Chapter 6

The Strong Interaction

The strong interaction is responsible for the binding of protons and neutrons in nuclei, and for the binding of quarks inside protons and neutrons. It is generated by the exchange of gluons between quarks. Quarks carry a new quantum number known as color. The strong force acting on quarks inside protons and neutrons prohibits their separation beyond distances larger than the diameters of protons and neutrons. This phenomenon explains why free quarks are not observed, but that quarks are conned in protons, neutrons, and various additional bound states denoted as baryons or mesons, which together are known as hadrons. Gluons carry color as well, and are likewise conned inside hadrons.

6.1 Quantum Chromodynamics

We have seen in the introduction that the strong interaction is responsible for the attractive force between quarks, the constituents of protons and neutrons. The attractive force between two protons or a proton and a neutron, etc. is just a secondary effect of this fundamental force between quarks. In quantum eld theory, the electromagnetic interaction (the force between two charged objects) is generated by the exchange of one or more photons. Likewise the strong interaction is generated by the exchange of particles with spin , the gluons, as illustrated in Fig. 6.1. We recall the Feynman rule according to which the electronphoton vertex (see Figs. 5.3, 5.4, 5.8, and 5.9) is proportional to the electron charge qe , see (5.26). In the case of the strong interaction, the quarkgluon vertex is proportional to a strong charge q s of the quarks, which is independent of its electric charge. There exist, however, three strong charges qis , i = 1, 2, 3, which are also denoted as colors. (For this reason this theory is also called quantum chromodynamics or QCD.) This color changes whenever a gluon is emitted or absorbed by a quark, as in Fig. 6.2. s Since we have q s j = qi , the gluons carry strong charges, i.e., colors, as well. (For this reason they are denoted by Gi j in Fig. 6.2.) The color of a quark can be
U. Ellwanger, From the Universe to the Elementary Particles, Undergraduate Lecture Notes in Physics, DOI: 10.1007/978-3-642-24375-2_6, Springer-Verlag Berlin Heidelberg 2012 73

74 Fig. 6.1 Feynman diagram describing the exchange of a gluon between two quarks; there exist, however, an innite number of additional relevant diagrams

6 The Strong Interaction

qj qi

G ij qj

or

G ij
Fig. 6.2 Emission or absorption of a gluon by a quark

qi

represented by a three-component vector q s in color space. After the emission or absorption of a gluon by a quark the direction of this vector has changed, but not its modulus. The result of a calculation of a probability as P ( ) (as, e.g., in (5.29)) depends only on the modulus q s = |q s | of the color vector of quarks, which remains invariant. (If all observables are independent of a variable such as the direction of the vector q s in color space, we talk about a symmetry with respect to variations of this variable. The components of the vector q s are complex numbers in general. In the case of three-component complex vectors with constant modulus, this symmetry is denoted by SU (3), see Chap. 9.) In analogy to (5.31) we dene a strong ne structure constant s : s = (q s )2 . 4 0 c (6.1)

The small value 1/137 of the electromagnetic ne structure constant signies that this interaction is relatively weak. In particular this small value implies that contributions to a given process from Feynman diagrams with a larger number of vertices are relatively small and normally negligible. In the case of the strong interaction, the numerical value of the ne structure constant is s 1. (6.2)

6.1 Quantum Chromodynamics

E pot

75

0.5 fm

Fig. 6.3 Schematic representation of the r dependence of the potential energy induced by the strong interaction (the exchange of an innite number of gluons) between two quarks

Consequently this interaction is relatively strong, and contributions to the interaction between two quarks from more complex Feynman diagrams (with more vertices) are not negligible. There exist an innite number of such diagrams, and the exact behavior of the strong interaction has not been computed to date. (Nowadays highperformance computers are used for this purpose, whose architecture is adapted to this objective.) The most important effect of diagrams with more vertices concerns the dependence of the strong force on the distance r between two quarks. The r dependence of the electric force between two charged particles is given in (5.10) and (5.34), according to which it decreases as 1/ r 2 . In the case of the strong interaction, we nd, on the other hand, that the modulus of the attractive force is approximately independent of r for r 0.5 fm = 0.5 1015 m. For such values of r its numerical value is Fstrong 1.8 105 kg m s2 . (6.3)

Accordingly, also the expression for the potential energy E pot (r ) differs from (5.40): for r 0.5 fm it behaves as E pot (r ) = r Fstrong . This behavior of E pot (r ) is sketched in Fig. 6.3. (6.4)

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6 The Strong Interaction

We recall that, in the absence of external forces, the total energy E tot = E pot + E kin of a system is conserved. If E pot increases with r as in (6.4) and in Fig. 6.3, the maximally possible distance between two quarks corresponds to the maximally possible value of E tot , given by E tot = E pot (rmax ) and E kin = 0. However, on average E pot and E kin are of the same order. Let us consider the orders of magnitude of these energies for quarks inside a proton. Given a distance of about 0.5 fm between two quarks the mean value of the potential energy is, according to (6.3) and (6.4), E pot 0.5 1015 m 1.8 105 kg m s2 0.9 1010 kg m2 s2 . (6.5)

For a rough estimate we can assume that the mean velocity of a quark inside a proton is on the order of the speed of light c. The mass of a quark is about a third of the mass of a proton. Correspondingly the mean value of the kinetic energy of 1 mp 2 1 2 1.7 1024 g and both quarks is on the order of 2 2 3 c = 3 m p c . With m p 8 1 c 3 10 m s this leads to E kin 0.5 1010 kg m2 s2 . (6.6)

Thus it is indeed of the same order as the potential energy (6.5). On the one hand this calculation explains (for given quark masses) the order of the radius of a proton, i.e., the average distance between two quarks. On the other hand we see the difculties to be overcome if we want to separate two quarks by a much greater distance: the necessary potential energy must be much larger, which rapidly becomes impossible: in order to separate two quarks by 1 mm = 1012 fm we would need 1012 times the energy available in a proton (or neutron)! (In contrast, the separation of an electron from a positron or a nucleus requires just a nite energy since, for r , the electric potential energy approaches a constant, as in Fig. 5.14.) The impossibility of separating quarks to distances much larger than a fermi is denoted as connement. It follows that free quarks do not exist. Quarks are always bound, either (a) in congurations of three quarks carrying three different colors, corresponding to a white or color-neutral state, or (b) in congurations of one quark and one antiquark of opposite colors, which also results in a color-neutral or white state.

6.2 Bound States of Quarks

These bound states of quarks are denoted as hadrons, the three-quark states as baryons and the quarkantiquark states as mesons. The proton (three quarks uud) and the neutron (three quarks ddu) are members of the family of baryons. The spin of a baryon is always a half-integer multiple of ; /2 in the case of protons and neutrons.

6.2 Bound States of Quarks

77

Fig. 6.4 Description of the decay of the baryon quarks and gluons

++

into a proton p and a pion + at the level of

There also exist baryons consisting of three u quarks (denoted as ++ ) or three d quarks ( ) with spin 3 /2 and about 1.3 times as heavy as a proton. The most important mesons are the pions + , 0 , and , consisting of the following quarks: + : ud + dd 0 : uu : du (6.7)

Here u denotes an anti-u quark, the antiparticle of a u quark with electric charge an anti-d quark with charge + 1 e. (This explains the electric charges of 2 e , and d 3 3 the pions.) The pions carry spin 0, and due to a large binding energy (see (1.7) for the mass of a nucleus) they are relatively light: m + m 0 m m p /6 ( is the antiparticle of the pion + of the same mass). It follows from m ++ > m p + m + that the baryon ++ can decay into a proton and a pion, as illustrated in Fig. 6.4. Accordingly the baryons ++ are unstable; owing to the strength of the strong interactions the probability of the decay of a ++ baryon is very large, and its mean lifetime is very small: 5.2 1024 s. (We can hardly speak of a particle; often the notion resonance is used.) We can detect such an unstable particle by its decay products: in Fig. 6.4 the sum of the momenta p of the pion and pp of the proton is equal to the momentum p of the baryon ++ , and the sum of the energies E + E p is equal to E . If, in an experiment, we nd a large number of pions and protons whose momenta and energies satisfy the relation ( E + E p )2 ( p + pp )2 c2 = m 2 c4 for a given value of m , we can conclude that particles ++ of corresponding mass had been produced, even if they decayed immediately thereafter. (In fact even the pions + are unstable due to the weak interaction, see the next chapter.) Before we give more precise values for the masses of these particles, it is helpful to introduce more convenient units than grams or kilograms for particle masses (elementary or composite).

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6 The Strong Interaction

Table 6.1 Masses and electric charges (in multiples of e) of the known quarks Quark Masses [GeV/c2 :] Electric charge: u 0 .3 +2 3 d 0 .3 1 3 s 0 .5 1 3 c 1.4 +2 3 b 4 .4 1 3 t 173 +2 3

First we use, as a unit of energy, 1 eV = 1 electron volt, which corresponds to the energy gained by a particle of electric charge e on passing through an electric potential difference of 1 volt. 1 volt is equal to 1 J C1 , where 1 J = 1 joule = 1 kg m2 s2 and C stands for Coulomb. Using e 1.6 1019 C we nd 1 eV 1.6 1019 J. (6.8)

Now we specify masses in multiples of eV/c2 , where c is the speed of light. (This allows us immediately to obtain the energy in electron volts stored in a mass m via the formula E = mc2 .) Expressed in kilograms, this unit is 1 eV/c2 1.6 1019 J 9 1016 m2 s2 1.78 1036 kg. (6.9)

As usual, 1 keV = 103 eV, 1 MeV = 106 eV, and 1 GeV = 109 eV. The masses of the pions and some baryons are m m 0 mp mn m
++

139.6 MeV/c2 , 135.0 MeV/c2 , 0.938 GeV/c2 , 0.939 GeV/c2 , 1.23 GeV/c2 . (6.10)

All these hadrons consist of u and d quarks with masses of about 300 MeV/c2 . (The mass of a particle, such as a quark, that is never observed free cannot be measured directly and is thus ill dened. Usually we determine the mass of a particle by the relation E 2 = m 2 c4 + p 2 c2 and independent measurements of the energy E and the momentum p , which is impossible for conned quarks. The value 300 MeV/c2 is used in so-called potential models, which describe quite well the measured masses of hadrons.) There exist additional heavier quarks, which are unstable due to the weak interaction (see the next chapter). All of them carry spin /2 and they are denoted as the s quark (s for strange), c quark (c for charm), b quark (b for bottom), and t quark (t for top). All quarks carry color, and their masses and electric charges (in multiples of the elementary charge e) are given in Table 6.1. All quarks form hadrons (apart from the top quark, which decays too fast), in particular mesons (with integer spin) consisting of a quark q and an antiquark q : the 0 (corresponding to us , respectively); , u s, d s , and ds mesons K + , K , K 0 , and K

6.2 Bound States of Quarks

79

Fig. 6.5 Emission and absorption of gluons by gluons

Fig. 6.6 The four-gluon vertex

); and more, with masses of about m meson m q + m q (s s ); J/ (cc ); (bb . The quarks u, d, and s alone form about 100 different hadrons. The 1969 Nobel prize was awarded to Murray GellMann for the relatively simple description of all these hadrons in the quark model [11]. We recall that the emission or absorption of a gluon changes the color of a quark (see Fig. 6.2) and, correspondingly, the gluons carry colors as well. Hence the gluons carry also a strong chargeunlike photons, which carry no electric charge. Owing to their strong charge, gluons can emit or absorb other gluons, as shown in Fig. 6.5. There even exists a four-gluon vertex, as illustrated in Fig. 6.6. Accordingly there exist Feynman diagrams contributing to gluongluon scattering as in Fig. 6.7 and, as in the case of quarkquark scattering, an innite number of diagrams with more vertices that are not negligible. These diagrams generate an attractive force, implying connement of gluons, just as for quarks. (Conned gluons exist inside hadrons, where their exchange between quarks generates the strong attractive force.) There probably exist bound (but very unstable) states of masses of about 1.5 GeV/c2 consisting only of gluons, called glueballs. However, owing to their extremely short lifetime, among other reasons, these states are very difcult to detect.

80 Fig. 6.7 Feynman diagram contributing to gluongluon scattering

6 The Strong Interaction

6.3 Summary
At rst sight the strong interaction is very similar to quantum electrodynamics: like quantum electrodynamics, it is generated by the exchange of massless particles of spin , the gluons. For the following reasons the strong interaction differs from the electromagnetic force, however: (a) Instead of a single electric charge there exist three strong charges, the three colors. Only quarks, not electrons or additional leptons, carry strong charges, i.e., colors. (b) The numerical value of the strong ne structure constant s is about one, much larger than 1/137. Accordingly the force is stronger, in particular its behavior at larger distances is very different from that of the electric force. This implies the connement of quarks, according to which the only observable states are white (or color neutral), i.e., baryons, consisting of three quarks (qqq), and mesons, consisting of a quark and an antiquark (qq ). Between hadrons, the strong interaction leads also to attractive forces, which are responsible for the binding of protons and neutrons in nuclei, but these forces decrease rapidly with increasing distance. (c) Gluons carry a strong charge and feel the strong force, and are also bound in hadrons. In contrast to photons they are not observed as free particles.

Exercise
6.1. The following baryons with corresponding masses consist only of u, d, and s quarks: neutron (0.939 GeV/c2 ), proton (0.938 GeV/c2 ), 0 (1.116 GeV/c2 ), + (1.189 GeV/c2 ), 0 (1.193 GeV/c2 ), (1.197 GeV/c2 ), 0 (1.315 GeV/c2 ), (1.321 GeV/c2 ). Estimate their quark content from their charges and mass differences.