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129A Lecture Notes
Strong Interactions II
1 Regge Trajectories and String
One organizing principle came out when people looked at the masses and the spins of hadrons of the same type (i.e., same isospin, same parity, etc). By plotting the masses and spins on the so-called Chew–Frautschi plot^1 on the (m^2 , J) plane, the hadrons of the same type fall on straight lines: J = α(0) + α′m^2. The intercept α(0) depends on the types, but α′^ came out more-or-less the same: α′^ ' (1.2–1.4 GeV−^2 ).
J
(^2) m [GeV
2 ]
Figure 1: Chew–Frautschi plot for mesons.
This fact led to the following picture: the hadrons are elastic strings, not particles. When the string is stretched, there is a constant tension T , giving the energy T r where the r is the length of the string. This was the beginning of the string theory. If you regard T r as a potential energy, a hand-waving analysis indeed gives the linear relation between E^2 = (mc^2 )^2 and J. Write down the relativistic kinetic energy cp and the linear potential T r:
H = cp + T r. (1) (^1) This is Jeff Chew of our Department.
In the spirit of Bohr’s argument, pr = l¯h. Therefore,
E =
¯hcl r
Minimizing it with respect to r, we find the average length of the string
r =
√ ¯hcl/T and its energy
E =
¯hcl √ ¯hcl/T
+ T
√ ¯hcl/T = 2
¯hclT , (3)
and hence
m^2 = (E/c^2 )^2 =
4 T
c
l. (4)
Indeed, the mass-squared is linear with the angular momentum l!^2
J
(^2) m [GeV
2 ]
Figure 2: Chew–Frautschi plot for baryons. (^2) The correct quantum mechanical treatment of a relativistic string turned out to be
much more difficult. You start with Nambu–Goto action and quantize it, and find that the quantization procedure is consistent only in 26-dimensional spacetime. Even that case predicts a tachyon, a particle with a negative mass-squared, whose presence violates causality because it would go faster than the speed of light. A supersymmetric version happily gets away with tachyons, but still live in 10-dimensional spacetime. But the interesting thing about it was that it predicts a massless spin-two particle, which we don’t see in the world of hadrons but can be identified with the graviton. Since then the string theory switched its gear from the would-be theory of hadrons to the “Theory of Everything,” including quantum gravity.
Nishijima was that all hadrons satisfy the following relationship,
Q = Iz +
B + S
Here, Q is the electric charge, Iz the 3rd component of the isospin, B the baryon number, and S the strangeness. You can verify this relation for all particles in Tables 1, 2, 3, 4. I have no idea how they came up with such a mysterious relation by looking at a big mess like this.
particles isospin strangeness quark content π+, π^0 , π−^1 0 u d¯, (uu¯ − d d¯)/
2, du¯ K+, K^0 1/2 +1 u¯s, d¯s K^0 , K−^ 1/2 − 1 s d¯, su¯ η 0 0 (u¯u + d d¯ − 2 s¯s)/
η′^0 0 (u¯u + d d¯ + s¯s)/
Table 1: Members of pseudoscalar (0−) meson nonet. The quark contents shown for η and η′^ is the idealized octet and singlet combinations, but they actually mix substantially in reality.
particles isospin strangeness quark content ρ+, ρ^0 , ρ−^1 0 u d¯, (uu¯ − d d¯)/
2, d¯u K∗+, K∗^0 1/2 +1 us¯, d¯s K∗^0 , K∗−^ 1/2 − 1 s d¯, su¯ ω 0 0 (uu¯ + d d¯)/
φ 0 0 ss¯
Table 2: Members of vector (1−) meson nonet. ω and φ are an “ideal mixing” of singlet and octet states such that φ is an ss¯ to a good approximation.
Gell-Mann and Zweig generalized the concept of isospin to include strangeness. What they noticed is that the baryons come in an octet, which is a natural combination in the group SU (3). In order not to get into too much complica- tion, I discuss the flavor SU (3) “top down,” i.e., going backwards in history knowing what the true theory turned out to be. The hadrons are composite objects built up of “quarks,” spin 1/2 fermions that are never observed in isolation.^3 The mesons are bound states of one quark and one anti-quark,
(^3) Apparently Gell-Mann took this name from Finnegans Wake by James Joyce, but I haven’t read the book.
particles isospin strangeness quark content p, n 1/2 0 uud, udd Λ^0 0 -1 uds Σ+, Σ^0 , Σ−^1 -1 uus, uds, dds Ξ^0 , Ξ−^ 1/2 -2 uss, dss
Table 3: Members of baryon octet of spin 1/2. They all have baryon number one.
particles isospin strangeness quark content ∆++, ∆+, ∆^0 , ∆−^ 3/2 0 uuu, uud, udd, ddd Σ∗+, Σ∗^0 , Σ∗−^1 -1 uus, uds, dds Ξ∗^0 , Ξ∗−^ 1/2 -2 uss, dss Ω−^0 -3 sss
Table 4: Members of baryon decuplet of spin 3/2. They all have baryon number one.
and baryons of three quarks. The three quarks, up, down, and strange are listed in Table 5.
name notation isospin strangeness electric charge up-quark u 1/2 0 +2/ 3 down-quark d 1/2 0 − 1 / 3 strange-quark s 0 − 1 − 1 / 3
Table 5: Quantum numbers of quarks. All of them carry baryon number 1/3.
The minute you postulate that all hadrons are made up of quarks, Gell- Mann–Nishijima relation all of a sudden becomes trivial. It is simply that three quarks satisfy this relation. Note that Q, Iz , B, and S are all additive quantum numbers, and Gell-Mann–Nishijima relation is a linear equation. Therefore, if the constituents (quarks) satisfy this relation, the bound states also satisfy the same relation. Nothing mysterious anymore. Within the quark model, the isospin invariance of the strong interactions can be understood as a consequence of an approximate degeneracy between up and down quarks. If their masses are not very different, to the extent you ignore the electromagnetism, you can freely interchange or rotate among up and down quarks. A general rotation among up and down quarks is a
the group. Due to the analogy to the case of the real spin, we take the generators to be normally 12 ~τ. The generators are traceless, as a consequence of the unit-determinant condition of the matrices. This can be seen using the Cayley–Hamilton formula, deteA^ = eTrA^ for any matrix A.^4 There are four linearly independent two-by-two hermitian matrices, and indeed three are left after requiring tracelessness. The one that is left out is the two-by-two identity matrix. Similarly, once we include the strange quark, the interchange among three quarks define a group SU (3), the group of unit-determinant three-by-three unitarity matrices. The strong interaction is invariant under this, except for the “small” difference in masses. It is actually not that small (they differ by about 150 MeV/c^2 ), but it can still be considered a small perturbation to the hadron mass, say baryon masses of 940 MeV/c^2 and above. Under the SU (3), three flavors of quarks transform as a triplet by definition. There are nine linearly independent three-by-three hermitian matrices, and eight of them are traceless. They are given by so-called Gell-Mann’s lambda matrices (the generalization of Pauli’s sigma matrices),
λ^1 =
, λ (^2) =
0 −i 0 i 0 0 0 0 0
, λ (^3) =
, (10)
λ^4 =
,^ λ^5 =
0 0 −i 0 0 0 i 0 0
,^ (11)
λ^6 =
, λ (^7) =
0 0 −i 0 i 0
, λ (^8) = √^1 3
(12).
All matrices are normalized as Trλaλb^ = 2δab, and we normally take 12 λa^ as generators of the SU (3) group. Using the lambda matrices, the octet of mesons can be easily constructed. They are given by the combinations ¯qλaq for a = 1, · · · , 8 with
q =
u d s
,^ q¯^ = (¯u,^ d,¯^ s¯).^ (13)
(^4) If you want to prove this formula, the easiest way is to first diagonalize the matrix A = P DP −^1 , and then compare both sides of the formula. It works even when D is not completely diagonal but is of Jordan’s standard form.
The quark vector change from q → U q under the SU (3), while the anti- quarks ¯q → qU¯ †. This way, we can identify
π+^ = ¯q
λ^1 − iλ^2 2
q = ¯ud, π^0 = ¯q
λ^3 √ 2
q =
uu¯ − dd¯ √ 2
, π−^ = ¯q
λ^1 + iλ^2 2
q = du¯
K+^ = ¯q
λ^4 − iλ^5 2
q = ¯su, K^0 = ¯q
λ^6 − iλ^7 2
q = ¯sd
K^0 = ¯q
λ^6 + iλ^7 2
q = ds,¯ K−^ = ¯q
λ^4 + iλ^5 2
q = du,¯ η = ¯q
λ^8 √ 2
q =
uu¯ + dd¯ − 2¯ss √ 6
The eight meson states in the octet transform among each other under the SU (3). The singlet state in the nonet is a state that does not change under the SU (3) group. There is a unique such combination,
η′^ =
qq¯ =
(¯uu + dd¯ + ¯ss). (14)
Now the interesting case is the baryons. Let us leave out the spin degrees of freedom for the moment. Because the quarks are fermions, their wave function must be totally anti-symmetric. Out from three quarks, we choose three quarks without allowing two of them are the same, and hence there is only one combination,
1 √ 6
(uds + dsu + sud − usd − dus − sdu). (15)
It does not appear that we can obtain decuplet or octet this way. Let us for the moment say that quarks have totally symmetric wave func- tion as if they are bosons. We will come back to the question why this is possible for quarks later. With this new rule, there are many combina- tions possible. We choose three out of three objects, allowing multiple uses of the same objects.^5 There are (^) 3+3− 1 C 3 = 10 combinations. This nicely
(^5) In general, if you choose r out of n objects allowing the multiple uses, the number of
combinations is given by (^) nHr ≡ (^) n+r− 1 Cr. This can be understood by preparing r boxes and n − 1 fences. You order boxes and fences in an arbitrary way. There are (^) n+r− 1 Cr ways to do so. Once you picked a particular order, you fill the first object in the boxes until you meet the first fence, and then fill the second object in the boxes until you meet the second fence, and so on, until you see the n − 1-th fence after which you fill the n-th object in the boxes. Because there are r boxes all together, you have made a choice of r out of n objects allowing multiple uses.
Another way to construct the proton wave function is by combining one up and down quark in I = 0, S = 0 combination first,
|ud − du〉| ↑↓ − ↓↑〉 = |u↑d↓〉 − |u↓d↑〉 − |d↑u↓〉 + |d↓u↑〉, (20)
and then combine it with u↑^ to form I = 1/2, S = 1/2 state,
|u↑d↓u↑〉 − |u↓d↑u↑〉 − |d↑u↓u↑〉 + |d↓u↑u↑〉. (21)
Finally we make it totally symmetric by adding two cyclic permutations,
|u↑d↓u↑〉 − |u↓d↑u↑〉 − |d↑u↓u↑〉 + |d↓u↑u↑〉
- |d↓u↑u↑〉 − |d↑u↑u↓〉 − |u↓u↑d↑〉 + |u↑u↑d↓〉
- |u↑u↑d↓〉 − |u↑u↓d↑〉 − |u↑d↑u↓〉 + |u↑d↓u↑〉. (22)
By reassembling them, we find exactly the same combination as before. All other baryon wave function can be constructed by similar methods. See Griffiths how the SU (3) wave functions can be used with the non- relativistic quark model to calculate the masses of mesons and baryons as well as baryon magnetic moments. The model works remarkably well.
3 Color SU (3)
The success of the non-relativistic quark model of baryons left a big question behind.
- Why do we construct a totally symmetric wave function when the spin-statistics theorem tell us that quarks (having spin 1/2) must obey Fermi-Dirac statistics?
There are also several other important questions about the quark model.
- Why don’t we observe isolated quarks? Nobody has detected a frac- tional electric charge. If quarks are indeed constituents of the hadrons, somehow we have to come up with an explanation why quarks are “confined” inside hadrons.
- What is the connection to the string-like behavior we saw in Regge trajectories?
- Why do “partons” behave as if they are free at high-energy collisions (Deep Inelastic Scattering), if they are actually quarks that are confined by a very strong force inside hadrons?
The answer to these questions emerged from the most unlikely one about the statistics. Greenberg pointed out that if the quarks carry additional degrees of freedom that takes three possible states, and if the baryon is to- tally anti-symmetric for this degree of freedom, it will explain the totally symmetric wave function of the quarks. Later people called this degree of freedom “color,” Red, Green, and Blue. They have nothing to do with the color we see optically with our eyes. The only connection is that there are three primary colors. Greenberg’s assumption can be rephrased as the follow- ing: only “white” states are allowed in nature. In the case of baryons, three quarks come in different colors so that the color part of the wave function is (RGB + GBR + BRG − RBG − GRB − BGR)/
- It is “white” because three primary colors cancel each other, while the wave function is totally anti-symmetric. Then the complete baryon wave function is the produce of totally symmetric ones we constructed in the previous section and the totally anti-symmetric color wave function, and hence is totally anti-symmetric un- der the interchange of quarks as a whole. Then it is consistent with the Fermi-Dirac statistics of quarks. In the case of mesons, quarks can come in R, G, B, and anti-quarks in R, G, B. The meson wave function has then the color part, (RR + GG + BB)/
- Again the wave function is “white” because the color is cancelled between quarks and anti-quarks. The proposal of the color appears an even more bizzarre “patch” to the already-bizzarre quark model. But this turned out to be the true choice by the Nature. We can freely rotate three colors among each other like we did with three flavors. This defines yet another SU (3) group, the color SU (3). Han and Nambu pointed out that maybe there is a color-octet vector (spin one) bosons coupled to the color degree of freedom, which explains why we see only color- neutral bound states in nature. These color-octet vector bosons are later called “gluons,” reflecting the strong glue these vector bosons provide to confine quarks eternally inside hadrons. Recall the QED. There, the photon couples to anything electrically charged, and the Feynman vertex is proportional to the charge, e for the electron and |e| for the proton, and 23 |e| for the up quark, and so on. e determines the strength of the electromagnetism in general, often quoted in terms of the
charge appear smaller at longer distances because of screening effect. The same effect happens in the QCD. The virtual pairs of quark-anti-quark screen the color you’ve put in the vacuum. However, there is an important differ- ence. The Coulomb field is a virtual gluon, and because the gluon itself is colored, it also produces virtual pairs of gluons, which further produces gluons, and so on. The net effect of it is that gluons “anti-screen” color in- stead of screening it, and makes the color appear larger and larger at longer distances. In the case of the QED, we can parameterize the Coulomb force as α(rr)¯ hc, where α(r) is the distance-dependent fine-structure constant. At r → ∞ limit, α(∞) = 1/137, while it “runs” to larger values for smaller distances. Similarly, we can parameterize the gluon-Coulomb force as αs(rr )¯hc, where αs(r) is a “small” perturbative value at short distances and hence the force is approximately Coulombic. However at larger distances, αs(r) keeps growing, and eventually the force approaches a constant value, the “tension” T. Once the force becomes constant, the potential energy is an integral of the force and hence grows linearly with distance. That is how quarks are confined inside hadrons.
Q (GeV)
α (Q)s^ NLO^ NNLO^ Lattice OPAL ALEPH DELPHI L Deep Inelastic Scattering e+e- (^) Annihilation Hadron Collisions Heavy Quarkonia ep → jets
0.
0.
0.
0.
0.
0.
0.
1 10 10 2
Figure 3: Experimentally measured values of αs as a function of the energy scale. Taken from OPAL Collaboration, Eur. Phys. J. C16, 185 (2000).
If you try to pull a quark inside a meson, the gluon Coulomb force from the anti-quark becomes constant beyond the distance of about 0.5fm, and
the potential energy between the quark and anti-quark keeps rising linearly. At some point, it becomes energetically favorable to create a pair of a quark and an anti-quark, so that the created quark binds with the original anti- quark to form a bound state meson, cutting off the linearly rising potential, and the created anti-quark binds with the original quark in the same way. This way, you never see an isolated quark. It is always in a bound state no matter what. This way, the QCD answers the second and third questions beautifully. On the other hand, as you go to small distances, or large momentum transfers, the strong coupling constant becomes weak. For instance, at the momentum transfer of 91 GeV, αs ≈ 0 .12. This is a small enough value so that perturbation theory can be trusted. Then it is no mystery that the quarks can behave as if they are nearly free inside hadrons in Deep Inelastic Scattering experiments. This is the answer to the fourth question. In the end, the theory of the strong interactions is based on the same principle as the QED. You fix the charge of the particle (electric charges for the QED and the multiplicity of color for the QCD), and the force carrier (spin one boson) couples according to the charge. Such theories are called gauge theories. The main difference is that gluons themselves carry color, while the photon does not have an electric charge. The latter type of theories is called Abelian gauge theories, while the former type the non-Abelian gauge theories. The important and immediate consequence of a gauge theory is that particles with the same charge should feel the same force. This point can be tested in the QCD by comparing the three-jet events for the bottom quark, charm quark, and other light quarks in electron-positron annihilation (see more on this later). The comparison is shown in Fig. 4. Despite these successes, the prejudice against fractionally charged and enternally confined consituents you can never see was so strong, that people still didn’t believe in quarks until November 11, 1974.
5 November Revolution
Only in the second half of 70’s, after the so-called November Revolution in Particle Physics when the teams at SLAC and Brookhaven independently discovered a particle now called J/ψ, people started to take the quark model seriously. The J/ψ is now understood as a boundstate of a charm quark and its anti-particle cc¯, an entirely new type of quark not seen earlier. See
Chapter 9 of Cahn–Goldhaber for the fascinating story of the discoveries, including the Italian team who completely missed the Nobel prize by a few days. Afterwards, many more new mesons, and eventually a new baryon (Λc(udc) baryon) was discovered, and everything fit together using the hy- pothesis of a new quark. People had to accept the quark hypothesis after a long period of skepticism. See also Griffiths for the brief discussion of the spectroscopy of char- monium states, namely c¯c bound states. People tried to fit the observed spectrum of the charmonium using a non-relativistic potential between two massive particles, which confirmed the picture of the Coulombic potential at the short distance and the linear potential at the large distance, interpo- lated by an approximately logarithmic behavior at the intermediate distance. Griffiths gives an argument how the logarithmic behavior can be seen by com- paring charmonium and later discovered bottomonium spectra.
6 Jets
Is there really no hope to “see” quarks and gluons? Actually there is. Because the quarks and gluons interact only weakly at high energies, you can create quarks and gluons in a perturbative process you can calculate, and “see” them after they “hadronize.” The best way to do so is using high-energy electron-positron colliders. We discussed before that the annihilation of an electron and a positron pro- duces an energetic photon “at rest,” a virtual photon with no momentum but with a large energy violating energy-momentum conservation. This virtual photon must materialize into something else quickly, but the photon doesn’t remember that it was created by an electron and a positron. It can material- ize into anything that has electric charge, as long as there is enough energy. For example, it can create a pair of muons μ−μ+. This way, you can produce particles that did not exist at all in the initial state. In the same way, the virtual photon may materialize into a pair of quarks, whose Feynman vertex is proportional to the charge of the quark. There is an important difference, however, that the quarks come with three colors. Therefore the probability for prodeucing a quark pair is proportional to the electric charge squared, and multiplied by three for the possible final color
ALEPH DALI^ Run=15995^ Evt=
Figure 5: A pair of muons produced in electron-positron annihilation. The beams are perpendicular to the plane. Two charged tracks penetrate all layers and hit the outmost part of the detector called the muon chamber. From ALEPH DALI data base, http://alephwww.cern.ch/DALI/
10 -
1
10
10 2
10 3
1 10 10 2
ρ
ω φ
J /ψ ψ(2 S )
R
Z
√ s (GeV)
Figure 6: The data on R. The horizontal axis
s is the center-of-momentum energy of the electron-positron collision. The resonances J/ψ, ψ(2S), and Υ(nS) for n = 1, · · · , 4 are also shown. The green dashed curve is the pre- diction of the native quark parton model Eq. (24), while the red solid curve is three-loop QCD prediction. Taken from http://pdg.lbl.gov.
ALEPH DALI^ Run=15995^ Evt=
Figure 7: A pair of electron and positron produced in electron-positron an- nihilation. Two charged tracks shower in the electromagnetic calorimeters and stop.
