields diverges due to integrating over gauge equivalent configurations.

a a 1 ab b A (x)  A (x)  D  (x) g0

Fixing the gauge means choosing one configuration out of gauge equivalent configurations. Quantum Chromo Dynamics (QCD) (Quantization)

Gauge fixing implies i dx4 (L L L  f  f  f  A J      ) Z( ,, J , , )  d[A ]e  QCD GF FPG a a a a a a   

1  2  a a abc  a b c Not gauge invariant due LGF   (  Aa ) ; LFPG  (  )  g0 f (  ) A to gauge fixing. 20

The generating function Z must be gauge invariant because gauge transformation amounts to redefine the variables integration.

This trivial requirement of Gauge invariance of the generating function translates into non-trivial identities which all Green functions carrying gauge field(s) must satisfy.

QED: QCD: Wards-Green-Takahashi identities (WGTI) Slavnov Taylor identities (STI) 2 2 1 q 1 q q D  q q D  q     0 0 2 1 1 1 iq  (k, p)(1 b(k ))  (1 B(k, p))S (k) iq  (k, p)  S (k)  S ( p)  S 1( p)(1 B(k, p)) Green Functions of QCD

2-point Green functions:

Ω 푇 휓(푥1), 휓 (푥2) Ω ≡

푎 푏 Ω 푇 (퐴휇(푥1), 퐴휐(푥2) Ω ≡

푎 푏 Ω 푇 (푐 (푥1), 푐 (푥2) Ω ≡

3-Point Green functions: 4-Point Green functions: Schwinger-Dyson Equations (SDEs) of QCD

• In QFT the Green functions satisfy a set of exact mathematical equations called SDEs.

• Corresponding to each green function we have a SDE.

• These are exact equations and can be used to find Green functions perturbatively as well as non-perturbatively.

Schwinger-Dyson Equations of QCD

SDE of Quark Propagator: Non Perturbative Truncation: One or more Green functions are modeled subjecting to some general field theoretical constraints. Or SDE of Gluon Propagator: Using the results obtained from lQCD. + 5 terms

SDE of quark-gluon vertex:

+ 7 terms

Perturbative Truncation:

푁푓 Feynman Rules of QCD QCD full Lagrangian:

f  f f f f f  f 1  1  a 2 LB  i     m    g0  ta Aa (x)  Fa Fa  ( A ) 4 20  a a abc  a b c   0  0  g0 f  0 0 A0 1 1 1 Where,  F F    ( A   A )(  A   A )  g f ( A   A )A A 4 a a 4  a  a a a 2 0 abc  a  a b c g 2  0 f f A A A A . 4 abc ab'c' b c b' c' Vertices:

ig  t a  g f abc[( p  q) g  (q  k) g cab ig 2{ f abe f cde (g g  g g ) 0 ij 0      g0 f p 0     abe cde  (k  p) g ]  f f (g g  g g )  f abe f cde (g g  g g )} Propgators:    

1 Free Quark propagator : S (k)   ij ij k  m  i 1  q  q   q  q Free gluon propagator : Dab (q)    g    0  ab 2   2  2 2 q  i  q  q  i q 1 Free ghost propagator : G (k)   ab ab k 2  i Color Algebra:

2 a a Nc 1 4 tiktkj  CFij , CF   2Nc 3

1 Tr(t at b )  T  ab , T  F F 2

acd bcd ab f f  CA , CA  Nc  3 Leading order QCD green functions Quark propagator:

dk 4 1 ∞ 푘3푑푘 S 1( p)  (  p  m )  iC g 2 D (k)  ∝ (Linearly divergent) 0 F  4 0 0   0 푘3 (2 ) p  k  m0  i0 (푘 → ∞) Gluon propagator:

1 1 (quark) (gluon) (ghost) D (q)  D0 (q)         

dk D  4D 1 1 quark  iT g 2Tr(  ),  F  D 0  f  f f (2 ) k  m0 q  k  m0

4   C dk  (2k  q) g  (k  q) g  (2q  k) g   gluon  i A g 2 D (k)D (q  k)   2  (2 )4 0 0 0        (2k  q) g  (k  q) g  (2q  k) g  dk 4 k (k  q) ghost  iC g 2    A  (2 )4 0 k 2 (k  q)2 ∞ ∝ 0 푘 푑푘 (Quadratically divergent) Quark gluon vertex (1PI):

∞ ∝ 0 푑푘 /푘 (Logarithm divergent)

If theory is renormalizable then at any order of perturbation, all the divergences can be absorbed by redefinitions of the coupling and fields.

Redefinitions of coupling and fields are made through multiplicative constants called renormalization constants. f 1/ 2 f f 1/ 2 f  0c (x)  Z F  c (x),  0c (x)  Z F  c (x), a 1/ 2 a A0 (x)  Z B A (x), 1/ 2 1/ 2 0a (x)  Z a (x), 0a (x)  Z a (x),

f f 1 Z g0  Z g g, m0  Zm, f m ,  0  Gauge invariance which is expressed in terms of STI’s enforce that i) All quarks are renormalized by same constant 푍퐹. ii) All gluons are renormalized by same constant 푍퐵. iii) All ghost are also renormalized by same constant 푍휔. End of Lecture 1 Summery:

• In QCD we have 5 different interactions i) Quark-Gluon interaction ii) 3 Gluon interaction iii) 4 Gluon interaction iv) Ghost-Gluon interaction

• QCD green functions satisfy SDE’s which are exact but unsolvable unless truncated.

• In perturbative truncation the amplitude of QCD green functions can be written directly from the Feynman diagram using Feynman rules.

• Naïve analysis show that these Green function carry ultraviolet divergences.

• Since QCD in renormalizable, therefore, the divergences can be absorbed by redefinition coupling and field of QCD.

Next Lecture: • We will continue renormalization program in QCD using dimensional regularization. • Discuss some non-perturbative aspects of QCD. Procedure of Renormalization:

• Regularize the field theory. (i.e., to modify it temporarily in such way that all divergent integrals become finite)

i) Dimensional regularization dk 4 dk D dk Dv4D   0 4  D  4    (2 )4  (2 )D  (2 )D (breaks only scale invariance)

ii) Pauli-Villars regularization 1 1 1   k 2  i0 k 2  i0 k 2  2  i0

(breaks Hermicity and gauge invariance in non-Abelian gauge theories)

iii) Lattice regularization continuous space-time is replace by discrete Lattice of space-time. Lattice spacing acts as regularizing parameter.

(breaks Poincaré invariance) 1 S ( p)  (  p  m0 )  ( p)

dk Dv4D 1 ( p)  iC 0 g 2 D (k)  F  D 0 0   (2 ) p  k  m0  i0

f 2 f 2 ( p)  ( p  m0 )A( p )  m0 B( p ) 2 2 g0   2  2  A( p )  CF 2      log 4  1 2I1( p ) 16      2 2 g0   2  2  B( p )  CF 2  3    log 4  1 2I2 ( p ) 16     

1  xm2  x(1 x) p2  I ( p2 )  (1 x)log 1   2  0  v0  1  xm2  x(1 x) p2  I ( p2 )  (1 x)log 2   2  0  v0  1 1 (quark) (gluon) (ghost) D (q)  D0 (q)         

2 quark g0 2 2N f  2  2    (q)  2TF 2 (g q  q q )     log 4   4I3 (q ) 16  3     2 gluon g0 19  2  1 2  2 11 2  2 2      CA 2      log 4    I4 (q )g q       log 4    I5 (q )q q  16  6    2   3    3   2 ghost g0 1  2  1 2  2  1  2  2      CA 2      log 4    I6 (q )g q       log 4   2I6 (q )q q  32 6    6   3     

q2 q  D1  q    q    0 0 gluon ghost The Π휇휈 and Π휇휈 do not satisfy the condition of transversality.

2 gluon ghost g0 2  10  2  62 10 2        CA 2 (g q  q q )     log 4    log(q ) 32  3    9 3 

all 2 all 2    (g q  q q ) (q )

2 g  4N f 10  2   31 5   all (q2 )  0 T  C     log 4  C  log q2  8T I (q2 ) 2  F A   A   F 3  16  3 6     9 3   a a (1)a (2)a ig0 ( p, p)  ig0t    ig0  ( p, p)  ig0  ( p, p)

3 (1)a  3 a  g0  2  ig0   3i CAt        log 4   (Finit Part)  2 16 2    3 (2)a a g0  2  ig0    CA / 2  CF  t i       log 4   (Finit Part) 16 2    3 a g0  2  ig0   (CA  CF )i       log 4   (Finit Part) 16 2    Renormalization: f 1/ 2 f f 1/ 2 f  0c (x)  Z F  c (x),  0c (x)  Z F  c (x), a 1/ 2 a A0 (x)  Z B A (x), 1/ 2 1/ 2 0a (x)  Z a (x), 0a (x)  Z a (x),

f f 1 Z g0  Z g g, m0  Zm, f m ,  0 

1. Multiplicative Renormalization Strategy: ~ 1 S ( p)  Z F S( p) ~ 1 D (q)  Z B D (q)

~ Z g    ( p, p )  1/ 2  ( p, p ) Z F Z B 2. Counter terms Strategy:

f  f f f f f  f 1  1  a 2 LB  i 0    0  m0  0  0  g0 0  ta 0 A0a (x)  F0a F0a  ( A ) 4 20  a a abc  a b c   0  0  g0 f  0 0 A0

Z Z Z L  Z i f     f  m f Z Z  f f  gZ Z Z 1/ 2 f  t  f A (x)  B F F   B  (  Aa )2 B F  m, f F g F B a a 4 a a 2   a a 1/ 2 abc  a b c  Z     g Z g Z Z A f    A

Z F  1 Z F , Z B  1 Z B , Z g  1 Z g

1 1 L  i f     f  m f f f  g f  t  f A (x)  F F   (  Aa )2 B  a a 4 a a 2   a a abc  a b c       g f    A Z  Z i f     f  (Z m f  m f ) f f  (Z Z Z 1/ 2 1)g f  t  f A (x)  B F F  F  F 0 g F B a a 4 a a  a a 1/ 2 abc  a b c  Z     (Z g Z Z A 1)g f    A Renormalization Schemes: A renormalization constant Z: ~ 1 Z  (divergent part)  (finte part) S ( p)  Z F S( p) ~ 1 • The divergent part is chosen in such a way that it cancels D (q)  Z B D (q) the pole in un-renormalized Green function. ~ Z g    ( p, p )  1/ 2  ( p, p ) Z F Z B • Each different way of choosing the finite part defines a different renormalization scheme.

Minimal subtraction (MS) scheme: The renormalization constants only cancel the divergent parts of the Green functions without affecting their finite part.

Modified Minimal subtraction (MS) scheme: The renormalization constant also cancel a finite part in the Green function 2 N     log 4  

These renormalization schemes are called mass independent renormalization schemes.

The results of perturbative calculations are scheme dependent. Renormalization constants of QCD in MS at 1 loop order: g 2 Z  1 C N Renormalized gluon propagator F F 16 2  at 1 loop order: g 2 1 1 (all) Zm  1 3CF N 16 2 D (q)  D0 (q)    2 g  5 4  2 1 2  q q  q q q Z B  1 2  CA  TR N f N D (q)  q  g    16 3 3 0   2  2    q   q 2 g 11 2  all (q)   q2 g  q q all (q2 ) Z g  1  CA  TR N f N       16 2  6 3  2 all 2 g0   31 5 2  2   (q )  2 CA   log q   8TF I3 (q ) 16   9 3   Renormalized quark propagator at 1 loop order: 1  q q   q q D (q)    g    1 f 2 all 2  2  2 2 S ( p)  ( p  m )  ( p) q 1  (q ) q  q q

f 2 f 2 ( p)  ( p  m )A( p )  m B( p ) g 2 A( p2 )  C 1 2I ( p2 ) F 16 2 1 g 2 B( p2 )  C 1 2I ( p2 ) F 16 2 2 1 A( p2 ) S( p)  f 2 p  m 1 B( p ) Scale dependence in QCD:

• Dimensional regularization necessitate the introduction of a scale parameter 휈0.

SD ( p,v0 ), DD (q,v0 ), D ( p, p,v0 )

• This dependence on 휈0 can easily be removed by redefining the renormalization constants Z’s.

 2 dk 4 dk Dv  v  2  v   0 Z(v)    Z N     log 4  log 0   4  D  v    2  (2 ) (2 )  0    v 

~ 1 S ( p,v)  Z F (v,v0 )SD ( p,v0 ) ~ 1 D (q,v)  Z B (v,v0 )DD (q,v0 )

~ Z g (v,v0 )    ( p, p ,v)  1/ 2 D ( p, p ,v0 ) Z F (v,v0 )Z B (v,v0 )

• Renormalized green functions are now function of arbitrary scale parameter 휈.

• However, the physical observables must not depend upon the choice of 휈. This is ensured by a correct 휈 dependence of fundament constants (m’s and g’s) of the field theory.

퐴 표푏푠. = 퐴 푚 휈 , 푔 휈 , gre. fun. 휈 Running coupling of QED The running of QCD coupling:

Z g (v)g(v)  g0

Taking the derivative w.r.t log(휈).

Z g 1 g g g(v)  Z  0   (g)  logv g  logv g  logv

Z  (g)  Z 1 g g  logv QCD is asymptotically free

g 2 11 2  2  v2  N     log 4  log 0  Z g (v)  1  CA  TR N f N   2  16 2  6 3    v 

g 2 11 4  11 4  (g)    CA  TF N f  0  CA  TF N f 16 2  3 3  3 3

1 g g 2 g(v)2     (v2 )  g  logv 16 2 0 s 4

2 2  (v )  s  s  (Q )  s   0 s 2 2 2  logv 2 1  s (v )0 log(Q / v ) /(4 ) END of Lecture 1

Lecture 2 (non-perturbative QCD) Quantum Chromodynamics (Non-perturbative aspects) Lecture 2

Faisal Akram

International Symposium on Physics Beyond the Standard Model NCP Islamabad 2015 Outlines

• Characteristics of QCD Asymptotic freedom Confinement QCD phase transition Dynamical breaking of chiral symmetry

• Non-Perturbative truncation of Schwinger-Dyson equations

• Models of quarks-gluon vertex and gluon propagator

• Numerical Solution of SDE of quarks propagator

• Results and comparison with the experiments Characteristics of QCD • Asymptotic freedom

• Quarks Confinement

• QCD phase transition

• Dynamical Breaking of Chiral Symmetry  Q2 s   ( )  logQ2 s   ( )   s   O(2) s 4 0 where   (33 2N ) / 3 0 f 푁푓 < 16.5 (Asymptotic freedom)

* D. J Cross and F. Wilczek, PRL. 30, 1343 (1973); H. D. Politzer, PRL 30, 1346 (1973). Characteristics of QCD • Asymptotic freedom

• Quarks confinement

• QCD phase transition

• Dynamical Breaking of Chiral Symmetry

푉 푟 ≈ 푏푟

휕푉

퐹 =− − ≈ −푏 )

휕푟 푟

( 푉

* K.J. Juge, J. Kuti, and C. Morningstar, Phys. Rev. Lett. 90, 161601 (2003). QCD phase transition

Confined State of Matter De-Confined State of Matter

12 QGP is expected to produce at Tc ≈ 170 MeV ≈ 2 x 10 K ≈ 130,000 T☼ and  ≈ 0

Evidence of QGP? Theoretical Evidence

 2   3 T 4 HG I 30

2 2 So far no conclusive  7   4  4 experimental proof.  QGP  2 f 2s 2q 3c  2s 8c  T  37 T  8  30 30 Quantum Chromo-dynamics • Asymptotic Freedom

• Quarks confinement

• QCD phase transition

• Dynamical Breaking of Chiral Symmetry

(2008 Nobel Prize)

Current mass: 푚푢/푑 = 3 − 6 MeV

Self interaction of quarks 푀푝 = 939 MeV

Constituent mass: 푚푢/푑 ≈ 350 MeV

u d s c b Current mass 3 6 100 1100 4200 (MeV) Constituent 350 350 530 1500 4600 mass (MeV) Dynamical Breaking of Chiral Symmetry in QCD

Fermionic part of QCD Lagrangian for light quarks:

L  iq D q  m q q i  u,d,s This Lagrangian is invariant under SU(3)L⊗SU(3)R i i i i i symmetry if the current masses of light quarks Free quark propagator: are zero. 1  Pole mass ip  m 푚푢/푑 = 3.7MeV Mass function 푀 푝2 = 푝2 Full propagator: (dynamical mass) F( p2 ) Pole mass, 2 ip  M ( p ) 푀푢/푑 = 390MeV

푚푢/푑 = 3.7MeV Perturbative expansion

m 2 M ( p 2 )  If the current mass m = 0 then M(p ) = 0  m  1 2 2  Schwinger-Dyson equations (A non-perturbative technique)  lnp /  QCD   2  2 2   qq 0 2 2 2 m Renormalization point independent Where, p > Λ =0.2 GeV M ( p )  1 QCD 3 1 m Quark condensate 2 2 2  2 2  What if, p < Λ QCD?? p  lnp / QCD   2    qq 0  (0.241)3 GeV 3

 m  4 / 0 2 11N  2N Non-zero value of M(p ) even when m = 0 means chiral symmetry   c f 0 3 is dynamically broken. Green Functions of QCD

2-point Green functions:

Ω 푇 휓(푥1), 휓 (푥2) Ω ≡

푎 푏 Ω 푇 (퐴휇(푥1), 퐴휐(푥2) Ω ≡

푎 푏 Ω 푇 (푐 (푥1), 푐 (푥2) Ω ≡

3-Point Green functions: 4-Point Green functions: Schwinger-Dyson Equations of QCD SDE of Quark Propagator:

SDE of Gluon Propagator:

푁푓 + 5 terms

SDE of quark-gluon vertex:

+ 7 terms

Perturbative Truncation: Non Perturbative Truncation: One or more green functions are modeled subjecting to some general field theoretical constraints.

푁푓 Or Using the results obtained from lQCD. Green Functions of QCD (continued) 푘 푘 Quark Propagator: Free Quark Propagator:

F(k 2 ) 1 1 S(k)   S (k)  ik  M (k 2 ) ikA(k 2 )  B(k 2 ) 0 ik  m

푞 Gluon Propagator: Free Gluon Propagator:

          1  qq qq   q q1 q q 1  q q  0 q q D ((qq))DDT((qq22)   D L0(q2 ) D (q)       2   22  2 2 2 2  2   2  2 2 q  qq  q q q q q  q  q q

2  1 q q D  q 퐿 2 0 WGTI of gluon propagator implies 퐷 푞 = 휉0

푝 • In covariant gauge Green functions are function Quark-Gluon Vertex: of arbitrary gauge fixing parameter 휉.

a 푆 푘, 휉 , 퐷휇휈 푞, 휉 , Γ휇(푘, 푝; 휉)  푘 ig0  (k, p) 2 • A necessary condition that physical observable 12 2 2 will not depend upon the choice of 휉 is that all  (k, p)   Pi (k , p ,k  p)Vi (k, p) i1 WGTI/STI’s should be satisfied.

−1 −1 푖푞휇Γ휇 = 푆 푘 − 푆 푝 WGTI of quark gluon vertex SDE of Quark Propagator:

dk 4 a a S 1( p)  (i  p  m )  g 2 D (q)  S(k) a (k, p) 0  (2 )4 0  2  2 

To solve the SDE of quark propagator we require the knowledge of full gluon propagator and quark-gluon vertex.

We model gluon propagator and quark-gluon vertex subjecting to some general field theoretical constraints or using results from lQCD. What necessary and sufficient knowledge of Field Theoretical Constraints: gluon propagator and quark-gluon vertex is required in order to describe the complete • Gauge Invariance (WGTI’s or STI’s). and correct behavior of quark propagator? • Gauge covariant relations. • Multiplicative renormalizability. • Agreement with perturbation theory in weak coupling limit.

• The model should also agree with lQCD in construction as well as in its implications. • Phenomenological agreement is an ultimate condition. Models of Quark-gluon Vertex

1. Replacing by free vertex

 (k, p)    Rainbow Truncation

This model breaks WGTI/STI of quark gluon vertex. −1 −1 푖푞휇Γ휇 = 푆 푘 − 푆 푝

Constraint on the vertex due to WGTI: 12 L T 2 2  (k, p)   (k, p)   (k, p)   Pi (k , p ,k  p)Vi (k, p) i1

퐿 −1 −1 푇 푖푞휇Γ휇 = 푆 푘 − 푆 푝 푞휇Γ휇 = 0

 A(k 2 )  A( p2 )   A(k 2 )  A( p2 )  (k  p) 8  L        (k  p) T 2 2      2 2      (k , p ,k  p)T  2   k  p  2   i i i1  B(k 2 )  B( p2 )    (k  p)  2 2    k  p  Multiplicative renormalizability of SDE 푇 Longitudinal part is called Ball Chiu (BC) part. can be used to constrain Γ휇 (푘, 푝) Renormalization of SDE of quark propagator SDE of Quark Propagator: The MR requires that all Green functions should be renormalizable by using finite number of multiplicative renormalization constant.

4 a Thea condition of MR cannot be satisfied by any arbitrary dk   S 1( p)  (i  p  m )  g 2 D (q)  S(k) quark (k-,gluonp) vertex ansatz in SDEs. 0  (2 )4 0  2  2 

−1/2 휓 = 푍 휓 ~ 1 How to do renormalization: 2 0 S ( p;)  Z (,)S( p;) S( p;) 푎 −1/2 푎 2 퐴휇 = 푍3 퐴0휇 ~ 1 Regularize the theory by D (q;)  Z (,)D (q;) D (q;) 1/2  3  applying hard momentum 푍2푍3 ~ 푔 = 푔  (k, p;)  Z (,) (k, p;)  (k, p;) 0  1  cutoff. 푍1

• Renormalized green functions are function of arbitrary scale parameter 휇.

• However, the physical observables must not depend upon the choice of 휇. This is ensured by a correct 휇 dependence of fundament constants (m’s and g’s) of the field theory.

퐴 표푏푠. = 퐴 푚 휇 , 푔 휇 , gre. fun. 휇

~ dk 4 ~ a ~ ~ S 1( p)  Z (i  p  m )  Z g 2 D (q)  S (k)a (k, p) 2 0 1  (2 )4  2   Multiplicative Renormalizability

The condition of MR cannot be satisfied by any arbitrary quark-gluon vertex ansatz in SDEs. (An example) F(k 2 ) Quark renormalization function in leading-logarithm approximation: S(k)  ik  M (k 2 ) 2 2 2 2 2 2 2 F(p ) 1A1,1 ln( p /  )  A2,2 ln (p /  ) 

2 MR requires that 퐴2,2 = 퐴1,1/1!

If we use bare approximation of quark-gluon vertex in quark propagator SDE we get

2 2 1   p k 2  p2  1  dk 2 F(k 2 )  dk 2 F(k 2 ) 2 2  2  2 2 F( p ) 4p  p 2 k  (where 푀 푝 is taken zero as an approximation)  0 p 

2   3   F( p2 ) 1  ln( p2 / 2 )    ln 2 ( p2 / 2 )  4 2 4     푇 MR can be used to constrain Γ휇 (푘, 푝) 3 2 In which 퐴2,2 = 퐴1,1/1! as the BC vertex itself doesn’t satisfy 2 the condition of MR. Models of Quark Gluon Vertices 2. Curtis Pennington (CP) Vertex: 1 T  (k, p)   BC (k, p)  A(k 2 )  A( p2 )  6   2 d(k, p) CP vertex satisfies the condition of MR in quench approximation for zero as well as non-zero quark mass.

3. Kizilersu Pennington (KP) Vertex: BC  (k, p)   (k, p)   iTi i2,3,6,8

4 1 1 1  A(k 2 )A( p2 )     A(k 2 )  A( p2 )  A(k 2 )  A( p2 ) ln , 2 4 4   2 2 2    2  3 (k  p ) 3 (k  p )  A(q )  5 1 1 1  A(k 2 )A( p2 )     A(k 2 )  A( p2 )  A(k 2 )  A( p2 ) ln , 3 2 2   2 2    2  12 (k  p ) 6 (k  p )  A(q )  1 1   A(k 2 )  A( p2 ), 6 4 (k 2  p2 )

 8  0.

KP vertex satisfies the condition of MR in un-quenched approximation only for zero quark mass. Models of gluon propagator SDE of Quark Propagator: Rainbow Truncation

4 a a 1 dk 2  a S ( p)  Z (i  p  m )  Z g D (q)  S(k) (k, p)   2 0 1  (2 )4  2   2 Maris-Tandy Model: It is based on Rainbow truncation of vertex. Gluon propagator: D(q2 )  q q  g 2 D (q)  g 2       2   2  q  q  (in Landau gauge) g 2 D(q2 ) g 2 D(q2 )   1 lim  ?? lim  4 m 2 2 2 2 2 2 2 2 2 q  q QCD q ln[q /  ] q QCD q QCD (Perturbative calculation)

2 2 2 2 2 q / 4mt g D(q ) 4 2 2   1 e  Dq2eq /  4 m 2 6 2 2 2 2 Maris-Tandy Model q  1/ 2ln[  (1 q / QCD ) ] q

Infrared dominant; A Model UV dominant; Given by perturbative QCD MT model 휔 = 0.4 so that the first term don’t perturb with 2nd in UV region. 퐷 = 0.93 GeV2 is fitted to lQCD result of quark condensate , Perturbative 3 3 − 푞 푞 휇=1 퐺푒푉 = 0.241 GeV .

This model does not include any Nf dependence in the infrared part. Solution of SDE of quark propagator in Rainbow truncation + MT model

4 dk 4 g 2 D(q2 ) q q a S 1( p)  Z (i  p  m )  Z (    )  S(k) 2 0 3 1  (2 )4 q2  q2 2  

• We need to first fix the renormalization constants Z1 and Z2. 푍2 = 푍1 (WGTI of quark-gluon vertex) • On-mass shell renormalization scheme. Renormalization BC: 1 S( p) | 2 2  p  Renormalized current mass i  p  m

2 2 F( p ) 2 2  1, F( p ) p  S( p)  2 2 M ( p ) 2 2  m i  p  M ( p ) p  

1  1 g 2 D(q2 ) F(k 2 )  2(k  q)( p  q)   Z 1 dk 4 k  p   2 2  4 2  2 2 2 2  2   F( p )  12 p q k  M (k )  q   M ( p2 )  1  g 2 D(q2 ) M (k 2 )F(k 2 )  Z m  dk 4 2 2  0 4  2 2 2 2  F( p )  4 0 q k  M (k ) 

1  1 g 2 D(q2 ) F(k 2 )  2(k  q)( p  q)  Z  1 dk 4 k  p   2  12 4 p2  q2 k 2  M 2 (k 2 )  q2       p2  2  1  g 2 D(q2 ) M (k 2 )F(k 2 ) m  Z m  dk 4   2 0 4 4  q2 k 2  M 2 (k 2 )  0  p2  2 Quark Propagator using MT model

b c s

u/d

푚푢/푑 = 0.00374 GeV How the model generates Dynamical mass? 푚 = 0.083 GeV 푠 휇 = 19 GeV • The 1st term of the MT model enhance 푚푐 = 0.88 GeV the coupling strength in IR region through 푚푏 = 3.8 GeV the controlling parameter D. • It is this enhancement which generates dynamical mass in IR region. • Gradual decrease in parameter D shows that DCS is MT model of gluon propagator function: • restored if D < 0.261.

2 2 2 2 2 q / 4mt g D(q ) 4 2 q2 / 2  m 1 e 2  6 Dq e  4 2 2 2 2 q  1/ 2ln[  (1 q / QCD ) ] q Comparison with Qu-lQCD results:

Pion form factor hBSE:

Ladder Truncation MT model Quark Mass functions using KP Vertex Troubles with MT model:

• Rainbow truncation breaks Gauge invariance as the WGTI/STI of quark-gluon is not satisfied.

As a result physical observable may develop Γ휇(푘, 푝) → 훾휇 dependence on gauge fixing parameter 휉.

• Violates multiplicative renormalizability.

As a result physical observable may develop dependence on arbitrary scale parameter 휇.

Kizilersu Pennington (KP) Vertex: BC  (k, p)   (k, p)   iTi i2,3,6,8

• Asymptotic freedom requires that Nf > 16.5 (at 1 loop order). • Dynamical breaking of chiral symmetry may also require a critical number of quark flavors.

SDE of Quark Propagator: MT model does not allow as to study the flavor dependence of quark propagator in the infrared region as the gluon propagator is itself taken Flavor independent.

dk 4 a S 1( p)  Z (i  p  m )  Z g 2 D (q)  S(k)a (k, p) 2 0 1  (2 )4  2  

Gluon propagator (in Landau gauge 휉=0):

2   D(q , N f )  q q  D (q, N )     푁푓 + 5 terms  f 2   2  q  q 

Fit with Refined Gribov-Zwanziger (RGZ) form z( 2 )q2 (q2  M 2 ) D(q2 , N )  f 4 2 2 2 2 2 2 q  q M 13g A / 24 M m0

1/ 2 m0 (N f ) 1.011(9.161 N f ) GeV 2 2 g A (N f )  0.474(16.406  N f )GeV M 2  4.85GeV

A. Ayala, A. Bashir, D. Binosi M. Cristofretti, and J. Rodriguez-Quintro, Phys. Rev. D 86, 074512 (2012) Solution of SDE of the Quark Propagator SDE of Quark Propagator:

dk 4 a a S 1( p, N )  Z (i  p  m )  Z g 2 D (q, N )  S(k, N )  (k, p) f 2 0 1  (2 )4  f 2  f 2  General form of quarks propagator: F( p2 , N ) S( p, N )  f f 2 KP Vertex i  p  M ( p , N f ) Lattice QCD

(Current mass 푚 = 0) Final remarks

• Asymptotic freedom in QCD requires that 푁푓 < 16.5.

• Latest lattice results of gluon propagator when used in SDE of quark propagator truncated by KP vertex shows that 푁푓 < 7.2 if QCD is to exhibit DCSB. Extra slides Elementary Particle Physics Today

푣푒 푣휇 푣휏 Meson: Baryon Leptons: 푒 휇 휏

Gauge Bosons: 훾, 푊±, 푍0, and 8 gluons

399 Mesons, 574 Baryons have been discovered Higgs Bosons: 퐻 (God/Mother particle)

• Strong interaction (Quantum Chromodynamics) • Electro-weak interaction (Quantum electro-flavor dynamics) The Standard Model

Explaining the properties of the Hadrons in terms of QCD’s fundamental degrees of freedom is the Problem laying at the forefront of Hadronic physics. Elementary Particle Physics

1. Scattering Experiments Cross sections, Decay Constants, 2. Decay Processes Couplings, Form factors, Masses, Spin etc 3. Study of bound states Models

Particle Accelerators Models and detectors

Experimental Particle Physics Theoretical Particle Particle Physicist Phenomenologist Physicist

Measured values of Calculated values of physical observables ≈ physical observables

Decades of observation and calculations show that the Standard Model of particle physics can describe almost every thing which we have observed in the labs of high energy physics. The Standard Model • The standard model is a field theory.

• In field theories we associate a field to every different kind of a particle. (e.g., electron field, proton field, photon field etc)

• Particles appear as quanta of field.

Quantization Quanta of field has particle properties. 휙(푥)

• Equation of motion of the fields (푠 = 0) (푠 = 1) Free Scalar Field: Free Abelian Vector Field: 휇 2 휕 휕휇퐴휈 + 휕휈(휕 퐴휇) = 0 (휕휇휕 + 푚 )휙(푥) = 0 휇 휇 † 휇 2 † 1 ℒ0 = 휕휇휙 휕 휙 − 푚 휙 휙 ℒ = − 퐹 퐹휇휈 0 4 휇휈 (푠 = 1/2) 퐹휇휈 = 휕휇퐴휈 − 휕휈퐴휇 Free Dirac Field: 휇 푖훾 휕휇 − 푚 휓(푥) = 0 휇 ℒ0 = 푖휓 훾 휕휇휓 − 푚휓 휓 The Standard Model

• Interaction is introduced by the coupling of fields .

휇 For example: ℒ퐼 = 푒휓 훾 휓퐴휇 for QED

휓 훾

휓

• Only Lorentz invariant couplings are allowed.

휇 휇 5 5 휇 ℒ퐼 = 휓 훾 휓퐴휇; 휓 훾 훾 휓퐴휇; 휓 휓휙; 휓 훾 휓휙; 휓 훾 휓휓 훾휇휓; . . . .

• Possible coupling are further constrained by the gauge symmetries.

i) 푆푈 2 퐿 ⊗ 푈(1)푌 for electroweak interaction. ii) 푆푈(3)푐 for strong interaction.