Emergence of large effective massesof quarks

A tribute to Yoichiro NambuTalk at the 2008 Brijuni Conference,NATO Advanced research workshop

”Hydrogen: A universal saga”

D. Klabucar (speaker) and D. Horvatic

Physics Department, University of Zagreb, Croatia

Brijuni Archipelago, Croatia, 25. – 29. August 2008

What is the problem, what will be the answer?◮ Fundamental theory of strong interactions = QCD, confirmed by

high-energy experiments. They and QCD say that protons p andneutrons n (and other light hadrons) contain light u and d quarkswhich are a couple of hundred times lighter than the proton.

◮ light, almost massless quarks ⇔ approximate chiral symmetry◮ BUT at low energies, p, n & other hadrons, seem to be built of

very different quarks - the so-called ‘constituent’ quarks withMu,d ∼ mp/3, required by baryon masses, magnetic moments ...

◮ i.e., light ‘constituent’ u and d quarks ∼ 100 times moremassive than the light ‘fundamental’ QCD u and d quarksconsistent with experiments at high Energies

◮ What are these “two kinds of quarks”? What is their relation?◮ We sketch the answer: at low E, QCD is strongly interacting

and generates strongly dressed quasiparticles - effectivelymassive ‘constituent’ quarks - through a nonperturbativeeffect called spontaneous or dynamical chiral symmetrybreaking (DChSB) ⇒ the Nobel prize 2008 to Y. Nambu

H nucleus = proton p = the lightest & only absolutely stable hadron

p and its neutral partner n, are composite p’cles of size ∼ 10−15 m.

◮ 1st indication that nucleons (p and n) 6= point particles, were theirmagnetic moments: µp 6= µN ≡ Dirac nuclear magneton = e

2mp

,while µn 6= 0. Experimentally, µp = 2.79µN , µn = −1.91µN

◮ Nowadays, huge evidence on quark substructure of hadrons

◮ Baryons: 3 quarks bound by gluons according to QCD, e.g.,p ∼ uud, n ∼ udd,Λ ∼ uds, ..., etc . Mesons: qq bound states.

uu

d

Figure: proton or ∆+

u d

Figure: π+ or ρ+ meson

QCD Lagrangian density contains gluons Aaµ(x) (a = 1, . . . , 8) and

quarks ψq(x) of flavors q = u, d, s, . . . with masses mq:

L(x)QCD =

{

ψq(x)

[

γµ

(

∂µ − igλa

2Aa

µ(x)

)

+mq

]

ψq(x)

+1

4F a

µν(x)F aµν(x) +

1

2ξ∂µAa

µ(x) ∂νAaν(x)

}

, (1)

where the repeated indices (q, a, µ, ν) are summed over, and

F aµν(x) ≡ ∂µAa

ν(x) − ∂νAaµ(x) + gfabcAb

µ(x)Acν (x), (2)

where {fabc : a, b, c = 1, . . . , 8} are the structure constants of SU(3),and { 1

2λa; a = 1, . . . , 8} are 3 × 3 Gell-Mann matrices/2 = generators

of SU(3) for the fundamental representation, ψTq = (ψred

q , ψgreenq , ψblue

q ).g is the coupling of the SU(3) generalized charges - “colors”.

Politzer, Wilczek, Gross - Nobel prize 2004 - for perturbative QCD

◮ pQCD applicable to high-energy processes (P 2 & 2 GeV2)◮ Asymptotic freedom : the QCD interaction strength falls with

growing spacelike P 2; the higher P 2 is, the better pQCD works.◮ pQCD works with mu ∼ (4 ± 2.5) MeV, md ∼ (6 ± 2.5) MeV.◮ But at low energies , many things (e.g., baryon masses and

magnetic momenta) require ‘constituent quark masses’ Mq

◮ For u, d quarks, Mu,d ∼ mp,n/3 ∼ 330 MeVHow to bridge this mass gap Mu,d −mu,d ≈ Mu,d ∼ 1/3GeV ???

Nuclei – consist of nucleons,which consist of u, d quarks ⇒Understanding masses of u, dquarks is the key to understandingof more than 98% of the mass ofthe “ordinary” matter in theUniverse! =⇒The key task of subnuclearphysics is to understand p & n interms of quarks and gluons!

... But, hadrons exist thanks to non- perturbative QCD = unsolved !

◮ 1. non-pert. effect: CONFINEMENT = no free quarks seen =verybig problem to understand... here just take it as an empirical fact

◮ 2. non-pert. effect to explain = generation of hadron masses:e.g., mp,n ≈ 940 MeV with mu,d ∼ 4−6 MeV. This also requiresexplanation of the relation between the pQCD (Lagrangian) quarkmasses mu,d and the spectrum/constituent quark masses Mu,d

What is the mass contribution of gluons?Gluons = massless, but as mediators of QCD interaction, also causethe self-energy contribution to quark masses: mq −→ mq + Σq

= dressing of quarks by gluons = the effect similar to the emergenceof a Debye mass MD of a photon propagating in a dense e− gas!

Massless photon thus acquires MD ∝ P 2F 6= 0

◮ screening of the interaction beyond roughly the length ∼ 1/MD

◮ The photon propagator changes, 1P 2 → 1

P 2+M 2D

Quark and gluon propagators modifiedin a similar way by a ‘cloud’ of virtualparticles – gluons and qq pairs ⇒complicated non-perturbative QCDvacuum = “propagation medium” ⇒Large quasiparticle masses at low P 2

⇒ P -dependent effective, quasiparticle masses Mq(P2) from mq:

quark: Sq(P )free =1

iγ ·P +mq

−→ Sq(P ) =1

iγ ·P +mq + Σq(P )

Sq(P ) =1

iγ ·P Aq(P 2) +Bq(P 2)=

1/Aq(P2)

iγ ·P + Mq(P 2), Mq ≡ Bq

Aq

Figure: Fundamental DOF resolved at high P 2

Figure: Effective quarks at low P 2

How to get this? In perturbation theory, Mq small: Mq ∼ mq, since

Bq(p2)pert = mq

(

1 − α(p2)

πln

[

p2

m2q

]

+ ...

)

→ 0 for mq → 0 (Chiral limit)

But pQCD anyway holds only for p2 ≫ 1 GeV2 (high-E processes)

⇒ in bound states, we need a non-perturbative method for Mq

⇒ use DS equations for propagators of a theoryFor Sq(p) - the “gap” DS equation, Sq(p)

−1 = [Sq(p)free]−1 + Σq(p)

Sq(p)−1 = [iγ · p+mq] +

∫

d4l

(2π)4g2Gµν(p− l)

λa

2γµ Sq(l) Γa

ν(l, p)

λa

2γµ Sq(l) Γaν(l, p)

= +

Gap eq. studied mostly in “ladder approximation”, Γaν(l, p) → λa

2 γν .

Solving the “gap” DS equation for Sq(p) exhibits a spectacular result:

DChSB = generation of mass from nothing :

i.e., even in the chiral limit (mq → 0), strong QCD interaction leadsto Mq 6= 0 ... and big Mq (∼ order of typical hadronic scales mp/3=⇒ more important than Higgs mechanism for mass generation!).

How do we know that? We do not really know QCD interactions atlow E! – Because it is confirmed by all QCD models as soon as theirinteractions have sufficient strength at low energies/momenta! (Morerecently, also confirmed by lattice Monte Carlo QCD simulations.)

We illustrate it by two simple, extremely different models for g2Gµν(p):

1.) Constant (step) in momentum space (cut-off at l2 = Λ2):2.) δ–function in momentum space

1.) g2GNJLµν (p− l) = δµν GNJL Θ(Λ2 − l2) , [⇔ δ(x) in x−space]

where the constant GNJL is the effective coupling strength in the modelwhich is basically the Nambu–Jona-Lasinio (NJL) model.

(For nucleons, only q = u, d needed ... since md ≈ mu =⇒ we dropq-subscript , i.e., mq, Sq → m,S and Aq, Bq,Mq → A,B,M)

For 1.), solutions for the functions in S(p) : A(p2) = 1 ⇒ M(p2) = B(p2),

B(p2) = constant ≡M = m+M1

3π2GNJL

(

Λ2 −M 2 ln[

1 + Λ2/M 2])

,

Solve the blue equation for M – first, in the chiral limit, m = 0, toexhibit exclusively dynamic mass generation: DChSB.

◮ Obviously, the solution M ≡ 0, consistent with perturbationtheory, exists always, for any interaction strength GNJL.

◮ The solution M 6= 0 also exists in the model 1.), but only for theinteraction stronger than the critical value,

GNJL >3π2

Λ2≈ 29.6

Λ2.

E.g., for GNJL = 4π2/Λ2 and the typical hadronic scale Λ ≈ 1 GeV,dressed quark mass M ≈ 0.33 GeV due to DChSB, although m = 0.

The realistic explicit ChSB, m = mu,d<∼ 0.01 GeV, gives only a small

correction to the DChSB-generated constituent quark mass:

Figure: The blue curves show the coupling-strength dependence of the dressedmass M generated by the “NJL” interaction 1.) The dashed curve is the chiral limitcase, m = 0, and the solid curve is the case of a small “bare” (i.e., Lagrangian) massm0 = m = 0.01Λ (red dotted line). Masses are in units of Λ, the coupling is in the unitsof Λ−2. The choice Λ = 1 GeV estimates the upper limit of fully nonperturbative QCDdynamics. The dashed vertical marks the critical coupling strength 3π2/Λ2

≈ 29.6/Λ2,i.e., the onset of the purely dynamically generated nonzero M .

2.) Munczek-Nemirovsky: exactly soluble model for low-energy QCD:

g2GMNµν (k) = (2π)4 G δ4(k)

[

δµν − kµkν

k2

]

. [⇔ constant in x−space]

G = const defines the interaction strength and model’s mass-scale.Meson masses reproduced well if

√G∼mp/3 ... nice, as

√G=M(0).

In the chiral limit ( m = 0), the solutions are in closed form:

A(p2) =

{

2 ; p2 ≤ G12

(

1 +√

1 + 8G/p2)

; p2 > G (3)

B(p2) = 2M(p2) =

{

2√

G − p2 ; p2 ≤ G0 ; p2 > G . (4)

◮ DChSB : M(p2) 6= 0 for all p2 below the interaction strength G◮ The above chiral-limit behaviour is modified by m 6= 0 only

quantitatively, as in the model 1.)◮ In MN model, propagator functions A & B, and resulting M,

differ strongly from their free-particle forms & are p2-dependent– same as in realistic DS approaches and in true QCD .

Lattice Monte Carlo simulations – access non-perturbative QCD without anymodeling, but (unlike DS approach) have difficulties at very low p and m.Realistic DS approaches to QCD (also incorporating precisely knownpQCD at high p2, where M(p2) → m) agree with lattice :

Figure: Lattice data for M(p2) compared with the numerical solutions of the gap equation forthe realistic QCD model used by Bhagwat et al., Phys. Rev. C 68, 015203 (2003). Dashed curve:the solution in the chiral limit, m = 0. Three solid curves: solutions for M(p2) for the respectivecurrent-quark masses m = 30 MeV, 55 MeV, and 110 MeV. [Adapted from Bhagwat et al.] Reddashed curve is the chiral-limit solution (4) for M(p2) from the MN model 2.) with G = 0.281 GeV2 ,and the solid green curve is the corresponding numerical solution with m = 5 MeV.

Conclusions

We have bridged the gap between mq and Mq, i.e., sketched theexplanation for:

◮ the emergence of the constituent quark (effective at lowenergies and momenta) from the fundamental, “bare” or currentquark of QCD Lagrangian – and thus, how the low-energyhadronic “constituent quark models” and QCD proper are related.The constituent quark is “constructed” through the gap equation.

◮ I.e., we have shown that dynamical dressing through the gapequation rises the quark mass from mu,d to Mu,d(0) by somemp/3.

◮ Thus the origin of mp mass is clarified – but also of all otherhadron masses, which no other appoach can (as DS approachwhich includes BS equations, is the only bound-state approachwith good chiral behavior and thus reproduces otherwisemysteriously low masses of pseudoscalar octet mesons.)

◮ Clarifying the origin of mp mass means clarifying the origin ofmore than 98% of the visible mass in the Universe!