sign in sign up
<<
Home , Color confinement

Running coupling constants Color confinement -antiquark potential

Properties of QCD at low energies

Harri Waltari

University of Helsinki & Helsinki Institute of Physics

Autumn 2018

H. Waltari Properties of QCD at low energies Running coupling constants Color confinement Quark-antiquark potential Content

The idea of this lecture is to explain qualitatively what is look qualitatively how the results of DIS (scattering off free ) can be understood even if free quarks are not seen look at interquark potentials and the spectrum Detailed computations/derivations are beyond the scope of this course. This lecture corresponds to chapters 10.4, 10.5 and 10.8 of Thomson’s book.

H. Waltari Properties of QCD at low energies Running coupling constants Color confinement Quark-antiquark potential Quantum corrections require renormalization

If you add loop corrections, you end up with infinities of two types — poles in the propagator (especially soft collinear photons/ in the infrared) and infinite loop momenta (ultraviolet) In the infrared end you can absorb the infinities to the parton distributions — you define that the soft gluons are a part of the quark, you can change the scale at which this separation is done by the DGLAP equations For this procedure to be successful, you must consider only infrared safe observables It took ∼ 20 years to learn how to handle UV divergences, the procedure is called renormalization In the renormalization procedure you first have to regularize the infinities, the simplest way is to change the upper limit of the momentum integral from infinity to a finite number (unfortunately this is not Lorentz or gauge invariant, the better technique is dimensional regularization, where the spacetime dimension is changed d = 4 → d = 4 − )

H. Waltari Properties of QCD at low energies Running coupling constants Color confinement Quark-antiquark potential Quantum corrections require renormalization

Once we have a finite result, we match it (renormalize it) with the experimental result at some scale (known as renormalization scale) The result will depend on the choice of scale at any finite order in perturbation theory, the infinite series should be independent of the choice

The perturbative result is reliable if g ln(Λ1/Λ2) is small, where Λi are the highest and lowest momentum scales in the problem, but if there are mass scales that are far apart problems arise It is possible to avoid large logarithms by making your couplings (and masses etc.) scale dependent, this can be done by so the called technique Details of renormalization will be considered in the QFT courses

H. Waltari Properties of QCD at low energies Running coupling constants Color confinement Quark-antiquark potential The running of gauge couplings comes from vacuum polarization diagrams

The first order perturbation to the running of the gauge coupling comes from so called vacuum polarization diagrams, which alter the propagator In QED only the fermionic corrections exist, in QCD the loops are also present

H. Waltari Properties of QCD at low energies Running coupling constants Color confinement Quark-antiquark potential The vacuum polarization diagrams form a geometric series

Denote the bare (unobservable) charge by e0, the photon propagator 2 2 (at Born level) would be P0 = e0 /q Denote the sum of all diagrams that start from a photon and end to a photon by π(q2), where q is the momentum of the (incoming or outgoing) photon The class of diagrams is called one particle irreducible (1PI) and they can be computed to a given number of loops by the methods of perturbation theory Now the full propagator becomes 2 2 2 2 2 P0 e (q ) P = P0 + P0π(q )P0 + P0π(q )P0π(q )P0 ... = 2 = 2 1−P0π(q ) q

H. Waltari Properties of QCD at low energies Running coupling constants Color confinement Quark-antiquark potential We may define a scale-dependent coupling

2 2 2 2 2 Now by writing P0 = e0 /q and defining Π(q ) = π(q )/q we get 2 2 2 e0 e (q ) = 2 2 1−e0 Π(q ) The problem is that Π(q2) is divergent at all q2 so the result in this form doesn’t make sense To get a finite result, we shall regularize Π(q2) by some means ⇒ finite and scale-dependent result for e(q2) Next we match this result to the measured value at some q2 = µ2, where µ is the renormalization scale 2 2 2 e (µ ) Then we may solve for e0 = 1+e2(µ2)Π(µ2) , this result depends on the chosen scale 2 2 2 2 e (µ ) Substituting this back gives e (q ) = 1−e2(µ2)[Π(q2)−Π(µ2)] The divergences in Π(q2) − Π(µ2) will cancel so the regulator can be removed

H. Waltari Properties of QCD at low energies Running coupling constants Color confinement Quark-antiquark potential In QED the gauge coupling becomes stronger at high energies

At one-loop order we get 2 2 1 2 2 Π(q ) − Π(µ ) = 12π2 ln(q /µ )

The running of the QED gauge coupling α

2 2 α(µ ) α(q ) = 2 α(µ ) 2 2 1− 3π ln(q /µ )

At low energies 2 α(4me ) ' 1/137, running leads 2 to α(mZ ) ' 1/128, a difference that can be measured

H. Waltari Properties of QCD at low energies Running coupling constants Color confinement Quark-antiquark potential In QCD the gluon loops give an opposite contribution

In QCD the gluon loops come with an opposite sign compared to the fermion loops

2 2 11Nc −2Nf 2 2 The full contribution is Π(q ) − Π(µ ) = − 48π ln(q /µ ), where Nc is the number of colors and Nf the number of quark flavors

For Nc = 3 the result has an opposite sign compared to QED running as long as Nf ≤ 16, so in QCD the gauge coupling is strong in the infrared and moderate in the ultraviolet

The most precise value for αs is obtained at the Z-pole, 2 αs (mZ ) = 0.1184 ± 0.007, at mτ the value is 2 αs (mτ ) = 0.3186 ± 0.056 Theories where the coupling is large in the infrared and becomes weak in the ultraviolet are called asymptotically free Gross, Politzer and Wilczek got the Nobel Prize of 2004 for discovering in QCD (1974)

The running of αs now known to five loops

H. Waltari Properties of QCD at low energies Running coupling constants Color confinement Quark-antiquark potential Asymptotic freedom explains why free quarks are not seen

From the running of αs we see that at high momentum transfers (i.e. the conditions of DIS) the quarks are bound rather weakly ⇒ Scattering off point-like objects ⇒ Bjorken scaling At lower energies coupling of quarks or gluons to gluons much stronger, hence a system of colored particles more tightly bound If you try to separate quarks, a flux tube of the color field appears between them ⇒ increasing potential energy Lattice computations compatible with a linear potential V (r) ' κr, with values of κ ' 1 GeV/fm

H. Waltari Properties of QCD at low energies Running coupling constants Color confinement Quark-antiquark potential The potential energy of the color field produces new

Separating quarks far apart would require a huge amount of energy ⇒ at some point enough to form new quark-antiquark pairs ⇒ flux tube breaks down to hadrons cannot be computed from perturbation theory, either lattice computations or phenomenological models need to be used

H. Waltari Properties of QCD at low energies Running coupling constants Color confinement Quark-antiquark potential Observable hadrons are color singlets

Since free colored particles cannot exist, we must assume that all observable hadrons are colorless, i.e. singlets of SU(3).

Mesons are singlets of 3 ⊗ 3 ψc = √1 (rr + gg + bb) 3 This state will be annihilated by all raising and lowering operators of SU(3).

Baryons are singlets of 3 ⊗ 3 ⊗ 3 ψc = √1 (rgb − grb + brg − rbg + gbr − bgr) 6 The color part of the baryon wave function is antisymmetric in the exchange of any two quarks.

H. Waltari Properties of QCD at low energies Running coupling constants Color confinement Quark-antiquark potential The interquark potential has a Coulomb part and a confining part

The most used quark-antiquark potential is the so called Cornell potential: Cornell potential (Eichten et al. 1975)

αs Vqq(r) = −C r + κr

Currently no reliable method for deriving this from QCD The confining part κr is inferred from lattice computations The Coulomb-like part is included due to the analogous form of the QCD gluon exchange vertex compared to the QED photon exchange The minus sign is due to the fact that the Born level potential between a fermion and an antifermion coming from a vector mediator is attractive (for quark-quark potential the sign would be positive) The factor C is a color factor, computed in the book to be 4/3

H. Waltari Properties of QCD at low energies Running coupling constants Color confinement Quark-antiquark potential The Cornell potential best describes with heavy quarks

The usage of the Cornell potential somewhat ambiguous, since the running of αs significant at the energy scales relevant for hadrons (rather good fits with data achieved by αs ' 0.2, but should not expect to give exact results) Also the assumption of nonrelativistic quarks used, hence should work best for mesons with b- or c-quarks (other techniques exist for lighter mesons)

H. Waltari Properties of QCD at low energies Running coupling constants Color confinement Quark-antiquark potential Summary

Gauge couplings run with energy scale, in QED the coupling is stronger at high q2, in QCD weaker at high q2; this property of QCD is known as asymptotic freedom

At low energies the large value of αs leads to color confinement and only color singlets exist as free particles Gluon self-interactions are essential in understanding the behaviour of hadrons Quark-antiquark interactions in mesons with heavy quarks can be described with the Cornell potential V (r) = a/r + br

H. Waltari Properties of QCD at low energies