QCD - properties confinement, Higgs mechanism more on confinement - - PowerPoint PPT Presentation

QCD - properties

confinement, Higgs mechanism

more on confinement

(i) the absence of free quarks in Nature

[but quarks could combine with a fundamental coloured scalar]

(ii) observable particles are colour singlets

[but this confuses confinement and screening (Higgs phase)]

(iii) quarks interact with a long range linear interaction

[obfuscated by string breaking]

(iv) the work required to separate quarks grows linearly as one takes the quark mass to infinity

[ok, but removes quarks from the definition!]

more on confinement

an instantaneous potential that depends on the gauge potential K is renormalization group invariant K is an upper limit to the Wilson loop potential

K(x − y; A)

H = 1 2

• dx
• E2 + B2

−1 2

• dxdyf abcEb(x)Ac(x)⟨xa|

g ∇ · D∇2 g ∇ · D|yd⟩f defEe(y)Af(y)

K generates the beta function K is infrared enhanced at the Gribov boundary

0.0 2.0 4.0 6.0 8.0 10.0 r/r0

• 10.0
• 5.0

0.0 5.0 10.0 15.0 r0( VQQ(r) - VQQ(2r0) )

Szczepaniak & Swanson

more on confinement

Ut(t0, x) → zUt(t0, x)

t0

x P(x) → zP(x) ∀x a global symmetry of QCD

more on confinement

either ⟨P(x)⟩ = 0

• r ⟨P(x)⟩ ̸= 0

broken Z phase symmetric Z phase

N N

⟨P(x)⟩ = e−FqT confinement iff QCD is in the symmetric Z phase

N

more on confinement

Coulomb gauge and the Gribov problem

· Aa = 0

det(∇ · D) = 0

more on confinement

Coulomb gauge and the Gribov problem

· Aa = 0

det(∇ · D) = 0

Gribov region Fundamental modular region

more on confinement

Fundamental modular region Gribov region FMR is convex GR contains the FMR FMR contains A=0 physics lies at the intersection

• f the FMR and GR

identify boundary configurations

Zwanziger; van Baal

det(∇ · D) = 0

Coulomb gauge and the Gribov problem

more on confinement

Kugo-Ojima confinement criterion

D(p) = − 1 p2 1 1 + u(p)

u(p) → −1 p → 0

Alkofer, von Smekal, Fischer

Vortex driven confinement

make a singular gauge transformation

Vortex driven confinement

make a singular gauge transformation vortex (locates infinite field strength caused by the sgt)

Vortex driven confinement

= e−σA = tr

A/A0

• i=1

⟨Z(i)⟩ ⟨trULR(C)⟩ = ⟨tr

A/A0

• i=1

Z(i)⟩

⟨Z(1)Z(2)⟩ ∼ ⟨Z(1)⟩⟨Z(2)⟩

σ = −log⟨Z(1)⟩ A0

more on confinement

Greensite; Weise; de Forcrand; Reinhardt; Langfeld;...

vortices!

Lattice Gauge Theory

confinement cartoons

17.5 20 22.5 25 27.5 30 32.5 6 8 10 12 14 16 18 20

`Casimir scaling’ c= 4/3 c= 3 c= 10/3 c= 16/3 c= 18/3

Lattice Gauge Theory

Lattice Gauge Theory

Properties: Confinement

Lattice Gauge Theory

Lüscher & Weisz

• π/12
• π/24

V = br + c/r

Lattice Gauge Theory

Juge, Kuti, & Morningstar

short range

}

}

long range intermediate

Lattice Gauge Theory

Confinement vs. Higgs Mechanism

• E. Fradkin and S. Shenker, PRD19, 3682 (1979)

QCD - functional approaches

what is Nonperturbative?

F = 1 + x + x2 + . . . F ≡ 1 1 − x F = e−1/x F = 0 + 0 + 0 + . . . x2F = F

not this...

nonperturbative quantities

• electron mass in a massless theory
• confinement
• symmetry breaking

the generating functional

Z[J] =

• DϕeiS+i

R d4x J(x)ϕ(x)

δ δf(x)f(y) = δ(y − x)

Z and its derivatives contain all the information in this quantum field theory

the generating functional

Z[J] =

• DϕeiS+i

R d4x J(x)ϕ(x)

S =

• d4x 1

2∂µϕ∂µϕ − 1 2m2ϕ2 − λ 24ϕ4

• Dϕe

i 2

R R ϕ1M(12)ϕ2+i R Jϕ = (detM)−1/2e− i

2

R R J1(M −1)(12)J2

(the only integral we can do!)

the two-point function

Z[J] =

• DϕeiS+i

R d4x J(x)ϕ(x)

ϕ1ϕ2J = 1 Z[0]

• Dϕϕ1ϕ2eiS+i

R Jϕ

= 0|T[ϕI(x1)ϕI(x2)e−i

R VI]|0

0|e−i

R VI|0

the two-point function

12J=0 = i

• d4k

(2)4 e−ik·(x1−x2) k2 m2 + i ∆(x1 x2) (∂2

x + m2)∆(x − y) = −iδ4(x − y)

(0) (0) (0)

connected greens functions

eiF [J] = Z[J]

F = −E0 ∆(x − y) = e−i

R V [

δ iδJ ] Z(0)[J] = + + + ...

• DϕeiS = Ψ0|e−iHT |Ψ0

legendre transformation

ϕcl(x) ϕ(x) 1 = 1 Z[0] δ iδJ(x)Z[J] = δ δJ(x)F[J] ϕ(0)

cl (x) = i∆(x − y)Jy

(∂2

x + m2)ϕ(0) cl (x) = Jx

Γ(0)[ϕ(0)

cl ] = F (0)[J] − Jxϕ(0) cl (x)

Γ(0)[ϕ(0)

cl ] = 1

2

• d4x
• ∂µϕcl∂µϕcl − m2ϕ2

cl

legendre transformation

ϕcl ≡ δ δJ F[J] Γ[ϕcl] ≡ F[J] − Jxϕcl(x) δΓ δϕcl(x) = −J(x)

the effective action

Γ[ϕcl] = −(TL3)Veff(ϕcl) ∂Veff ∂ϕcl |J=0 = 0. Veff ϕcl

the effective action

Γ[ϕcl] = −(TL3)Veff(ϕcl) ∂Veff ∂ϕcl |J=0 = 0. Veff = 1 4ϕ4

cl

• λ +

λ2 16π2 ((N + 8) log(λϕ2

cl/M 2) − 3

2) + 9 log 3

• .

V (0)

eff = 1

4λϕ4

cl

• 0.2

• 1
• 0.5

Veff

ϕcl

1PI greens functions

δ(x−z) = δφc(x) δφc(z) =

• d4y

δ2F δJ(x)δJ(y) J(y) δφc(z) = −

• d4y

δ2F δJ(x)δJ(y) δ2Γ δφc(z)δφc(y)

δ2Γ δφc(x)δφc(z) = −

• δ2F

δJ(x)δJ(z) −1 δ2Γ δφc(x)δφc(z)|φc=0 = iS−1(x − z)

1Pi Greens functions

=

“one-particle irreducible”

Schwinger-Dyson Master Equation

• Dϕ

δS δϕ + J

• eiS+iJxϕx = 0

e−iF δS δϕ(x) δ iδJ

• eiF = −J(x)

δS δϕ(x) δ iδJ + δF δJ

• · 1 = −J(x)

δ iδJ = δϕcl(z) iδJ δ δϕcl(z) δϕcl(z) δJ = δ2F δJδJ(z)

Schwinger-Dyson Master Equation

δS δϕ

• ϕcl + i
• δ2Γ

δϕclδϕcl −1

.z

δ δϕcl(z)

• · 1 = δΓ

δϕcl δS δϕ = −(∂2 + m2)ϕ − λ 6 ϕ3 −(∂2 + m2)ϕcl(x) − λ 6 (ϕcl(x) + ∆xz δ δϕcl(z))3 · 1 = δΓ δϕcl(x)

Schwinger-Dyson Master Equation

−(∂2 + m2)ϕx − λ 6

• ϕ3

x + 3∆xxϕx + ∆xa∆xb∆xciΓabc

• = δΓ

δϕx = 0 Γabc = 0

Propagator

δ2Γ δϕxδϕy |J=0 = −(∂2

x + m2)δxy − λ

2 ∆xxδxy − λ 2 ∆xb∆xc∆xdiΓabc∆eaiΓdye − λ 6 ∆xa∆xb∆xciΓabcy

−(∂2 + m2)ϕx − λ 6

• ϕ3

x + 3∆xxϕx + ∆xa∆xb∆xciΓabc

• = δΓ

δϕx

δ δϕ(z)∆xy = i δ δϕ(z) δ2Γ δϕδϕ −1

xy

= ∆xa(iΓazb)∆by

no sum over x, sum over a,b,c

review

δS δϕ

• ϕcl + i
• δ2Γ

δϕclδϕcl −1

.z

δ δϕcl(z)

• · 1 = δΓ

δϕcl

δ δϕ(z)∆xy = i δ δϕ(z) δ2Γ δϕδϕ −1

xy

= ∆xa(iΓazb)∆by

Propagator

= −1 −1 − − −

δ2Γ δϕxδϕy |J=0 = −(∂2

x + m2)δxy − λ

2 ∆xxδxy − λ 2 ∆xb∆xc∆xdiΓabc∆eaiΓdye − λ 6 ∆xa∆xb∆xciΓabcy

• bservations
• diagram topology is specified by following indices in the Schwinger-

Dyson equation

• factors of i, 1/2, -1, etc are absorbed into the definition of the

diagrams except as indicated. This is because the perturbative Feynman rules are sufficient to determine all of these factors uniquely. The explicit minus signs then make everything work out.

• the master equation implies that there must be exactly one bare vertex

in every Schwinger-Dyson equation

• the master equation restricts the form of possible diagrams that

appear in Schwinger-Dyson equations.

Diagrammatic Approach

+ + +

... ... = y y = x = delta(x,y)

δΓ δϕ =

δ δϕ(y)

−(∂2 + m2)ϕx − λ 6

• ϕ3

x + 3∆xxϕx + ∆xa∆xb∆xciΓabc

• = δΓ

δϕx

δ δϕ(y) δ δϕ(y)

n-point functions

= + + + + + + +

+ + +

δΓ δϕ =

notation shift!

n-point functions

= + + + + + + +

= + + + + + + + + + + + + + + + + + + + +

J=0 n-point functions

= −1 −1 − −

= + + +

= + + + + ... +

+ perms

Solving Schwinger- Dyson Equations

∆p = i A(p2)p2 − B(p2) Γ(1, 2, 3, 4)(2)4(1 + 2 + 3 + 4)

Ap2 − B i = p2 − m2 i − i 2

• d4q

(2)4 i Aq2 − B − i 6

• d41

(2)4 d42 (2)4 d43 (2)4 i A12

1 − B1

i A22

2 − B2

i A32

3 − B3

Γ(1, 2, 3, −p)(2)4(1 + 2 + 3 − p)

= −1 −1 − −

• we need to deal with divergent integrals
• we need an expression for . Approximating

this as the bare vertex is a typical.

• we need to evaluate some pretty nasty

integrals

• we need to solve a(many) nonlinear integral

equation(s).

Solving Schwinger- Dyson Equations

q0 → iq4 −A(−p2

E)p2 E − B(−p2 E) = −p2 E − λ

2

• d4qE

(2π)4 1 Aq2

E + B

A = 1

α = 2π2 Λ q3

EdqE

(2π)4 1 q2

E + m2 + λα/2

α Λ2 = π2 1 − 1

2λπ2

m2 = m2

0 + 1 2λπ2Λ2

1 − 1

2λπ2

Solving Schwinger- Dyson Equations

Wick rotate to Euclidean space

Ladder QED

−1 − = −1

Dµν = −i k2 + i

• gµν − (1 − )kµkν

k2

• A/

p − B = / p − m − e2

• d4q

(2π)4 γµDµν(p + q)(A/ q + B)γν A2q2 − B2

S(p) = i A(p2)/ p − B(p2)

Ladder QED

B(p2) = m − ie2

• d4q

(2π)4 (3 + 0)B(q2) (q2 − B2)(p − q)2

• dΩ4

1 (p − q)2 = 2π2 1 q2 θ(q > p) + 1 p2 θ(p > q)

• A/

p − B = / p − m − e2

• d4q

(2π)4 γµDµν(p + q)(A/ q + B)γν A2q2 − B2 A(ξ = 0) = 1

ladder QED

B(−p2

E) = 3e2

8π2

• 1

p2

E

pE dqq3 B(q) q2 + B2 + Λ

pE

dqq B(q) q2 + B2

• p4

EB = − 3e2

8π2 pE dqq3 B q2 + B2 (p4

EB) = − 3e3

16π2 p2

E

B p2

E + B2 =

d dp2

E

ladder QED

B(−p2

E) = 3e2

8π2

• 1

p2

E

pE dqq3 B(q) q2 + B2 + Λ

pE

dqq B(q) q2 + B2

• (p4

EB) = − 3e3

16π2 p2

E

B p2

E + B2

(p4B)|p=0 = 0 (B + p2B)|p=Λ = 0 B → p−1±√

1−3e2/(4π2)

α > α ≡ π 3

numerical methods

expand A(p) in a convenient basis discretise

A(p) =

• i

ciTi(p) A(p) → Ai = A(pi) xi = fi({x})

numerical methods

(i) iterate (ii) iterative Newton-Raphson (iii) minimise

G({x}) =

• i

(xi − fi({x}))2 xi|n+1 = fi({x}n)

• j
• −δij + ∂fi

∂xj |xi

• δxj = xi − fi({x})

xi|n+1 = xi|n + δxi

xi = fi({x})

final words

Thus it is vital that the practitioner not abandon theoretical investigations too early. One must carefully track and deal with singularities in the equations, understand asymptotic behaviour, and develop decent analytic approximations to have any hope

• All of the techniques discussed here will fail

miserably unless one starts very close to the solution.

• how does one truncate SD equations

(beyond convenience)?

• be prepared for heartbreak

• 1

=

• 1
• =
• 1
• 1
• 1

=

• 1
• ghost

quark gluon

QCD

Exotic Theory: Schwinger-Dyson Equations

• J. Meyers, PhD Thesis, Pittsburgh, 2014.

Bethe-Salpeter

= = =

+ regular

2VPI 2VPI

Exotic Theory: Schwinger-Dyson Equations

= −1 −1 − −1 = −1 −1 −

electron photon

E.S. Swanson, arXiv 1008.4337

¯ ψψ(Nf) = aNf exp

• 2π
• N/Nf 1
• CBC

RL BC BC+CP

N(CBC) = 1.00 · N A

• N(RL)

= 1.10 · N A

• N(CP)

= N(BC) = 1.21 · N A

• Results: QED3

P.M. Lo, E.S. Swanson, PRD83, 065006 (2011) P.M. Lo, E.S. Swanson, PRD81, 034030 (2010)

no solution solution

parity preserving

η = N+ − N− N+ + N−

maximal parity breaking RL CBC

Results: QED3

P.M. Lo, E.S. Swanson, PRD83, 065006 (2011) P.M. Lo, E.S. Swanson, PRD81, 034030 (2010)

parity symmetric: solns are (M,-M) maximally broken (M,M) A solution exists for eta<0.4, implying parity symmetry breaking!

0.05 0.1 0.15 0.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Tc µ

Results: QED3, finite temperature and density

P.M. Lo, E.S. Swanson, PLB697, 164 (2011)

First calculation with full frequency depencence in a gauge theory. First calc in QED3 the treat the IR-div seriously.

P.M. Lo, E.S. Swanson, PRD89, 025015 (2014)

symmetric broken

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 3.5 4 M [GeV], 1/A p [GeV]

0.5 1 1.5 2 2.5 3 3.5 1 2 3 4 5 6 7 8 Z p [GeV] 0.5 1 1.5 2 2.5 3 3.5 4 0.01 0.1 1 10 100 h p2 [GeV2] 1 2 3 4 5 6 7 8 9 10 11 0.01 0.1 1 10 100 G [GeV-2] p2 [GeV2

Results: QCD propagators

• J. Meyers, E.S. Swanson, PRD90, 045037 (2014)
• 1
• 1

NOT TRUE that the mass of the visible universe comes from the Higgs… it comes from this —->

=

P k+;¹ k¡ ;º

= + + + +...

2VPI 2VPI

Exotic Theory: Schwinger-Dyson and Bethe-Salpeter Equations

• J. Meyers, E.S. Swanson, PRD87, 036009 (2013)

χµν(k+, k−) = ig2N

• d4q

(2π)4 χαβ(q+, q−) Cαβ

..µν G(q+)G(q−)

+ig2N

• d4q

(2π)4 χαβ(q+, q−) T αβ

..µν(q+, q−, k+, k−) G(q+)G(q−)G(Q)

+ig2N

• d4q

(2π)4 χ(q+, q−) Gµν(q+, q−, k+, k−) H(q+)H(q−)H(Q) +ig2 2

• tr [γµS(q+)χ

χ(q+, q−)S(q−)γνS(Q)] χ(k+, k−) = ig2N

• d4q

(2π)4 χ(q+, q−) H(q+, q−, k+, k−) H(q+)H(q−)G(Q) +ig2N

• d4q

(2π)4 χαβ(q+, q−) Bαβ(q+, q−, k+, k−) G(q+)G(q−)H(Q) χ χ(k+, k−) = g2CF

• γαS(k− + q−)γβG(q+)G(q−)χαβ(q+, q−)

+ig2CF

• γµS(q+)χ

χ(q+, q−)S(q−)γνG(Q)Pµν(Q)

Results: Glueballs

0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 4 1/ |P| (GeV) 0++ 0-+

Results: Glueballs

• J. Meyers, E.S. Swanson, PRD87, 036009 (2013)

= −1 −1 − −1 = −1 −1 −

iΓµ

RL(k, p) = γµ

iΓµ

CBC(k, p) = 1

2(A(k) + A(p))γµ

iΓµ

BC(k, p) = 1

2 (A(k) + A(p)) γµ + 1 2 A(k) − A(p) k2 − p2 (/ k + / p)(kµ + pµ) − B(k) − B(p) k2 − p2 (kµ + pµ) iΓµ

CP (k, p) = 1

2 A(k) − A(p) d(k, p)

• γµ(k2 − p2) − (k + p)µ (/

k − / p)

• d(k, p) = (k2 − p2)2 + (M(k)2 + M(p)2)2

k2 + p2

Schwinger-Dyson Equations Vertex Ansa̎tze

iS−1 = A/ p − B

+ = + =

+

=

+ P k+;¹ k¡ ;º

1 1.5 2 2.5 3 3.5 4 2 4 6 8 10 12 14 16 18 20 h p2 (GeV2)

results: ghost

I.L. Bogolubsky, E.M. Ilgenfritz, M. Muller-Preussker and

• A. Sternbeck, Phys. Lett. B676, 69 (2009)

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 3.5 4 Z, M (GeV) p (GeV) M Z

¯ ψψ(1 GeV) = (251 MeV)3 fπ = 240 MeV

results: quark

P.O. Bowman et al., Phys. Rev. D71, 054507 (2005).

S(k) = i Z(k) / k − M(k)

more on confinement

Kugo-Ojima confinement criterion

D(p) = − 1 p2 1 1 + u(p)

u(p) → −1 p → 0

Alkofer, von Smekal, Fischer

Heavy Quarks

modelling, quarks, heavy quarks

• many body
• relativistic
• strong coupling (contrast to QED)
• quantum
• nonlinear

challenges

QCD is

• physical pictures can change depending on the
• scale
• observables
• gauge

ρ decay vs. ρ scattering quark mass, glue confinement in Coulomb gauge vs. Weyl gauge

Modelling QCD

• we seek to understand low energy hadronic physics
• - we are fortunate that we have the theory, but it is

not always helpful!

• cf. the theory of DNA:
• to make progress we need to identify the

appropriate effective degrees of freedom and their interactions

H =

• i

−¯ h2 2m ∇2

i +

• i<j

qiqj rij

ex: the bag model fermions current quarks bosons bag pressure (+ perturbative one gluon exchange) ex: the flux tube model fermions constituent quarks bosons flux tubes

Modelling QCD

• spontaneous chiral symmetry breaking implies

both the existence of Goldstone bosons and constituent quarks

• it is the structure of the vacuum that gives chiral

symmetry breaking and confinement

• effective degrees of freedom should be derived

from QCD to the extent possible

it is desirable to incorporate the physics of the vacuum and chiral symmetry breaking into the model from the beginning

• nly in this way can we recover perturbative QCD in the high energy regime

current quarks evolve into constituent quarks at scales < Λ QCD

Modelling QCD

Constituent Quarks

pre-QCD quarks: m~ 5 GeV Copley, Karl, & Obryk: m ~ 330 MeV QCD: m(2 GeV) ~ 4 MeV but recall that quarks are not observable ⇒ different kinds of quark masses exist: current/constituent

.1 ((((((((

~ ~ ~

1)+ 1)+ 1)+ 1)+

• )
• ✝

✝ *

k,m

• k,m

✝

• ✝

✝

k,m

• k,m

✝ 2 2 3 3 4 4

CbpmC† = ηCdpm Cd†

pmC† = ηCb† pm

L S JP C

(2S+1)LJ example

0 0 0−+

1S0

π 0 1 1−−

3S1

ρ 1 0 1+−

1P1

h1 1 1 (0, 1, 2)++ 3P(0,1,2) a0, a1, a2 2 0 2−+

1D2

π2 2 1 (1, 2, 3)−− 3D(1,2,3) ρ, ρ2, ρ3

not in the list: 0 , (even) , (odd)

• +-
• +

‘(quantum number) exotics’ discovering such a state would be the first time a meson has been observed with no qq content

• Bohr levels with a Bohr radius (2/3mα ) ~ 0.01

fm

• L , L , L & L splittings are due to tensor

and spin-orbit interactions

• S - S splittings are due to the contact

interaction

ψ, Υ

s 3 1 3 3 3 1 1 L L L+1 L-1

Heavy Quarkonia

Lorentz structure:

sources of spin-dependence are (i) gluon exchange (ii) corrections to the static potential (iii) meson exchange (Fock sector mixing) (iv) instanton forces

Constituent Quarks (heavy)

spatial regimes:

r < 1/10 fm r ∼ 1/2 fm r > 1 fm

• ne gluon exchange

confinement

• ne pion exchange

make a (field-theoretic) Foldy-Wouthuysen transformation

σ.B σ.B (a)

σ.B σ.B (b)

D2 σ.B (c)

D2 σ.B (d)

zero hyperfine + tensor V V

1 2

Interactions

sig1_i*sig1_j = del_ij+ eps_ijk sig k so = B^2 so not spin- dependent

spin-dependence in the confinement potential

examine in Coulomb gauge via the Foldy-W

• uthuysen transformation

VSD(r) = σq 4m2

q

+ σ¯

q

4m2

¯ q

• · L

1 r dVconf dr + 2 r dV1 dr

• +

σ¯

q + σq

2mqm¯

q

• · L

1 r dV2 dr

• +

1 12mqm¯

q

• 3σq · ˆ

r σ¯

q · ˆ

r − σq · σ¯

q

• V3 +

1 12mqm¯

q

σq · σ¯

qV4

+1 2 σq m2

q

− σ¯

q

m2

¯ q

• · L +

σq − σ¯

q

mqm¯

q

• · L
• V5.

(1)

VSI(r) = −3 4 αs r + br

model building — more later

Eichten & Feinberg Ng, Pantaleone, & Tye

U = (VC + Vso + Vhyp)

⃗ λ1 2 · −⃗ λ2

∗

2

VC = α

r − απ 2 ( 1 m2

1 +

1 m2

2 )δ(⃗

r) Vhyp =

α 4m1m2

• ⃗

σ1·⃗ σ2 r3

− 3(⃗

σ1·⃗ r)(⃗ σ2·⃗ r) r5

− 8π

3 ⃗

σ1 · ⃗ σ2 δ(⃗ r)

• Vso = −

α 2m1m2r

• ⃗

p1 · ⃗ p2 + ⃗

r·(⃗ r·⃗ p1)⃗ p2 r2

• −

α 4r3 (⃗ r×⃗ p1·⃗ σ1 m2

1

− ⃗

r×⃗ p2·⃗ σ2 m2

2

) −

α 2m1m2r3 (⃗

r × ⃗ p1 · ⃗ σ2 − ⃗ r × ⃗ p2 · ⃗ σ1)

spin dependence

• ge approximation/model

U = (VC + Vso + Vhyp)

⃗ λ1 2 · −⃗ λ2

∗

2

VC = α

r − απ 2 ( 1 m2

1 +

1 m2

2 )δ(⃗

r) Vhyp =

α 4m1m2

• ⃗

σ1·⃗ σ2 r3

− 3(⃗

σ1·⃗ r)(⃗ σ2·⃗ r) r5

− 8π

3 ⃗

σ1 · ⃗ σ2 δ(⃗ r)

• Vso = −

α 2m1m2r

• ⃗

p1 · ⃗ p2 + ⃗

r·(⃗ r·⃗ p1)⃗ p2 r2

• −

α 4r3 (⃗ r×⃗ p1·⃗ σ1 m2

1

− ⃗

r×⃗ p2·⃗ σ2 m2

2

) −

α 2m1m2r3 (⃗

r × ⃗ p1 · ⃗ σ2 − ⃗ r × ⃗ p2 · ⃗ σ1)

One Gluon Exchange

F . F

i j

U = (VC + Vso + Vhyp)

⃗ λ1 2 · −⃗ λ2

∗

2

VC = α

r − απ 2 ( 1 m2

1 +

1 m2

2 )δ(⃗

r) Vhyp =

α 4m1m2

• ⃗

σ1·⃗ σ2 r3

− 3(⃗

σ1·⃗ r)(⃗ σ2·⃗ r) r5

− 8π

3 ⃗

σ1 · ⃗ σ2 δ(⃗ r)

• Vso = −

α 2m1m2r

• ⃗

p1 · ⃗ p2 + ⃗

r·(⃗ r·⃗ p1)⃗ p2 r2

• −

α 4r3 (⃗ r×⃗ p1·⃗ σ1 m2

1

− ⃗

r×⃗ p2·⃗ σ2 m2

2

) −

α 2m1m2r3 (⃗

r × ⃗ p1 · ⃗ σ2 − ⃗ r × ⃗ p2 · ⃗ σ1)

One Gluon Exchange

Darwin term

U = (VC + Vso + Vhyp)

⃗ λ1 2 · −⃗ λ2

∗

2

VC = α

r − απ 2 ( 1 m2

1 +

1 m2

2 )δ(⃗

r) Vhyp =

α 4m1m2

• ⃗

σ1·⃗ σ2 r3

− 3(⃗

σ1·⃗ r)(⃗ σ2·⃗ r) r5

− 8π

3 ⃗

σ1 · ⃗ σ2 δ(⃗ r)

• Vso = −

α 2m1m2r

• ⃗

p1 · ⃗ p2 + ⃗

r·(⃗ r·⃗ p1)⃗ p2 r2

• −

α 4r3 (⃗ r×⃗ p1·⃗ σ1 m2

1

− ⃗

r×⃗ p2·⃗ σ2 m2

2

) −

α 2m1m2r3 (⃗

r × ⃗ p1 · ⃗ σ2 − ⃗ r × ⃗ p2 · ⃗ σ1)

contact hyperfine term tensor hyperfine term

U = (VC + Vso + Vhyp)

⃗ λ1 2 · −⃗ λ2

∗

2

VC = α

r − απ 2 ( 1 m2

1 +

1 m2

2 )δ(⃗

r) Vhyp =

α 4m1m2

• ⃗

σ1·⃗ σ2 r3

− 3(⃗

σ1·⃗ r)(⃗ σ2·⃗ r) r5

− 8π

3 ⃗

σ1 · ⃗ σ2 δ(⃗ r)

• Vso = −

α 2m1m2r

• ⃗

p1 · ⃗ p2 + ⃗

r·(⃗ r·⃗ p1)⃗ p2 r2

• −

α 4r3 (⃗ r×⃗ p1·⃗ σ1 m2

1

− ⃗

r×⃗ p2·⃗ σ2 m2

2

) −

α 2m1m2r3 (⃗

r × ⃗ p1 · ⃗ σ2 − ⃗ r × ⃗ p2 · ⃗ σ1)

One Gluon Exchange

spin orbit term spin independent term

Gupta & Radford, PRD33, 777 (86)

Constituent Quarks (heavy)

perturbative potentials

V1(mq, m¯

q, r)

= −br − CF 1 2r α2

s

π

• CF − CA
• ln
• (mqm¯

q)1/2r

• + γE
• V2(mq, m¯

q, r)

= −1 r CF αs

• 1 + αs

π b0 2 [ln (µr) + γE] + 5 12b0 − 2 3CA + 1 2

• CF − CA
• ln
• (mqm¯

q)1/2r

• + γE
• V3(mq, m¯

q, r)

= 3 r3 CF αs

• 1 + αs

π b0 2 [ln (µr) + γE − 4 3] + 5 12b0 − 2 3CA+ + 1 2

• CA + 2CF − 2CA
• ln
• (mqm¯

q)1/2r

• + γE − 4

3

• V4(mq, m¯

q, r)

= 32αsσ3e−σ2r2 3√π V5(mq, m¯

q, r)

= 1 4r3 CF CA α2

s

π ln m¯

q

mq (1)

Constituent Quarks (heavy)

perturbative potentials

Constituent Quarks (heavy)

V = 1 2

• d3x d3y ¯

ψΓψ(y)K(x − y) ¯ ψΓψ(x)

Γ = 1 1 Γ = γµ Γ = γ5

scalar vector pseudoscalar

model the 1/c^2 spin-dependence in the confinement interaction… NB: there is no reason for QCD to take on this simple form!

spin-dependence in the confinement potential

Dieter Gromes

Constituent Quarks (heavy)

ρ = 2++ − 1++ 1++ − 0++

‘scalar’ confinement is preferred compare to experimental spin splittings

0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5

• <r> (GeV-1)
• scalar

vector

Constituent Quarks (heavy)

Howard Schnitzer

H =

• i

mi +

• i

p2

i

2mi + C +

• i<j
• −
• − αs

rij + 3 4brij

• ⃗

Fi · ⃗ Fj + V oge

SD (rij) + V conf SD

(rij)

• variants:

running coupling smeared delta functions relativized perturbative corrections Mercedes baryon potential instanton potential flip flop potential apply to light quarks?

Constituent Quarks (heavy)

L*S = [J(J+1) - L(L+1)/2 - S(S+1)]/2 L*S requires L=J and S=0,1 so 1P1+3P1; 1D2+3D2; 1F3+3F3 for tensor: 3S1-3D1; 3P2-3F2; 3D3-3G3

E(2++) = E + S/4 - T/5 + L E(1++) = E + S/4 +T - L E(0++) = E + S/4 - 2T - 2L E(1+-) = E - 3S/4 S = <V >

con ten

T = <V > L = <V >

so

T= 13 MeV, L=28 MeV T= 20(2) MeV, L=34(3)MeV

Heavy Quarkonia

Charmonium Spectrum

expt GI BGS

nonrelativistic quark models

Charmonium Spectrum

lattice gauge theory

3000 3200 3400 3600 3800 4000 4200 4400 4600 4800 expt BGS 1S 2S 1D 3S 2D 4S 3D 5S 4D Y Y Y Y

Charmonium V ectors -- Constituent Quark Model

3000 3200 3400 3600 3800 4000 4200 4400 4600 4800 expt BGS 1S 2S 1D 3S 2D 4S 3D 5S 4D Y Y Y Y DD DD* D*D* DsDs DsDs* Ds*Ds* DD1 D*D0 D*D2 DsDs1 Ds*Ds0 Ds*Ds2 DsDs1

Charmonium V ectors -- Constituent Quark Model

DD DD DsDs DsDs

0-+ 0-+ 1-- 1-- 2-+ 2-+ 1-+ 1-+ 0++ 0++ 1+- 1+- 1++ 1++ 2++ 2++ 3+- 3+- 0+- 0+- 2+- 2+-

500 1000 1500 M-Mhc HMeVL

• range is hybrids (later)

black = expt

DD DD DsDs DsDs

0-+ 0-+ 1-- 1-- 2-+ 2-+ 1-+ 1-+ 0++ 0++ 1+- 1+- 1++ 1++ 2++ 2++ 3+- 3+- 0+- 0+- 2+- 2+-

500 1000 1500 M-Mhc HMeVL

perturbative QCD

Bf(J/ψ → γη) Bf(J/ψ → γη) = 11.01 ± 0.29 ± 0.22 52.4 ± 1.2 ± 1.1 = 0.21 ± 0.04 Bf(ψ(2S) → γη) Bf(ψ(2S) → γη) = < 0.02 1.19 ± 0.08 ± 0.03 < 0.018

why the difference? Speculate that it is due to interference with hybrids?

*10-4

• T. Pedlar [CLEO], Moriond, 2009
• M. Shepherd [CLEO], GHP09

perturbative QCD

CLEO, PRD78, 091501 (2008)

R = Γ(χc2 → γγ) Γ(χc0 → γγ) = 4 15(1 − 1.76αs) R = 0.66 ± 0.07 ± 0.04 ± 0.05 keV 2.36 ± 0.35 ± 0.11 ± 0.19 keV = 0.278 ± 0.050 ± 0.018 ± 0.031 = 0.12 (αs = 0.32)

note: 4/15 = 0.27!

• W. Bardeen et al. PRD18, 3998 (78)

perturbative QCD

e+e− widths

Γ(3S1 → e+e−) = 16α2

sQ2 |ψ(0)|2

M 2 Γ(3D1 → e+e) = 50α2

sQ2 |ψ⇥⇥(0)|2

M 2m4

c

van Royen and Weisskopf

state qn thy (keV) expt (keV) J/ψ 13S1 12 5.40(17) ψ 23S1 5 2.12(12) ψ(3770) 13D1 0.06 0.26(4) ψ(4040) 33S1 3.5 0.75(15) ψ(4159) 23D1 0.1 0.77(23) ψ(4415) 43S1 2.6 0.47(10)

} mixing?

perturbative QCD

the pi-rho puzzle

Qh ≡ Bf(ψ⇥ → h) Bf(J/ψ → h) = Bf(ψ⇥ → e+e) Bf(J/ψ → e+e) ≈ 12.7%

Appelquist and Politzer, PRL34, 43 (75)

5 10 15 20 25 30

Mo et al., hep-ph/0611214

πρ KK p¯ p p¯ pπ

4π 6π

3(π+π−)π 2(π+π−)π 2(π+π−π 2(K+K−) K+K−π+π− K+K−π+π−π0 K+K−π+π−π+π− p¯ pπ+π−

agrees with LO pQCD, but NLO is negative

perturbative QCD

• T. Pedlar [CLEO], Moriond, 2009

PRL101, 101801 (2008)

BF(J/ψ → γγγ) = (1.17 ± 0.3 ± 0.1) · 10−5

a word on the ηb

BaBar, PRD78, 091501 (2008)

Υ(3S) → γηb Υ(2S) → γηb now confirmed

arXiv:0903.1124

M = 9388.9 ± 3 ± 3 M = 9392.9 ± 5 ± 2 Mcomb = 9390.4 ± 3 ∆Mhyp = 69.9 ± 3

• hyperfine splitting agrees with lattice
• hyperfine splitting disagrees with pNRQCD
• hyperfine splitting gives
• test NRQCD
• test mixing with A0

αs(Mηb)