Vacuum$polariza1on$in$QED$ force$between$two$electrons$ (in$natural$units)$ +! p e = 4⇡↵ -!
+! -! +! -! +!
-! +!
The$interac1on$strength$of$the$two$electrons$gets$stronger$as$the$distance$between$ them$becomes$smaller$ Electric charge is screened; interaction becomes weak at large distances Vacuum$polariza1on$in$QCD$
The left diagram is shared by QED and QCD which renders the interaction stronger at shorter distance (screening). The second diagram arising from the nonlinear interaction between gluons in QCD has the antiscreening effect, which makes the coupling weaker at short distance.
• Color$is$an1Mscreened$ • Color$builds$up$away$from$a$source$ • Interac1on$becomes$strong$at$large$distances$(low$ momenta)$ • Confinement$of$quarks;$quarks$are$not$observed$as$ isolated$par1cles$ g2 Strong coupling constant α = s 4π In quantum field theory, the coupling constant is an effec ve constant, which depends on four-momentum Q2 transferred. For strong interac ons, the Q2 dependence is very strong (gluons - as the field quanta - carry color and they can couple to other gluons). A first- order perturba ve QCD calcula on (valid at very large Q2) gives:
2 12π running coupling α Q = s ( ) 2 2 constant! (22 − 2n f )⋅ ln(Q / ΛQCD )
nf =6 − number of quark flavors ΛQCD − a parameter in QCD (~0.22 GeV), an infrared cutoff
The spa al separa on between quarks goes as " ! = Q2
Therefore, for very small distances and high values of Q2, the inter-quark coupling decreases, vanishing asympto cally. In the limit of very large Q2, quarks can be considered to be “free” (asympto c freedom). On the other hand, at large distances, the inter-quark coupling increases so it is impossible to detach individual quarks from hadrons (confinement).
Asympto c freedom was described in 1973 by Gross, Wilczek, and Politzer (Nobel Prize 2004). It is customary to quote αs at the 91 GeV energy scale (the mass of the Z boson) Mesons quark wave function of the pion
- Consider π (t=1 and t0=1). The only possible combina on is π − = ud In general, it is possible to find several linearly independent components corresponding to the same t and t0. The appropriate combina on is given by isospin coupling rules. Furthermore, the wave func on must be an symmetric among the quarks. This problem is similar to that of a two-nucleon wave func on! 1 1 0 − uu dd π = τ − π = ( − ) T=1 triplet: 2 2 π + = − ud
What about the symmetric combina on? 1 T=0 singlet: η0 = uu + dd 2 ( )
To produce heavier mesons we have to introduce excita ons in the quark- an quark system or invoke s- and other more massive quarks The lightest strange mesons are kaons or K-mesons. Since s-quark has zero isospin, kaons come in two doublets with t=1/2: {K + (us ), K 0 (ds )}, {K − (us), K 0 (ds)}
Y = A + S +C +B +T hypercharge 1 Q = −t0 + Y 2 the SU(3) symmetry limit is met for massless u,d,s quarks π − (ud)+ p(uud) → K 0 (ds )+ Λ(uds) strangeness is conserved! Pseudoscalar mesons ! J total angular momentum ! ! ! ! ! ! ! J = ℓ + S, S = s + s ℓ orbital angular momentum q q ! S total spin
S can be either 0 or 1. The mesons with the relative zero orbital angular momentum are lower in energy. For the pion, S=0, hence J=0. Consequently, pions are “scalar” particles. But what about their parity? The parity of the pion is a product of intrinsic parities of the quark (+1), antiquark (-1) and the parity of the spatial wave function is 1. Hence, the pion has negative parity: it is a pseudoscalar meson. With (u,d,s) quarks, one can construct 9 pseudoscalar mesons (recall our earlier discussion about the number of gluons!): 9 (nonet)=8 (octet)+1 (singlet) Members of the octet transform into each other under rota ons in flavor space (SU(3) group!). The remaining meson, η0, forms a 1-dim irrep. (us ) 1 497.61 π 0 = uu − dd (ds ) 2 ( ) 1 493.68 η = uu + dd − 2 ss 8 6 ( ) 1 η = uu + dd + ss 547.86 134.98 (ud ) 0 3 ( ) (du )
139.57 957.78 In reality, since the SU3 (flavor) symmetry is not exact one, the observed mesons are:
(su ) (sd ) η = η8 cosϑ +η0 sinϑ η' = −η8 sinϑ +η0 cosϑ
ϑ - Cabibbo angle, ~11o for pseudoscalar mesons
masses are in MeV/c2 The eta was discovered in pion-nucleon collisions at the Bevatron (LBNL) in 1961 at a me when the proposal of the Eigh old Way was leading to predic ons and discoveries of new par cles.
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€ € Vector mesons
Here S=1, hence J=1. They have nega ve parity. The vector mesons are more massive than their pseudoscalar counterparts, reflec ng the differences in the interac on between a quark and an an quark in the S=0 and S=1 states.
J π =1− 896 (us ) (ds ) 1 892 ρ 0 = uu − dd 2 ( ) 775 1 783 (ud ) ω = uu + dd (du ) 2 ( ) 775 ϕ = ss 1019
(su ) (sd )
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€ € Baryons (qqq)
With three flavors, one can construct a total of 3×3×3=27 baryons
baryon singlet t = 0
Completely asymmetric under flavor 1 Λ1 = { uds + dsu + sud − dus − usd − sdu } rota ons 6 The color and flavor wave-func ons should be an symmetric and thus zero orbital angular momentum and spin-1/2 are not possible if the wave-func ons is to be overall 1 − an symmetric as required by Fermi–Dirac sta s cs. Hence, L=1 J π = 2 Λ (1405) baryon decuplet
(ddd) (udd) (uud) (uuu)
3 + (dds) (uds) (uus) J π = 2
(dss) (uss)
(sss)
The discovery of the omega baryon was a great triumph for the quark model of baryons since it was found only a er its existence, mass, and decay products had been predicted by Murray Gell-Mann in 1962. It was discovered at Brookhaven in 1964.
€ baryon octet The remaining 16 baryons constructed from u-, s-, and d-quarks have mixed symmetry in flavor. The lower energy octet contains protons and neutrons as its members. The wave func ons for each member in the group is symmetric under the combined exchange of flavor and intrinsic spin (the quarks are an symmetric in color!)
939.6 938.3 (udd) (uud) (ddd) (uuu)
1115.7 (uds) 1 + 1197.4 J π = 2 (dds) 1192.5 (uus) 1189.4
1321.3 1314.9 (dss) (uss)
1 p = 2 u↑u↑d ↓ + u↑d ↓u↑ + d ↓u↑u↑ 18 { ( ) example: proton wave function −( u↑u↓d ↑ + u↑d ↑u↓ + d ↑u↑u↓ ) +( u↓u↑d ↑ + u↓d ↑u↑ + d ↑u↓u↑ )} New relatives of the proton
h p://www.fnal.gov/pub/ferminews/ferminews02-06-14/selex.html
Baryon Supermul plet using four-quark models and half spin
Two New Particles Enter the Fold http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.114.062004 The Roper resonance [N(1440)P11] is the proton's first radial excitation. Its lower- than-expected mass owes to a dressed-quark core shielded by a dense cloud of pions and other mesons.