LLeeccttuurree 1133 SSyymmmmeettrriieess ooff QCD

WS2010/11: ‚Introduction to Nuclear and Particle Physics‘ 1 SSyymmmmeettrriieess aanndd ccoonnsseerrvvaattiioonn llaawwss

° The interactions are defined by symmetry principles

° Symmetries imply conservation laws, in particular conserved currents

From the invariance with respect to

1) continuous transformations, i.e. shifts in space, Ï momentum conservation

2) time shift transformations Ï energy conservation

3) rotations in space Ï angular momentum conservation

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Consider the free fermion Lagrangian µ (1) L = i ψ γ ∂ µψ − m ψψ

° I) This Lagrangian has the global U(1) invariance: i.e. an invariance under global phase transformations ψ ( x) → e iαψ ( x) (2) α – arbitrary real constant

iα Phases transformations U (α ) ≡ e with α=const and real Ï (3) correspond to the unitary abelian group U(1)

Abelian means that the commutative multiplication low holds:

U (α1 ) U (α 2 ) = U (α 2 ) U (α1 ) (4) since the complex numbers (3) commute.

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Consider now an infinitely small transformation of the U(1) group: ψ ( x) → e iαψ ( x) → (1 + iα + ...) ψ ( x) iα (5) ∂ µψ ( x) → e ∂ µψ ( x) → (1 + iα + ...) ∂ µψ ( x) ψ ( x) → e −iαψ ( x) → (1 − iα + ...) ψ ( x)

Keep only the first term in the Taylor expansion: ψ ( x) → (1 + iα ) ψ ( x) (6)

The invariance of the Lagrangian under the transformation (6) means that δ L = 0 (7)

µ δ L = δ (i ψ γ ∂ µψ − m ψψ ) Ω (8) δ L(ψ ,∂ µψ ,ψ ,∂ µψ )

∂L Note: δ L( f ) = δf ∂f 4 SSyymmmmeettrriieess aanndd ccoonnsseerrvvaattiioonn llaawwss

∂L ∂L ∂L ∂L δ L = δψ + δ (∂ µψ ) + δψ + δ (∂ µψ ) (9) ∂ψ ∂(∂ µψ ) ∂ψ ∂(∂ µψ )

Substitute (6) in (9) : ∂L ∂L (10) δ L = (iαψ ) + (iα∂ µψ ) + [conjugated ψ ] ∂ψ ∂(∂ µψ )

≈ ’ ≈ ’ use that ∆ ∂L ÷ ∂L ∆ ∂L ÷ (11) ∂ µ ∆ ψ ÷ = ∂ µψ + ∂ µ ∆ ÷ψ « ∂(∂ µψ ) ◊ ∂(∂ µψ ) « ∂(∂ µψ ) ◊

Substitute (11) in (10) :

» ∂ ≈ ∂ ’ÿ ≈ ∂ ’ … L ∆ L ÷Ÿ ∆ L ÷ δ L = iα − ∂ µ ∆ ÷ ψ + iα ∂ µ ∆ ψ ÷ + ... (12) …∂ψ « ∂(∂ µψ ) ◊⁄Ÿ « ∂(∂ µψ ) ◊ || due to the Euler-Lagrange equation! 0 5 SSyymmmmeettrriieess aanndd ccoonnsseerrvvaattiioonn llaawwss

To fulfill δ L = 0 ≈ ’ ∆ ∂L ∂L ÷ µ (13) iα ∂ µ ∆ ψ −ψ ÷ ≡ ∂ µ j = 0 « ∂(∂ µψ ) ∂(∂ µψ ) ◊ ∂L = iψγ µ ∂(∂ µψ ) ∂L = −iγ µψ ∂(∂ µψ ) substitute in (13) => ≈ ’ ∆ ∂L ∂L ÷ µ µ iα ∂ µ ∆ ψ −ψ ÷ = ∂ µ (− αψγ ψ − αψγ ψ ) « ∂(∂ µψ ) ∂(∂ µψ ) ◊ µ µ µ = ∂ µ (−2αψγ ψ ) = ∂ µ (eψγ ψ ) = ∂ µ j = 0

using e.g. –e for the real constant α ‰e/ 2

Introduce the charge 4-current: j µ = −eψγ µψ

6 SSyymmmmeettrriieess aanndd ccoonnsseerrvvaattiioonn llaawwss C °charge 4-current: j µ : ( j 0 , j ) with j µ = −eψγ µψ

(13) = current conservation law :

µ ∂ µ j = 0

3 0 °Electric charge: Q = — d x j ( x) j 0 = −eψγ 0ψ

From (13) follows dQ = 0 Q = const dt

! Charge should be conserved due to the global U(1) invariance !

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Noether Theorem: A global invariance leads to the existance of a conserved current!

Global invariance means that the phase α is a real constant, which is unique for all space and time points Ï one cannot measure the phase α itself !

II) Let‘s now consider symmetry transformations where α can depend on the space-time coordinate α Ω α ( x) (17) ° Local U(1) invariance: i.e. invariance of the Lagrangian under the local gauge group U(1):

iα ( x ) (18) U (α( x)) ≡ e iα ( x ) ψ ( x) → e ψ ( x)

and ψ ( x) → e − iα ( x )ψ ( x) (19)

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iα ( x ) iα ( x ) (20) ∂ µψ → e ∂ µψ + ie ψ ∂ µα( x)

This term violates the invariance of the Lagrangian under local U(1) transformations!

•In order to construct a Lagrangian, which is invariant under local U(1) transformations, one has to introduce the covariant derivative:

Dµ ≡ ∂ µ − ieAµ (21)

where Aµ(x) follows the transformation: 1 (22) A ( x) → A ( x) + ∂ α ( x) µ µ e µ Aµ(x) is called a gauge vector field , which interacts with the dirac field ψ(x).

Thus, µ L = i ψ γ Dµψ − m ψψ

µ µ (23) =ψ (iγ ∂ µ − m)ψ + eψγ ψ Aµ

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The request for local gauge invariance leads to the introduction of a gauge vector field Aµ(x)

In order to identify the field Aµ(x) with real particles (photons), one has to add the kinetic energy term: 1 − F F µν 4 µν with Fµν = ∂ µ Aν − ∂ν Aµ - field strength tensor

• The local gauge invariance of L requires a massless vector field Aµ(x), 1 2 µ since a mass term m Aµ A violates the local gauge invariance of the Lagrangian 2

10 AAbbeelliiaann aanndd nnoonnaabbeelliiaann llooccaall ssyymmmmeettrriieess

Local symmetry: QED QCD local U(1) gauge transformation local SU(3) color gauge transformtion iα a ψ ( x) → e ψ ( x) ψ ( x) → e iα a ( x )t ψ ( x) Introduce one gauge vector field Introduce 8 gauge vector fields a Aµ(x) (photons) Aµ (x) (gluons) a Aµ(x) – massless Aµ (x) – massless (a=1,…,8) QED – abelian theory QCD – nonabelian theory: = [U(α1),U(α2)]=0 [ta , tb ] = if abc tc a no self-interaction of the field Aµ(x) self-interaction of field Aµ (x) since since the photons do not have a charge the gluons do have a charge Infinite range of interaction, however:

α (r) α S (r)

Thus: Local invariance Ï introduction of gauge fields! 11 SSyymmmmeettrriieess ooff QQCCDD

QCD Lagrangian:

1 µν a L ( x) = ψ ( x) (iγ µ [∂ − igt a Aa ]− Mˆ 0 )ψ ( x) − G a ( x)G ( x) QCD µ µ 4 µν

Gluonic field strength tensor: a a a abc b c Gµν (x) = ∂ µ Aν − ∂ν Aµ + gf Aµ ( x)Aν

ψ(x) - quark field flavor space Dirac space color space q = u, d , s µ = 0,1,2,3 c = r , b, g

In flavor space (3 flavors): Mass term: ≈ ’ ≈ m 0 0 0 ’ ∆ u ÷ ∆ u ÷ ˆ 0 ∆ 0 ÷ ψ ( x ) = ∆ d ÷ M = 0 m d 0 ∆ ÷ ∆ 0 ÷ « s ◊ « 0 0 m s ◊ 3x3 diagonal matrix in flavor space with the bare quark masses on the diagonal 12 SSyymmmmeettrriieess iinn ffllaavvoorr ssppaaccee Symmetries in flavor space: From the decay of particles we know that the flavor is conserved, i.e. the invariance of the Lagrangian under rotations in flavor space (for the strong Interaction)

++ + e.g. ∆ (uuu) → p(uud)+π (du)

implies the conservation of flavor currents Ï no need to introduce gauge fields in order to construct a conserved current in flavor space.

Thus: the flavor symmetry is a global symmetry

3 flavors u,d,s Ï SU(3) flavor group for 6 flavors Ï SU(6) flavor group

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°The global flavor symmetry

Ï The QCD Lagrangian is invariant under SU(3) flavor transformations:

≈ 8 ’ ∆ ÷ ψ ( x) → exp∆ − iƒ abtb ÷ψ ( x) (24) « b=1 ◊ flavor space a a a λ t are 8 generators of the SU(3)flavor group t = 2

The parameters ab in (24) are constants, but an arbitrary vector in flavor space

Since the transformation (24) is global, it holds for massless and massive quarks

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° Furthermore we consider the transformations: ≈ 8 ’ ∆ 5 ÷ ψ ( x) → exp∆ − iƒ abtb γ ÷ψ ( x) (25) « b=1 ◊ act on the flavor, Dirac components

The transformation (25) mixes the upper and lower component of the Dirac spinor

Ï The QCD lagrangian is invariant under the transformation (25) only in case of massless quarks !

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° Now define right and left-handed quarks by the linear combination: 1 1 ψ = (1 ± γ 5 )ψ ψ =ψ (1 @ γ 5 ) R / L 2 R / L 2 (26)

ψ = ψ R +ψ L , ψ = ψ R +ψ L (27)

• Handedness, or chirality – projection of the spin s on the momentum p (direction of motion)

right-handed quarks left-handed quarks

C C s s C C p p q q

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° For M ˆ 0 = 0 the left- and right-handed quarks are not mixed dynamically and conserve their handedness (chirality).

0 Ï chiral invariance of QCD in the limit Mˆ = 0 Ω SU (3) flavor SU (3)R × SU (3)L

It expresses the physical effect that the sign of projection of the fermion spin on its momentum direction can not be changed by dynamics if M ˆ 0 = 0 ,

µ a a i.e. the interaction in QCD igψ ( x) γ t Aµ ψ ( x) does not break the left-right symmetry

° A mass term M ˆ 0 ≠ 0 breaks the chiral symmetry of the QCD Lagrangian: ˆ 0 ˆ 0 M ψψ = M (ψ L +ψ R )(ψ L +ψ R ) ˆ 0 ˆ 0 = M (ψ Lψ L +ψ Rψ R ) + M (ψ Lψ R +ψ Rψ L ) the mass term plays the role of an interaction that mixes left- and right-handed quarks 17 SSyymmmmeettrriieess iinn ffllaavvoorr ssppaaccee

°Thus, the global chiral transformations ≈ 8 ’ ∆ 5 ÷ ψ ( x) → exp∆ − iƒ abtb γ ÷ψ ( x) « b=1 ◊ Ω SU (3) flavor SU (3)R × SU (3)L

are valid only for massless quarks (or fermions as neutrinos) !

°A mass term Mˆ 0 ≠ 0 Ï phenomenon of symmetry breaking

As a consequence of chiral symmetry breaking the bound states of QCD, i.e. mesons and baryons with opposite parity do not show the same mass !

A prominent example: the vector meson ρ and the axial vector meson a1 differ in mass by more than 500 MeV !

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