15 Asymptotic Freedom in QCD
From the Bjorken scaling of the nucleon structure functions, we learn that the hadronic constituents probed at small distances or at high energies behave as if they were almost noninteracting or ‘free’. We are confronted with an apparent paradox, since in quantum field theory virtual particles exchanged between partons can have arbitrarily high momenta, quantum fluctuations associated with them naturally occur at short distances. Why do these fluctuations turn themselves off, and the partons behave as if they were free at high energy, whereas at low energy they are strongly bound? How can a model of noninteracting quarks be reconciled with a force that is extremely strong in other circumstances? The resolution to this paradox lies in the asymptotic freedom property of QCD. We first show that in field theory, quantum effects cause the coupling constants to vary with energy; they are no more constant but running. In QED we will find that the charge, i.e. the coupling becomes stronger at large momenta and weaker at small momenta. The opposite behavior happens in QCD, its interaction is strong at low energy and vanishes asymptotically at infinitely high energy, hence the name of asymptotic freedom. ’t Hooft, Politzer, and Gross and Wilczek discovered that non-Abelian gauge theories are asymptotically free. Later, Coleman and Gross show that only Yang– Mills fields possess this unique feature in four-dimensional space-time. The success of QCD as the fundamental theory of strong interactions is due to this discovery. Here the Bjorken scaling finds its natural explanation: at infinite energy, the strong coupling vanishes and the quarks are noninteracting. However, the slowly logarithmic variations of the strong coupling αs(q 2 ) should induce smooth violations of scaling since, as large as q 2 may be, αs(q 2 ) is small but still nonzero, and perturbative calculations with the strong coupling are fully justifiable. This violation, expressed by a smooth q 2 -evolution of the structure functions, is governed by the Gribov-Lipatov-Altarelli-Parisi (GLAP) equation. As discussed in Chaps. 14 and 16, the use of perturbative QCD in heavy particle decays or in the e+ + e− annihilation into hadrons at high energy is legitimate, since αs (q 2 ) is small in these reactions. On the other hand, at low energy and for sufficiently large αs (q 2 ), QCD exhibits confinement of color. The only finite-energy asymptotic states of QCD are color singlets. If one attempts to separate a color singlet state into
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15 Asymptotic Freedom in QCD
its color constituents, for instance by breaking a meson into a quark and an antiquark, a tube of gluon would form between these two color sources. With sufficiently strong coupling, this tube would have a fixed radius, so the energy cost of separating color sources would grow proportionally with the separation distance. This picture is similar to that of a magnetic bar in which the north and south poles can never be isolated.
15.1 Running Coupling Constant We show how a coupling constant can be running by an example of the Abelian QED interaction between the electron and the photon. The variation with energy of the coupling constant is entirely due to quantum effects provided by loop diagrams. To lowest order of the coupling constant e > 0, the photon–electron vertex is given by −i(−e)u(p0 )γ µ u(p) = +i e u(p0 )γ µ u(p) .
(15.1)
Up to order e2 , four diagrams contribute to the corrections of the charge in (1). Three of them (Fig. 15.1a–c) are similar to Fig. 14.2a–c respectively. The new one Fig. 15.2 is related to the photon self-energy (also called the vacuum polarization) Πµν (q). p0 .. ..... ..... .... .............. . . . .. .... ............. ................ .. ... ................ ... ............. .. . ... ........................... . . . . .. .... . . .. .. .. .... ...... ....... ... ... ... ........... ............. .. .. .. .. .. .... ..... ..... .... .... .......... . ..... ..... ..... .... ..... ..
p...0
.... ..... ... ......... .......................... ..... ............. . .... .... .............. .... . . . .. .............. . . . . . . . . . . . . .. ... ...... ...... .................. .......... .............. .. .. .. .. .. .... ..... .............. ..... ......... ................ . ...... ..... ............. ..... ...... ........... .... .... ..... .
q
q
p
(a)
µ
(b)
0
0
Γ (p , p)
0
p
{Σ(p )/(6 p − m)} γ
µ
p0
.... ..... .... .... .............. ..... . .... ..... .... . . . .. .. .. .. .. .. .... .. ....... ....... .................................. .. .. .. .. .. .... ..... ..... ............... .... ........... . ... .. ............. ................ ... ............ ... .......... .... ....... ...... .... ...
q
(c)
p
µ
γ Σ(p)/(6 p − m)
Fig. 15.1. (a) Correction to the charge by the vertex function; (b, c) correction to the charge by the fermion self-energy 0 .... ... .... ..... . . . ............. .. .... ..... ..... .................................... ..... . ........... . . . . . . . . . . . . ...... .. .. .. .. .... .. .. .. .. .. .. .......... ....... ...... .............. ......... ............................................. . . . .. .. .. .... . . . . ... ..... . .. ... ν ..... . µ ................ . . . . . ........ .................................... ...... . ..... ..... ..... ..... ..... .....
p
q
k−q
γ
Πµν (q) γν /q 2
γ
k
p
Fig. 15.2. Correction to the charge by the photon self-energy
The first thing to do is to calculate these four diagrams. In the first three sections of Chap. 14, we have studied in detail the diagrams of Fig. 14.4b
15.1 Running Coupling Constant
507
and Fig. 14.5 which are the same diagrams of Fig. 14.2a–c without the τ –ντ vertex. The result can be directly used for the three diagrams of Fig. 15.1a–c with the replacement 34 gs2 ↔ e2 , p2 ↔ p0 , and p3 ↔ −p. Some formulas for F1 (q 2 ) and Zq , such as (14.12), (14.14), (14.23), (14.31) and the Ward identity (14.42), are relevant to this chapter. The quark masses m2 , m3 taken in Chap. 14 to be m are understood here as the electron mass. There, the Γµ (p2 , p3 ) is the weak current corrected by QCD. Here, Γµ (p0 , p) is the electromagnetic current corrected by QED. Besides the γ5 terms in (14.11) which are absent here, formally these Γµ are identical after the replacement 43 gs2 ↔ e2 . Also in Chap. 14, Σ(p) is the quark self-energy induced by QCD, here it represents the QED corrected electron self-energy. The Zq is now denoted by Z2 , a notation commonly used for the field-strength renormalization of fermions. Z2 is given by the derivative of Σ(p) taken at 6 p = m according to (14.31). Also, δ2 = Z2 − 1.
15.1.1 Vacuum Polarization Our next step is to compute the photon self-energy Πµν (q) associated with the diagram of Fig. 15.2. For this loop integral, Feynman rules give Z d4 k i i def µ ν +i Πµν (q) = (−1) Tr (+ieγ ) (+ieγ ) , (15.2) (2π)4 6k − m 6 k− 6 q − m the factor (−1) comes from the anticommuting property of the fermionic loop. Using again dimensional regularization and Feynman auxiliary variable x for the product of two propagators in (2), the latter can be re-expressed as Z 1 Z dn k 2k µ k ν − k µ q ν − k ν q µ − gµν (k 2 − k · q − m2 ) µν 2 Π (q) = 4i e dx (2π)n [k 2 − 2k · qx + q 2 x − m2 ]2 0 Z 1 2 −4e n N µν (q, x) = ) Γ(2 − dx (15.3) n . 2 0 (4π)n/2 [m2 − q 2 x(1 − x)]2− 2 n
The dn k integration gives C ≡ i π n/2 Γ(2 − n2 ) [m2 − q 2 x(1 − x)] 2 −2 times N µν (q, x), where N µν (q, x) is the sum of all the terms inside the curly brackets on the right-hand sides of the following equation: Z µ ν Γ(1 − n2 ) n 2k k µ ν 2 µν 2 2 d k = C 2q q x − g [m − q x(1 − x)] , D(k) Γ(2 − n2 ) Z −k µ q ν − k ν q µ dn k = −C {2q µ q ν x} , D(k) Z µν 2 Γ(1 − n2 ) n 2 n −g k µν 2 2 2 d k = −C g q x − [m − q x(1 − x)] , D(k) 2 Γ(2 − n2 ) Z gµν (k.q + m2 ) dn k = C gµν (q 2 x + m2 ) , D(k)
with D(k) ≡ [k 2 − 2k.qx + q 2 x − m2 ]2 .
(15.4)
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15 Asymptotic Freedom in QCD
Grouping together all the terms of the right-hand sides of (4), we obtain ( Z −4e2 n 1 2x(1 − x)[q 2 gµν − q µ q ν ] µν Π (q) = Γ 2− dx n n/2 2 0 (4π) [m2 − q 2 x(1 − x)]2− 2 h i Γ(1− n ) gµν 1 − (1 − n2 ) Γ(2− n2 ) 2 + . (15.5) n [m2 − q 2 x(1 − x)]1− 2 Γ(1− n )
The coefficient 1 − (1 − n2 ) Γ(2− n2 ) of gµν turns out to be identically zero since 2 a Γ(a) = Γ(a + 1), so the final expression of Πµν (q) is proportional to the gauge-invariant tensor (q 2 gµν − q µ q ν ), in accordance with the conservation of the electromagnetic current which implies qµΠµν (q) = qν Πµν (q) = 0. Thus, Πµν (q) = (q 2 gµν − q µ q ν )Π(q 2 ) , Z −4e2 n 1 2x(1 − x) Π(q 2 ) ≡ Γ 2 − dx n n/2 2 (4π) 2 0 [m − q 2 x(1 − x)]2− 2 m2 −e2 2 = − γE + log(4π) − log 2 2 12π ε µ Z 1 2 q −6 dx x(1 − x) log 1 − 2 x(1 − x) + O(ε) . m 0
(15.6)
Note that −e2 Π(0) = 12π 2
4π m2
ε
2
Γ
ε 2
, ε = 4−n .
To arrive at the two last lines in (6), we have used 2 −ε m2 − q 2 x(1 − x) ε m − q 2 x(1 − x) 2 = 1 − log + O(ε2 ) . 2 µ2
(15.7)
Like (14.17), we notice that beside the pole ε, an arbitrary mass scale µ must enter in (6) to make the regular part dimensionally correct. A nice feature of the dimensional regularization is that it automatically satisfies the gauge invariance of the theory. Other regularizations (such as the Pauli–Villars procedure with a covariant cutoff of the integral d4 k by a large mass Λ) give the same result for Πµν (q) but only after a long manipulation. Naively the integral in (2) ∼ d4 k/k 2 ∼ Λ2 /µ2 ∼ Γ(1 − n2 ) is quadratically UV divergent, however it disappears because it is contained in the gauge noninvariant gµν term, the coefficient of which is identically zero, the remaining divergence is only logarithmic [d4 k/k 4 ∼ log(Λ2 /µ2 ) ∼ Γ(2 − n2 )]. Gauge invariance constraint usually alleviates UV divergences. Due to nonvanishing fermion mass in the loop (m 6= 0), infrared divergences are absent in the
509
15.1 Running Coupling Constant
vacuum polarization Πµν (q). In (6), when x → 0 or 1, the integral does not blow up. Note that the pole Γ(2 − n2 ) is identified with log(Λ2 /µ2 ) in the Pauli–Villars cutoff (Problem 15.1). If we write the tree vertex of (1) as γ µ to be inserted between +i eu(p0 ) and u(p), then the corrections to γ µ , as represented by the four diagrams of Fig. 15.1a–c and Fig. 15.2, are respectively Γµ (p0 , p) ,
Σ(p0 ) µ γ , 6 p0 − m
γµ
Σ(p) , 6p − m
Πµν (q)γν . q2
(15.8)
These four operators are also inserted between +i eu(p0 ) and u(p); all of them are O(e2 ). Here q = p0 − p.
15.1.2 Dressed and Renormalized Photon Propagator In Fig. 14.7, the dressed electron propagator SD (p) was obtained by summing the geometric series of Σ(p). For photon, the corresponding lowest 1PI is the vacuum polarization Πµν (q), and the dressed photon propagator Dµν (q) can be similarly generated by summing the geometric series of Πµν (q). Thus −i gµν −i gµρ −i gσν ρσ + [+i Π (q)] + q2 q2 q2 −i gσλ −i gαν −i gµρ ρσ λα + [+i Π (q)] [+i Π (q)] + · · · (15.9) q2 q2 q2
Dµν (q) =
This summation is visualized in Fig. 15.3: . . . . . ........................................... . .. . . . ..
............................. . . . . . .... .. . . . .
+ .......................................................... Πρσ ............................................................ + .... ..... ..........................
≡
........ ............................. ......... .............. . . . .... .. . ... . .. .. .. .. .... ....... ........ ................................ ρσ ................................................................. λα ...... ....... ....... .......................... ... ... .. .. .... .... . . .... ........ .... ......... ............ . . . . . . . . . . . . . . . . . . . . ..........
Π
Π
+···
J •
......... ..... ..... .. .. .. .... .. .. ... .. .. .. ...... ... ... ... ........................................................ .. .. .. ... . ... . . . ..... ..... ........
Fig. 15.3. Dressed photon propagator Dµν (q)
The second term on the right-hand side of (9) can be written as −i gµρ ρσ q ρ q σ −i qµ qν 2 (g − 2 )gσν Π(q ) = 2 gµν − 2 Π(q 2 ) . q2 q q q q q The third term is found to be −i gµν − µq2 ν Π2 (q 2 ), then q2
−i (gµν − qµ qν /q 2 ) qµ qν [1 + Π(q 2 ) + Π2 (q 2 ) + · · ·] − i q2 q4 −i (gµν − qµ qν /q 2 ) qµ qν = −i . (15.10) q 2 [1 − Π(q 2 )] q4
Dµν (q) =
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15 Asymptotic Freedom in QCD
In any computation of physical amplitudes, the photon propagator is always attached to at least one fermion line, so in (10), the qµ and qν terms vanish when they are contracted with a fermionic conserved current. Therefore we can omit them for our purposes of calculating physical amplitudes, i.e. we only keep −i gµν . The bare and dressed propagators can be abbreviated as −i gµν −i gµν −→ Dµν (q) = 2 . q2 q [1 − Π(q 2 )]
(15.11)
The above equation indicates that as long as the dimensionless quantity Π(q 2 ) 2 is regular at q 2 = 0, i.e. Π(q 2 ) 6= µq2 , the dressed propagator Dµν (q) always has a pole at q 2 = 0, implying that the photon remains massless. The only source for Π(q 2 ) to be singular at q 2 = 0 would be the existence of a single massless particle coupled to the photon in the intermediate state. This does not happen in any 1PI diagram, at least in four-dimensional space-time. The photon cannot acquire mass in spite of quantum corrections in contrast to the fermion case. Close to the pole q 2 = 0 in (11), the residue of the dressed propagator no longer equals 1 but becomes Z3 defined below. Indeed, the analog of (14.29) for the photon case is Bare propagator −i gµν q2
−→ −→
Dressed propagator Dµν (q) 2=
−i gµν −i gµν ≡ Z3 , − Π(0)] q2
q →0 q 2 [1
(15.12)
where Z3 is given by (Z3 )−1 = 1 − Π(0). For the same reason already explained in (14.31), the perturbative expression of Z3 is Π(0) = 1 −
1 Z3 − 1 = = Z3 − 1 + O(e4 ) . Z3 Z3
(15.13)
From (6) and (13), we get 1ε e2 4π 2 e2 2 4πµ2 1 Z3 = 1 − Γ( 2 ε) = 1 − − γE + log + O(e4 ) , 12π 2 m2 12π 2 ε m2 p 1 e2 2 4πµ2 Z3 = 1 − ( ) − γE + log + O(e4 ) . (15.14) 2 12π 2 ε m2 Let us relate the pole in Γ( 12 ε) to the divergence log(Λ2 /µ2 ) found in the Pauli–Villars procedure
4π m2
1ε 2
Γ( 12 ε) ←→ log
Λ2 e2 Λ2 , so Z3 = 1 − log 2 + O(e4 ) .(15.15) 2 2 µ 12π µ
15.1 Running Coupling Constant
511
√Now from (14.36) we recall that the bare fermion field ψ0 (x) is absorbed by Z2 to become the renormalized field ψ(x), from which the renormal√ ized fermion propagator with residue equal to 1 is built. Similarly, Z3 is absorbed by the bare photon field A0µ (x) which becomes the renormalized √ Aµ (x). With A0µ (x) = Z3 Aµ (x), the renormalized photon propagator has residue equal to 1 at the pole q 2 = 0. We have
−i Z3 gµν Dressed propagator Dµν (q) = ←→ 0 T (A0µ (x) A0ν (y)) 0 , 2 q −i gµν ←→ h0 | T (Aµ (x) Aν (y)) | 0i . (15.16) Renormalized propagator q2 Now the renormalized charge e appears accompanied by Z3 . As already noted, since the fermionic charges are always attached to each end of the photon propagator in all amplitudes, the infinite Z3 in (14) can be absorbed by the √ infinite bare charge e0 to become the finite renormalized charge e: e0 → e = e0 Z3 . Indeed, as shown in Fig. 15.4, when a fermion line is attached to each end of the propagator in (12), the amplitude has an additional factor e20 or e2 times the corresponding propagator: Bare −i gµν 2 e0 q2
√
Interaction Π(q 2 )
−→
Z3 e20 ... ... ... ... q2 ... ... ... ... ... ... . . . . . . . . . . . . .... .. ... .... . . .. . . . . . . . .. .. ......................................................................................... ... . . . ... .. ... . . . ... ..... ..... ... ... ......... .... .... ... ... ... ... ... ...
Z3 e0 •
J •
•
√
Z3 e0
−→
Dressed Renormalized −i gµν Z3 2 −i gµν 2 e0 −→ e . (15.17) q2 q2
−→
e
e2 ... ... ... ... q2 ... ... ... ... ... ... ... .. . . . .. . . . . . .. .............................................. ........................................ ... . . . . .. .. .. . . ... ... ... .... .... ... ... ... ... ... ...
e
Fig. 15.4. Dressed and renormalized photon propagators attached to the bare and renormalized fermion charges respectively
Note that e√ 0 ψ0 (x) = e ψ(x). In spite of its apparent simplicity, the relation e = e0 Z3 has a far-reaching consequence, since it implies that the renormalized charge e depends only on the universal photon field-strength renormalization Z3 and not at all on other properties of the charged fermions to which the photon is coupled. In particular, in spite of the huge mass difference between the heavy tau lepton and the electron, their renormalized charges are identical, because they depend only on the photon Z3 . If the renormalized charge e depended also on the fermion self-energy Σ(p) and its electric form factor F1 (q 2 ), the universality of the renormalized charge e would be lost. In fact, e does depend on Σ(p) and F1 (q 2 ) via the three diagrams of Fig. 15.1. To recover universality, there must exist a cancelation among the quantities related to Σ(p) and F1 (q 2 ), so that the charge renormalization is only due to Z3 . We now show that the cancelation indeed takes place, due to the Ward identity.
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15 Asymptotic Freedom in QCD
15.1.3 Vertex Renormalization Mass and field renormalizations have been considered in Chap. 14. The last but not the least important quantity to undergo renormalization is the coupling constant. Let us proceed in three steps. A – Similar to (14.35), we start with the bare quantities e0 , m0 , ψ0 (x), and A0µ (x) of the QED Lagrangian 1 L = − (∂µ A0ν − ∂ν A0µ )2 + ψ 0 (i 6∂ − m0 )ψ0 + e0 ψ0 γ µ ψ0 A0µ . 4
(15.18)
Then we define the renormalized quantities e, m, ψ(x), and Aµ (x) by introducing at first three counterterms Z2 , Z3 , and ∆m0 with which we are already acquainted in Chap. 14 and (14). Z2 and Z3 are used to cancel the UV divergences of the fermion and photon self-energies respectively, while ∆m0 renormalizes the fermion mass by absorbing the bare mass m0 . We put p p ψ0 (x) = Z2 ψ(x) , A0µ (x) = Z3 Aµ (x) , m0 = m − ∆m0 . (15.19) √ √ Because the ψ0 (x) and A0µ (x) fields are scaled by Z2 and Z3 respectively, when we substitute (19) in the interaction e0 ψ0 γ µ ψ0 A0µ of (18), we have a √ factor e0 Z2 Z3 in front of ψγ µ ψAµ . Let us multiply this factor by 1: 1 δ1 Z1 = + , δ1 = Z1 − 1 , then Z1 Z1 Z1 √ √ p e0 Z2 Z3 e0 Z2 Z3 e0 Z2 Z3 = (1 + δ1 ) = e(1 + δ1 ) , where e ≡ . (15.20) Z1 Z1 1=
We will see [the last line of (24)] that in addition to the counterterms Z2 and Z3 , one more counterterm Z1 is needed to subtract the UV divergence of the QED vertex function, i.e. the UV divergence of the form factor F1 (q 2 ). The latter has already been computed in (14.17), we only need to replace 34 gs2 of (14.17) by e2 in our case here. When we put (19) into (18) and use (20), the original L in (18) can be rewritten as a sum of two parts having exactly the same forms, differing only in their coefficients. They are Lren (the renormalized Lagrangian) and Lct (the Lagrangian of counterterms): L = Lren + Lct ; 1 Lren = − (∂µ Aν − ∂ν Aµ )2 + ψ(i 6∂ − m)ψ + eψγ µ ψAµ , 4 1 Lct = − δ3 (∂µ Aν − ∂ν Aµ )2 + ψ(i δ2 6∂ − δm )ψ + eδ1 ψγ µ ψAµ , 4 δ1 = Z1 − 1 , δ2 ≡ Z2 − 1 , δ3 ≡ Z3 − 1 , δm ≡ mδ2 − Z2 ∆m0 .
(15.21)
Lren becomes formally identical to the bare one in (18) by interchanging m , e , ψ , Aµ with m0 , e0 , ψ0 , A0µ . Therefore the quantities Γµ (p0 , p), Σ(p),
15.1 Running Coupling Constant
513
and Π(q 2 ) that we have computed with the bare Lagrangian can be identified with the ones derived from Lren with the interchange (m0 , e0 ) ↔ (m , e). As for the counterterm Lagrangian Lct in (21), all are of second- or higher-order terms in e. B – The Feynman rules with these renormalized fields are: in addition to the familiar propagators and vertex from Lren , all are expressed in terms of the physical mass m and charge e, there are three contributions coming from Lct . To first order in δ1 , δ2 , δ3 , δm , they can be easily obtained from the three terms of Lct and visualized in Fig. 15.5: i (−q 2 gµν + q µ q ν )δ3 , . . . . .N . . . . ..... ..... ..... ........... ..... ..... ..... ..... .
i eγ µ δ1 . ...... (15.22) L....................... ... ... ... ... .... .........
i (6 pδ2 − δm ) , N
.. ... ... ... ... ...... ............................. .. .. .. .. .. ......... ......... ......... .
.........................................................
.... .. . .. . ...... .. . .. . .. . .. . .. . .. .. .. .. .. .. .. .. ..
Fig. 15.5. Counterterm contributions
The counterterms of the photon self-energy, of the electron self-energy, and of the electron–photon vertex are shown from left to right in (22). For instance, the term (−q 2 gµν + q µ q ν ) δ3 is obtained when we integrate by parts the kinetic term −( 14 )(∂µ Aν − ∂ν Aµ )2 δ3 of (21) to get ( 12 )Aµ (q)(−q 2 gµν + q µ q ν )Aν (q) δ3 in momentum space. Lastly, we multiply by two because of the symmetry µ, ν of the two-photon line. The two other terms in (22) are straightforward from (21). Their unique role is to cancel the UV infinities in loop integrals computed from Lren. In parallel with the three quantities from left to right in (22), we have also three corresponding terms coming from Lren, which we have already computed. They are given by (6), (14.23) and (14.11): i (q 2 gµν − q µ q ν )Π(q 2 ) ,
−i Σ(p) ,
i eΓµ (p0 , p) .
(15.23)
C – The sum of the two contributions from Lren and Lct are the renormalµ 0 e ren (q 2 ) , Σ e ren (p) , and Γg ized quantities Π ren (p , p), which are free of UV divergences. Taken from (22) and (23), they are e ren (q 2 ) = Π(q 2 ) − δ3 , Π e ren (p) = Σ(p) − (6 pδ2 − δm ) = Σ(p) − δ2 (6 p − m) − Z2 ∆m0 , Σ ren µ 0 µ 0 µ f (q 2 ) = F (q 2 ) + δ . Γg ren (p , p) = Γ (p , p) + δ γ =⇒ F 1
1
1
1
(15.24)
As a general rule stated in Chap. 14, these renormalized quantities obey e ren (q 2 ) and Σ e ren (p), the conditions are renormalization conditions. For Π
e ren (6 p) d Σ e ren (q 2 = 0) = 0 , e ren (6 p = m) = 0 , equivalently Π =0, Σ d 6p 6p=m dΣ(p) Π(q 2 = 0) = δ3 , = δ2 , Σ(6 p = m) = Z2 ∆m0 . (15.25) d 6 p 6p=m
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15 Asymptotic Freedom in QCD
From left to right in (25), the first two terms fix the residues of the renormalized photon and electron propagators to be 1, and we recover (13) and (14.31) respectively. The third term sets the pole of the renormalized electron propagator to be m , and as expected we recognize (14.46), remember that Z2 = 1 + O(e2 ) and ∆m0 = m − m0 = O(e2 ). See also (14.47). Concerning the vertex part, the third line of (24) clearly indicates the role of δ1 . It is used to cancel the UV divergence of Γµ (p0 , p), that of F1 (q 2 ) precisely. The renormalization condition for this vertex part is ren µ µ µ f (0) = F1 (0) + δ1 = 0 .(15.26) Γg ren (p, p) = Γ (p, p) + δ1 γ = 0 =⇒ F1
This condition defines the renormalized charge as the electron–photon couren f1 (q 2 ) pling at vanishing four-momentum transfer [q µ = (p0 −p)µ = 0], i.e. F must vanish at q 2 = 0. For this specific value q 2 = 0, the charge does not receive quantum corrections, it is nonrenormalized. See also Problem 10.2. On the other hand, direct computations – done in Chap. 14 through (14.20), (14.41) resulting in (14.42) – give the Ward identity (see Problem 14.3): dΣ(p) F1 (0) + ≡ F1 (0) + δ2 = 0 , d 6 p 6p=m
so that with (26), we have δ1 = δ2 =⇒ Z1 = Z2 , then using (20), we have e = e0
p
Z3 .
(15.27)
These detailed studies show that Z1 = Z2 for any type of charged fermions. The three diagrams of Fig. 15.1a–c do not contribute to the charge renormalization, the latter depends only on Z3 . The Ward identity actually is a nonperturbative result, valid for all perturbative orders. This completes the renormalization of the coupling constant, the sum of the four terms in (8) results in the QED renormalized electric form factor ren f1 (q 2 ) which is free of UV divergences, and satisfies the renormalization F ren f1 (0) = 0. We have from (8) condition F f1 F
ren
e ren (q 2 ) , (q 2 ) = F1 (q 2 ) − F1 (0) + Π
(15.28)
e ren (q 2 ) is given below in (29). Compared with the QCD the last term Π e ren (q 2 ) is missing. This is corrected weak form factor (14.43), the last term Π expected since the equivalent of Fig. 15.2 is absent from Fig. 14.2.
15.1 Running Coupling Constant
515
e ren (q 2 ) 15.1.4 Renormalized Vacuum Polarization Π
e ren (q 2 ) = Π(q 2 ) − δ3 written in (24). Together The total Lren + Lct gives us Π e ren (0) = 0 in (25), we get with the renormalization condition Π e ren (q 2 ) = Π(q 2 ) − Π(0) . Π
(15.29)
While both Π(q 2 ) and Π(0) are UV divergent, their difference is unambigue ren (q 2 ) is given by ously finite. Indeed, from (6), Π 2 e ren (q 2 ) = +e Π 2π 2
Z
0
1
q2 dx x(1 − x) log 1 − 2 x(1 − x) . m
(15.30)
The renormalization condition fixes the subtraction point at q 2 = 0, i.e. we subtract the divergence of Π(q 2 ) by the divergence of Π(0). This choice ensures that at the q 2 = 0 pole, the residue of the renormalized photon propagator is 1. In practice, the renormalization can be seen as a simple e ren (q 2 ) ≡ Π(q 2 )−Π(0) replacement of the divergent Π(q 2 ) in (6) by the finite Π in (30) symbolized by the following picture: Dressed Propagator
Renormalized Propagator
−i gµν −i gµν −i gµν −i gµν −→ , −→ . e ren (q 2 )] q2 →0 q 2 q 2 [1 − Π(q 2 )] q2 →0 q 2 [1 − Π(0)] q 2 [1 − Π
(15.31)
e ren (q 2 ). The analytic property of Π e ren (q 2 ) in (30) is illuAnalyticity of Π 2 minating. For spacelike photon q < 0, the argument 1−q 2 x(1−x)/m2 of the e ren (q 2 ) is real. For timelike photon, the logarithm in (30) is positive, and Π e ren (q 2 ) argument is negative for q 2 ≥ 4m2 [since 0 ≤ x(1 − x) ≤ 1/4], hence Π 2 2 becomes complex with a branch cut starting at q = 4m which is the threshold for the creation of an on-shell electron–positron pair. The imaginary part e ren (q 2 ) can be directly obtained from that of the logarithm, using the of Π relation Im [log(−(X 2 ± iε))] = ∓π. We have Z + e2 (∓π) X 2 e dx x(1 − x) Im [Πren (q ± i)] = 2π 2 − sX e2 4m2 2m2 =∓ 1 − 2 (1 + 2 ) , 12π q q
where X ± =
1 2
1± 2
q 1−
4m2 q2
(15.32)
. This is an example of singularities or cuts
in the complex q plane when the particles in any intermediate state are on-shell. It also illustrates the analytic property of physical amplitudes from which dispersion relations can be derived (Chap. 10).
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15 Asymptotic Freedom in QCD
e ren (q 2 ) is (with ρ = q22 > 1) From (30), the real part of Π 4m ( ) √ √ 2 2 ρ+ ρ−1 −e 1 + ρ − 2ρ 3 + 5ρ 2 e p √ Re [Πren (q )] = + log √ . (15.33) 2π 2 18ρ ρ− ρ−1 12ρ ρ(ρ − 1)
e ren (q 2 ) is found to be For 0 < q 2 ≤ 4m2 , the explicit expression of Π ( ) r 2 2 −e 3 + 5η 2η − η − 1 η q2 −1 e ren (q 2 ) = p Π + tan , η ≡ ≤1. 2π 2 18η 1−η 4m2 6η η(1 − η) 2
Q For spacelike q 2 ≡ −Q2 ≤ 0, we get (with ξ = 4m 2 ≥ 0) ( √ √ ) 2 2 −e −3 + 5ξ 1 − ξ − 2ξ 1 + ξ + ξ 2 e ren (Q ) = p √ . (15.34) Π + log √ 2 2π 18ξ 1+ξ − ξ 12ξ ξ(1 + ξ)
e ren (q 2 ) can also be easily The two extreme limits Q2 m2 and Q2 m2 of Π obtained from (30): 2Z 1 e2 −q −α q 2 2 2 2 e dx x (1 − x) = , (15.35) Πren (q ) −→ q2 →0 2π 2 m2 15π m2 0 2 Z 1 e2 −q e ren (q 2 ) −→ Π dx x(1 − x) log + log [x(1 − x)] q2 →−∞ 2π 2 0 m2 e2 Q2 5 +α Q2 = log − = log , (15.36) 12π 2 m2 3 3π C m2 where C = exp 53 . The (+) sign in (36) for the asymptotic limit of the vacuum polarization is an important property of QED. Aside the common e ren (q 2 ) factor e2 /2π 2 = 2α/π, the real and imaginary parts of the function Π 2 are plotted in Fig. 15.6 for both spacelike and timelike q . 0.8
Real Part Imaginary Part 0.6
0.4
PI(q^2)
0.2
0
-0.2
-0.4
-0.6 -10
-5
0 q^2/4m^2
5
10
e ren (q2 ) (without the overall factor 2α/π) Fig. 15.6. Real and imaginary parts of Π 2 2 are plotted as functions of q /4m
15.1 Running Coupling Constant
517
e ren (q 2 ) 15.1.5 Physical Effects of Π
We recall that a charged fermion line is always attached to each end of a photon propagator. To lowest order of the coupling constant, the electron scattering amplitude is given by e2 times the photon propagator. How does e ren (q 2 ) modify this amplitude? The effect of replacing the free photon Π propagator by the fully interacting photon propagator is equivalent to the substitution of e2 by e2 (q 2 ). The latter is defined following (31): −ie2 gµν −ie2 gµν −ie2 (q 2 ) gµν −→ , ≡ e ren (q 2 )] q2 q2 q 2 [1 − Π e2 α where e2 (q 2 ) = or α −→ αeff (q 2 ) ≡ . e ren (q 2 ) e ren (q 2 ) 1−Π 1−Π
(15.37)
At small distance or high −q 2 ≡ Q2 > 0, using (36) we get αeff (Q2 ) =
1−
α 3π
α , C = exp log(Q2 /C m2 )
5 3
.
(15.38)
We can also write αeff (r) as a function of the distance r by performing a Fourier transformation of αeff (q 2 ). Because of the minus sign in the denominator of (38), the effective electromagnetic coupling constant αeff (q 2 ) becomes larger at higher energies or at smaller distances, as we penetrate the screening cloud of virtual fermion pairs considered as a dielectrically polarized medium filled up with effective dipoles of length ∼ 1/m. An intuitive interpretation is represented in Fig. 15.7. ..................... .... ... .. − ... .... + .. ... ... .. ...... ....... ...........
e
......... ... .. .. −... .. .... ... +.... . .. ... .... ........
e e
e
......... ...... ........ ... −.... .... . .. + ... .. .... ... ....................
e
e
e−+ . ....... .. ..... +e− e ....... ............................................. .e.. ....... .. ........ .................. ... .......... .......... ... ............. .. . . .... + −... .....e 0 ...... ....... .... . ... .............e ............e ........... . ................... ..
...................... ...... . .... − +.... ......... ......... .........
e e
... ............................................ . .. . .. ...
+..................... + e − −e
.. ......... ............ ... .. +...... .... ... − .. .. ... ...... ....... ...........
e
e
e
e
....... ... ... .. +... .... . ... −..... ... .. ..... ..... ...
e e
(a)
..................... ... ... .. .. .... + . .. ... .. − .. .... . . . . ..................
e
e
αeff (r) .
... ....... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. ... ... ... ... ... ... .... ... .. ... ... ... ... .... ... .. .... .... ... ...... ....... .... .......... .. ............ ... .................. .................................................................. .... .................... . 1 ... . . . . . . . . . . . . . . . . . . . . . . . . . ............................. . .. . 137 ... .......................................................................................................................................................................................................... ....
r
(b)
Fig. 15.7. (a) Screening of the bare charge e0 by virtual pairs e+ e− ; (b) qualitative dependence of the QED αeff (r) on the distance r
518
15 Asymptotic Freedom in QCD
The quantum vacuum is not empty but filled with virtual charged fermion pairs. The introduction of a bare electron polarizes the vacuum in much the same way as a classical charge polarizes a dielectric medium. Attracted by e0 , the virtual positrons spend more time near the negative bare charge before annihilation, while the virtual electrons are repulsed and spend more time away. The cloud of virtual positrons that surrounds e0 screens the bare charge and acts like a dielectric medium with a dielectric constant > 1. This makes the charge stronger at shorter distances. The fact that e0 is larger than √ the effective renormalized charge e can be seen indirectly through e = e0 Z3 < e0 . In √ (15), we notice the sign of the coefficient of log(Λ/µ) in Z3 which implies Z3 = 1 − O(e2 ) log(Λ/µ) < 1. Its justification will be given later by the renormalization group equation. The minus sign of the coefficient of log(Λ/µ) is crucial to the asymptotic behavior of the QED coupling. This minus sign is intimately related to the anticommutation property of the fermionic loop in (2) which gives Z3 . The q 2 -dependence effect of the QED coupling αeff (q 2 ) is experimentally observed in high energy Bhabha scattering.1 Between q 2 = 0 and −q 2 = (30 (GeV)2 , the effective coupling increases by about 5%. Also at the weak boson Z0 mass, αeff (MZ2 ) is ≈ 1/128.896 > 1/137.036. On the opposite side, for small q 2 (mr 1), the effective charge αeff (r) decreases like exp(−2mr)/(mr)3/2 . It modifies the Coulomb potential and contributes to lowering the energy levels of atomic states. This is called the Uehling effect (Problem 15.3). The concept of a running coupling, which is no longer constant but energy-dependent, emerges from these remarkable quantum effects and constitutes a major theme of QCD. As we will see, due to the gluonic non-Abelian interaction, the color dielectric constant color < 1 has an antiscreening effect and makes the running QCD coupling weaker at shorter distances.
15.2 The Renormalization Group We have illustrated the notion of the running coupling constant by the effects of quantum corrections. The problem can be better understood in the most generality using the remarkable renormalization group concept. From the example in (6) we notice a crucial point already mentioned after (14.17): there always exists an arbitrariness (via an arbitrary mass scale µ) in the finite parts of F1 (q 2 ) and Π(q 2 ), after their UV divergences are removed. In fact since loops always have q 2 and m2 terms in the integrals, they must be scaled by some parameter µ2 to make the answers dimensionally correct for ε 6= 0. An explicit example is shown in (7). This arbitrariness is unavoidable in any regularization method, as can be seen merely on dimensional grounds. For example, in the Pauli–Villars regularization by a large mass Λ cutoff of the divergent integrals d4 k, a scale µ must enter via the factors (Λ/µ)l and 1
HRS collaboration, Derrick, M. et al., Phys. Rev. D34 (1986) 3286.
15.2 The Renormalization Group
519
log(Λ/µ) in order to make dimensionally correct the quantities we compute. When Λ is removed by renormalization (just as the ε poles are removed in dimensional regularization), some functions of q 2 /µ2 , m2 /µ2 , . . . in the finite parts always remain. Of course we can take a specific choice µ2 = m2 , but why m2 and not 8 m2 ? And what happens in the massless case? This case is not merely academic, because we are considering the asymptotic behavior of physical quantities for which m2 q 2 . The arbitrariness associated with the mass scale µ is reflected in the renormalization conditions which fix the finite part of the renormalized physical quantity at some kinematic point chosen arbitrarily (Sect. 8.4). The renormalization conditions for coupling constants are usually made by convention, sometimes guided by experiments. For instance, the renormalization condition (26) for the electric form factor Fe1ren (0) = 0 is dictated by the definition of the charge e as the coupling of an electron to a photon at zero momentum transfer. This coupling is given by the Thomson formula (4.223) of the e–γ cross-section measured at q 2 = 0. But we can equally define the charge – denoted now by e0 – as extracted for instance from the cross-section e+ + e− → µ+ + µ− measured at q 2 6= 0, say at 30 (GeV)2 , for which a posteriori we know that e02 /4π = α[30 (GeV)2 ] > α(0) = 1/137.0359895. The cross-sections, calculated by two physicists using two different definitions of the coupling constant, appear to be different by an overall constant. But it is immaterial, since the coupling constant is extracted from experiments by the value of the cross-section measured at some energy scale. In the present state of our knowledge, we cannot calculate the coupling constant, we only define it at some kinematic point, and then use this definition to compute other physical quantities like decay rates, etc. How about QCD? To obtain the coupling constant αs , it does not make sense to measure the quark–gluon cross-section at the Thomson limit. Obviously, as any dimensionless coupling constant, αs needs a scale. Let us briefly summarize the concept before considering its realization through the renormalization group equation. We start with a bare Lagrangian, then we compute quantum effects. To handle UV divergences in loop integrals, we choose some regularization scheme. After the subtraction of UV infinities, we impose renormalization conditions on the finite parts of the computed quantities at some kinematic point. Clearly, physics should not depend on this arbitrary choice of either the regularization procedures or the subtraction points. Any choice is as good as any other. This apparent arbitrariness in fact is subtle and turns out to imply powerful constraints on the asymptotic behavior of the theory. There must exist physical quantities invariant under the transformations which merely change the renormalization conditions as well as the regularizations. This is the gist of the renormalization group independently formulated by Callan and Symanzik in a concrete equation. The original work went back to Stueckelberg and Petermann, and exploited by Gell-Mann and Low to get information on the asymptotic behavior of the photon propagator, and developed by Bogoliubov and Shirkov.
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15 Asymptotic Freedom in QCD
15.2.1 The Callan–Symanzik Equation We start by considering a bare Lagrangian with Φ0 and g0 as bare field and coupling constant. Φ0 can be a self-interacting real scalar field with one coupling [for instance λφ4 /4!], or Φ0 can designate collectively many fields, for example the fields of a charged fermion ψ and a photon Aµ in QED (g0 = e0 ), or the fields of color quarks and color gluons in QCD. Also, g0 may generically designate one or many dimensionless couplings. Exactly as in (14.36) and√ (19), we renormalize the interacting field Φ0 by a multiplicative counterterm Z and get the renormalized field Φ, i.e. Φ0 = √ Φ Z. If there are many different fields φj , there are as many Zj . By the explicit examples of Z2 and Z3 for fermion and photon fields respectively in (14.41) and (14), we are familiar with the dependence of Z on an arbitrary scale µ. Z is also a function of g0 , and we write Z(µ, g0 ). The bare coupling constant g0 accordingly becomes the renormalized coupling constant g which is g0 times a counterterm denoted by Zcpl : g ≡ g0 Zcpl . The crucial role of Zcpl will be clear later in the renormalization group equation. The important point is the general formula (39) given below, which tells us how to build Zcpl . This formula is√a generalization of (20) and √ √ is √ illustrated √ in Fig. 15.8. For QED, e = e0 Z2 Z2 Z3 /Z1 , so Zcpl = Z2 Z3 /Z1 = Z3 (remember Z1 = Z2 ). ....... .. .............. ..... .. ............... 3 ..... .............. ..... ................ . . . . . . . . . . . . . . . . . . . . . .. ....... .. ..... .................................................... .................. 1 .................. .............. Z ............ ................. .1 ................. ... ................................. ... ... .............. ..... ... ... . . ... . ... ... ... ... ... ... . . . . . . . 2.. 2 .....
•
√
Z
•
√
Z
√ • Z
(a) QED
p p
Zφ
.....
Zφ .....
.... .
• .....
.....
.• ....
... ... ....................... ....... ...... ...... 1 ......... φ ....... ....... .......Z1 ......... ........................ ......... . . . .... . . ...
p
. ....
• p •..... .. Zφ ...
. ....
(b) φ4
Zφ
............ .... ....................... ........... ........................ ........... . ...................... .. .... . . ........ .......... .... 1 ..... ...... glu.. . .. .. Z ... ..1 .. .. . .. . . ........... . . . . . ... . . . ..................... . ............................. ........................... ............................. .......................... .. .. . .................
••
•• p Zglu
p
Zglu
•• p Zglu
(c) QCD
Fig. 15.8. Coupling constant counterterm Zcpl in (a) QED; (b) φ4 ; and (c) QCD
√ Q In general, Zcpl is the product ext ( Zext ) of all the external lines which are connected to the vertex loop divided by the vertex counterterm generically e1 naturally contribute denoted by Ze1 . The vertex loop and its counterterm Z to the renormalized coupling constant g because the zero-loop vertex is by definition the bare coupling constant g0 . e1 in the denominator of Zcpl comes about exactly as The presence of Z e1 = (1/Z e1 ) + (δe1 /Z e1 ). The first term (1/Z e1 ) in (20) by the trick 1 = Ze1 /Z e enters Zcpl to define the renormalized coupling constant g; the second (δ1 /Ze1 ) is a counterterm used to subtract the vertex loop divergence, similar to (26).
521
15.2 The Renormalization Group
Thus, the most general form of the renormalized coupling constant is g = g(µ) = g0 Zcpl (µ) =
g0 Y p Zext . Ze1
(15.39)
ext
4 Two more examples are shown in Fig. 15.8b, c respectively p for 4φ φand QCD. 4 For the φ self-interacting scalar field, we have Zcpl = ( Zφ ) /Z1 . In nonAbelian Lagrangian with only bosons, e.g.pQCD without quarks, via the three-boson self-coupling, we have Zcpl = ( Zglu )3 /Z1glu . Each Lagrangian has its own Zext and Ze1 to be calculated. We are not without experience of their tedious computations in (14.41) and (14). We go now to the next step by introducing the Green’s functions which are convenient for the discussion of quantum loop effects. A k-point function or Green’s function is defined by
Gk (x1 , · · · , xk ) = h0 | T [Φ(x1 ) · · · Φ(xk )] | 0i , and its Fourier transform in momentum space is denoted by Gk (p1 , · · · , pk ). Φ(x) may designate a single field or many different fields. .. .... ..... .... ............. . ... . ..... ......... .... ..... .... . . . .. .... ....... ....... ............. ........... . . .. ....... ........ .................................. .......... ........................................... .. ... ... .. . .. ........ ... .. .. .. .. .. . . . ....... ....... .............. .... ..... .... ..... ..... ..... ................ .... ... ........... ..... ..... .... .
•
•
1PI
....... ....... ...... .... .. .... .. .. ....... ....... .......
1PI
=
... ..... ..... ...... .... .............. .... . ........ . . ......... ............. .... ..... .............. .... ....... .......... .....
+···
•
Fig. 15.9. QED three-point function
A typical fermion two-point function is its full (dressed) propagator, i.e. the geometric sum of 1PI fermionic self-energy with two external fermion lines, similar to Fig. 14.7a. Note that a 1PI self-energy is already an infinite sum of graphs in perturbative calculations. A QED three-point function (with two charged fermion fields and one photon field) is the full vertex function which includes both a three-leg tree diagram and a 1PI closed loop with three external lines, the latter are full propagators of these fields (schematically drawn in Fig. 15.9 with • ). When these external propagators are removed, we have an amputated three-point function. The k-point Green’s function is defined similarly with k external propagators connected to closed loops. The S-matrix element with which we compute cross-section or decay rate is an amputated Green’s function. As it stands, it is a herculean task to compute Green’s functions in their most general form. In perturbative orders, the simplest Green’s functions are the tree amplitude, the one-loop amplitude, and so on. Renormalization group equation provides, as we will see, a powerful method for ‘partial
522
15 Asymptotic Freedom in QCD
summation’ over all perturbative orders; by partial summation we mean the leading logarithm terms of Green functions. Let us now consider a bare k-point function Gk0 (pi , g0 ) derived from a bare Lagrangian. For simplicity, we take Φ as a single field, for √ instance the scalar field φ in the g φ4 /4! Lagrangian. The changes Φ0 = Φ Z and −1 g0 = g Zcpl induce the change of Gk0 (pi , g0 ) into the renormalized Green’s √ function Gkren (pi , µ, g). After the rescaling by k powers of Z for Φ0 , and the elimination of g0 in favor of g, we obtain Gkren (pi , µ, g) which is numerically equal to the bare Gk0 (pi , g0). The former depends explicitly and also implicitly through g(µ) on µ. All of these operations are equivalent to a reparameterization. Thus, k
Gkren (pi , µ, g) = [Z(µ, g0 )]− 2 Gk0 (pi , g0 ) .
(15.40)
Equation (40) is the generalization of (16) from a two-point function to a k-point function. We first remark that the bare Gk0 (pi , g0 ) makes no reference to the scale µ, it depends only on g0 . The independence with respect to the variable µ of Gk0 (pi , g0 ) implies that [Z(µ, g(µ))]k/2 Gkren(pi , µ, g) is independent of µ too. So we have k k d d k 1 dZ k Z 2 Gren = Z 2 + Gkren = 0 . (15.41) dµ dµ 2 Z dµ Applying the differentiation d ∂ ∂ ∂g(µ) = + dµ ∂µ ∂g ∂µ on (41) and multiplying it by µ to make operators dimensionless yields the Callan–Symanzik (CS) equation ∂ ∂ k µ +β + γ Gkren (pi , µ, g) = 0 , where (15.42) ∂µ ∂g 2 ∂g(µ) β=µ , (15.43) ∂µ g0 fixed ∂ log Z(µ, g0 ) γ=µ . (15.44) ∂µ g0 fixed
Equation (42) expresses the fact that any change in the subtraction point µ amounts to a change in the coupling constant and a change in the fieldstrength, i.e. the physical content of the theory is not affected by a mere change of this parameterization. For each type of Lagrangian, there exists two corresponding functions β and γ, called Callan–Symanzik functions; the former governs the evolution of the coupling constant and the latter that of
15.2 The Renormalization Group
523
the field strength. The method can be easily generalized to other theories with many fields and dimensionless couplings. For instance in QED, (42) becomes ∂ ∂ 1 µ + β(e) + 2 [Ne γ2 (e) + Nph γ3 (e)] Gkren (pi , µ, e) = 0 , (15.45) ∂µ ∂e where Ne and Nph are respectively the number of external electron and photon fields in the k-point Green’s function Gkren(pi , µ, e), with k = Ne + Nph . As in (44), γ2 and γ3 are respectively the derivatives of the field-strength counterterms Z2 and Z3 of the electron and photon fields. According to (43), the β-function represents the rate of change of g with respect to µ. A positive sign for the β-function indicates that g increases at large momenta and decreases at small momenta, and inversely for a negative sign of β. Therefore the β-function, in particular its sign, is of great importance for the evolutionary behavior of the coupling. Before looking for the solution of (42), let us see how we can compute the β- and γ-functions.
15.2.2 Calculation of the β - and γ -Functions In free field theory without interactions (g = 0, Z = 1), the β and γ functions obviously vanish. If the interacting Lagrangian represents a finite theory, i.e. if all loop integrals were finite, then the renormalization group equation (42) is trivial and empty. Since Z is finite, we do not need regulators and the presence of µ is superfluous. For a finite theory, Z(µ, g0 ) and g(µ) do not depend on µ, the CS functions are also identically zero. This observation clarifies the role of counterterms, since nonzero β- and γ-functions appear when the theory, although not primitively finite, is renormalizable by the absorption of the counterterms. The infinities encountered in loop integrals turn out to be providentially useful. The counterterm Z and consequently the dimensionless g are now functions of µ and of the regulator 4 − n = ε; the latter can be replaced by a large mass scale Λ [the equivalence between the pole 1/ε and log(Λ/µ) is explicitly written in (15)]. In fact, since the function Z is dimensionless, its dependence on µ must be through the ratio Λ/µ, more precisely through log(Λ/µ). The β-function is obtained from (39) and (43) by taking the derivative of g = g0 Zcpl (µ) with respect to µ. We immediately recognize that via the ratio Λ/µ, the dependence of Zcpl on Λ is essential. To compute the β-function, it suffices to keep only the Λ dependence of the Zs , i.e. only their UV divergent parts; their finite terms are irrelevant. Note that whereas Z ∼ log(Λ/µ) is divergent, the derivative of Z is finite, so too is the β -function. Let us write ∂ ∂ Λ µ +Λ log = 0 , (15.46) ∂µ ∂Λ µ ∂ Λ ∂ Λ using µ log = −Λ log = −1 . ∂µ µ ∂Λ µ
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15 Asymptotic Freedom in QCD
We have from (39) ∂ Λ ∂ Λ β(g) = g0 µ Zcpl ( , g0 ) = −g0 Λ Zcpl ( , g0 ) ∂µ µ ∂Λ µ g0 ,Λ fixed g0 ,µ fixed ∂ Λ = −gΛ log[Zcpl ( , g0 )] (15.47) ∂Λ µ g0 ,µ fixed
using g d log Zcpl = g0 dZcpl . With Z ≡ 1 + δ and log Z ≈ δ, i.e. to lowest orders in g for the β-function, we get from (39) and (47) " # X ∂ e1 + 1 β(g) = −g Λ − log Z log Zext 2 ∂Λ ext " # X ∂ 1 e δext (µ, g) . (15.48) =gµ −δ1 (µ, g) + 2 ∂µ ext
The existence of β and its dependence on the renormalized coupling g are due to the renormalizability of the theory. When the cutoff Λ goes to infinity, Green’s functions remain finite when expressed in terms of the renormalized parameters, in particular of the renormalized coupling g. Since the CS functions β and γ are finite, they cannot depend on the cutoff Λ, hence they cannot depend explicitly on µ via the ratio Λ/µ; they depend only on the dimensionless renormalized coupling g, and implicitly on µ via g(µ). The CS function γj (g) defined in (44) for each field Φj , can be similarly obtained by γj (g) = µ
∂ δj (µ, g) . ∂µ
(15.49)
In brief, the computation of the β- and γ-functions reduces to the calculation of the vertex counterterm Ze1 and the field-strength counterterms Zext .
One-Loop β- and γ-Functions in QED. As the first √ illustration, let us compute β(e) and γ(e) in QED. From (48) and e = e0 Z2 Z3 /Z1 , ∂ 1 β(e) = e µ −δ1 + (2δ2 + δ3 ) . (15.50) ∂µ 2 We have already computed the δ2 of QCD in Chap. 14 and given it in (14.41). We now get the δ2 of QED by a simple substitution: 4 2 g ↔ e2 3 s
n 2 Λ2 and Γ 2 − ∼ ↔ log 2 . 2 ε µ
As for δ1 and δ3 , they are given by (26) and (15) respectively, δ1 = δ2 =
−e2 Λ2 −e2 Λ2 log 2 + ( ift ) , δ3 = log 2 + ( ift ) , 2 2 16 π µ 12π µ
(15.51)
15.2 The Renormalization Group
525
where ( ift ) means ‘irrelevant finite terms’. We get from (50) ∂( 21 δ3 ) ∂ −e2 Λ2 +e3 β(e) = e µ = eµ log = . ∂µ ∂µ 24π 2 µ2 12π 2
(15.52)
To check that the pole Γ(2 − n2 ) can be identified with log(Λ2 /µ2 ) in (51), we directly compute β(g) from the δ’s which are written in terms of Γ(2 − n2 ) without passing by the cutoff Λ. With (14) for QED, −e2 Γ(2 − n2 ) + ( ift ) , (15.53) n 12π 2 [µ2 ]2− 2 ∂ −e2 Γ(2 − n2 ) n i +e3 −e3 h n−4 β(e) = e µ (−2) µ Γ(3 − ) → . = n 2− 2 2 2 ∂µ 24π [µ ] 2 24π 2 12π 2 δ3 (µ) =
Similarly, we obtain γj via (49):
∂ −e2 Γ(2 − n2 ) e2 γ2 (e) = µ = ∂µ 16π 2 (µ)4−n 8π 2
, γ3 (e) =
e2 . 6π 2
(15.54)
As expected, the positive sign of β(e) results in an increase of the running charge with energy.
15.2.3 Running Coupling from the Renormalization Group Now that we know how to compute the β- and γ-functions, we go further by looking for solutions of (42). In fact, we obtain a relation connecting the Green’s function evaluated at two different scales µ1 and µ2 , valid for all perturbative orders (but limited only to the leading logarithmic terms). The CS equation is an exact consequence of the renormalizability of the theory. An amplitude computed perturbatively to a given order does not in general satisfy (42), therefore the renormalization group equation may complement perturbative calculations, and can be used to ‘improve’ them. We first give the solution of (42), the demonstration will follow in the next subsection. We start with an amplitude A(pi , g) evaluated at the momenta pi . Then at a different scale η pi of the momenta (η is an arbitrary number), the amplitude A(η pi , g) is related to the initial A(pi , g) by replacing the constant g with the running g(t): A(η pi , g) = A(pi , g(t)) exp
Z
0
t 0
γ [g(t )] dt e
0
, where t = log η , (15.55)
where the running g(t) is a solution to the equation d g(t) = β (g) , with the initial condition g(0) = g . dt
(15.56)
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15 Asymptotic Freedom in QCD
In this form, (55) is easy to interpret. The effect of rescaling the momenta pi in the amplitude is equivalent to replacing the coupling g by the running coupling g, apart from an overall multiplicative factor represented by the exponential of e γ (g). The latter will be defined later. So the most important quantity is the running coupling g(t) which can be computed from (56), once the β-function is known. We now discuss the physical implication of the solution (55). If η ∼ O(1), t ∼ 0, (55) would essentially be the ordinary perturbative evaluation of the amplitude. On the other hand, if η 1, the replacement of g by g(t) in (55) is a powerful method which tells us how to sum large logarithm terms of the amplitude. Indeed, in many circumstances, the Feynman perturbation series depend not only on g but also on the product gn [log (p2 /µ2 )]l . Even when g is small, but log (p2 /µ2 ) large, the perturbative result is not reliable. The result (55) of the renormalization group equation solves this problem by reorganizing the dependence of the amplitude into a function of the running g and an exponential scale factor. The separate criteria g2 1 and g2 log (p2 /µ2 ) 1 – which could fail in conventional perturbation theory when large logarithm factors appear at each order – are saved if g2 (p2 ) 1. This running coupling g(p2 ) enables us to sum the leading logarithm terms of Green’s functions to all perturbative orders, as illustrated in Fig. 15.10. ( 2 2 2 ) p p g 3 5 g + O g log + O g log + ·· = = g. 2 2 2 2 µ µ 1 + b g log ( µp2 ) ... .. ...... ... ..... .. ... . ... .. . ... ... ... ... . ... .. ... . ... .. . ... . . . . .
... .. ... ..
... .. ... ..
... .. ... ..
+
.. ... ... ... ... .... .. ... . . ... ... ... ... . ......... ....... .......... . ... .. . ... .. . ... . . ... . .
Fig. 15.10. The meaning of g as
+· · · P
n
.. ... ... ... ... ..... .. ... . . ... ... ... ... ......... ....... ........... . .. .. . ......... ....... ....... ............ . . ... ... .
+· · ·
g2n+1 [log( µp )]n through the β-function
The fact that the above equation is a geometric series in g2 log(p2 /µ2 ), reminiscent of (38), would be impossible to verify by a direct calculation of Feynman diagrams, while it is an elementary consequence of the renormalization group equation, as we will see later. The exponential factor of e γ (g) in (55) also has a simple interpretation. It represents the accumulation of the field-strength rescaling from µ to the actual momentum p at which the amplitude is evaluated.
15.2.4 Solution of the Renormalization Group Equation We first remark that the Green’s function, which is built up from coupling d constants, masses, external fields, etc., has the dimension (mass) , where d is the canonical or ‘naive’ dimension. For instance the naive dimension of the strong coupling gs in QCD is d = 0, a fermion field has d = 23 , a boson
15.2 The Renormalization Group
527
field has d = 1. A two-point Green’s function of the photon field Πµν (q) in (2) has d = 2. Its amputated two-point Green’s function, or the photon propagator, has d = −2. A QED three-point function has d = −4; when the three external propagators are removed, it has d = 0. Next, to cast (42) in its most useful form, instead of varying µ and fixing pi , we change pi into ηpi (where η is an arbitrary large number) and keep µ fixed in the Green’s function G(pi , µ, g). One may always write pi d G(pi , µ, g) = µ F ,g , µ where F (pi /µ, g) is an arbitrary dimensionless function, and the naive dimension µd of G(pi , µ, g) is explicit. For instance, the d = −2 propagator i/p2 can be written as i i = µ−2 , p2 (p/µ)2 and so on for other Green’s functions. Next, we have pi d G(η pi , µ, g) = µ F η , g . µ
(15.57)
Since the dimensionless F depends on η and µ only through the ratio η/µ, we can check from dimensional analysis that the following equation holds: ∂ ∂ pi +η µ F η ,g = 0 . (15.58) ∂µ ∂η µ This familiar equation is already seen in (46). Combining (57) with (58) and putting t = log η, one finds ∂ ∂ + − d G(η pi , µ, g) = 0 , µ ∂µ ∂t the substitution of the above equation into (42) gives ∂ ∂ − β(g) − γe(g) G(η pi , µ, g) = 0 , ∂t ∂g
(15.59)
where γe(g) = d + k2 γ(g), the e γ (g) is called the anomalous dimension since the amplitude has no longer the canonical naive dimension d. To find the solution to (59) with two independent variables t = log η and g , let us consider a similar differential equation with two variables t and x of a function D(t, x), where the initial condition D(0, x) is given. Thus ∂ ∂ − β(x) D(t, x) = R(x)D(t, x) . (15.60) ∂t ∂x
528
15 Asymptotic Freedom in QCD
The solution to the inhomogeneous differential equation (60) will immediately follow from the solution to the corresponding homogeneous equation. To find it, we first introduce a new variable g(t, x) obeying d g(t, x) = β(g) , with the initial condition g(0, x) = x . dt
(15.61)
R g(t,x) d y/β(y) . Next, Then we integrate (61) with respect to t and get t = x we differentiate both sides of the above equation with respect to x, dt dg 1 dg 1 dg =0= − =⇒ β(x) = β(g) = , dx β(g) dx β(x) dx dt which shows that, in addition to its own equation (61), g(t, x) equally obeys the homogeneous equation ∂ ∂ − β(x) g(t, x) = 0 . (15.62) ∂t ∂x Once g(t, x) – which is the solution to the homogeneous equation (62) – is known, the solution to the inhomogeneous equation (60) can be derived by replacing in the initial function D(0, x) the argument x by g. After this replacement, the result is multiplied by an exponential of R[g(t0 , x)] integrated from 0 to t. We have Z t D(t, x) = D[0, g(t, x)] × exp R[g(t0 , x)]dt0 . (15.63) 0
By the substitution g ↔ x, e γ (g) ↔ R(x) in (63), the solution (55) of the renormalization group equation (42) is then obtained. The QED Running Charge e(t). Once β(e) is known, we determine the running coupling e(t) of QED by solving (56) with (52): e3 de(t) = β (e) , where β(e) = . dt 12π 2 The solution is de(t) dt 1 1 −t e2 (0) 2 = −→ − = −→ e (t) = , 6π 2 e3 (t) 12π 2 e2 (t) e2 (0) 1 − e2 (0) t/6π 2 αem q αem(q 2 ) = for q 2 µ2 , t = log η = log , (15.64) q2 αem µ 1 − 3π log( µ2 ) and we recover our previous result (38) if we identify µ2 with Cm2 .
15.3 One-Loop Computation of the QCD β -Function
529
15.3 One-Loop Computation of the QCD β -Function We have seen in (50) that the β-function of QED is derived from the divergent counterterms Z1 (the electron–photon vertex), Z2 (the electron self-energy), and Z3 (the photon self-energy). Since the first two terms cancel each other by the Ward identity, only Z3 contributes to the QED β-function. In the non-Abelian QCD case, with quark and gluon replacing electron and photon respectively, all of these three terms contribute. Thus from (48) ∂ 1 βQCD (gs ) = gs µ −δ1 (µ) + δq (µ) + δglu (µ) . (15.65) ∂µ 2 When we compute these δ(µ) functions separately, they depend on the gauge parameter ξ of the gluon propagator, however their ξ dependences cancel each other in their combination (65), so that β(gs ) is gauge-independent as it should be. As in Chap. 14, we first use the ξ = 1 Feynman–’t Hooft gauge, then step by step show that the final result for βQCD (gs ) is independent of ξ.
15.3.1 Quark Self-Energy Counterterm Zq The quark self-energy Σ(p) is given in (14.23) and its derivative δq = Zq − 1 at 6 p = m in (14.41). We now rewrite δq as Γ(2 − n2 ) −gs2 −gs2 Nc2 − 1 Γ(2 − n2 ) δq (µ) = C = . (15.66) (N ) 2 c 16π 2 µ4−n 16π 2 2 Nc µ4−n For a general ξ gauge, in fact, we get Γ(2 − n2 ) −gs2 Nc2 − 1 δq (µ) = ξ , 16π 2 2Nc µ4−n
(15.67)
which tells us that in the Landau gauge (ξ = 0), δq vanishes at one-loop level.
15.3.2 Quark–Gluon Vertex Counterterm Z1 For the quark–gluon vertex Z1 , there are two diagrams, and not only one as in the Abelian QED case. In addition to Fig. 15.11a – for which the external as well as the internal photons of Fig. 15.1a are replaced by gluons – there is one more graph shown in Fig. 15.11b. ............ ........... ...................... ...... ..................... . .......... 1 .. ................... i . 2 .......... .. .... . . . .... ... ... ... .... .. .... 1 .......... .......... .......... .......... ................ 1 . . . j 2 ...j........... ................ ............... ................ ................ .......................2 ..... . . . . ......... 0 .. ... .... . . . . .... ... ... ...
λ
λ
λ p
p
(a)
µ
.......... ........... ........................ . ......... ............................... 1 ...... ...................... i ............. 2 ............................................. .. ..... . .......................... ..... ................... 0 .... ......................... ............ ...................... . . . . . . . . . ......... ..... .. ........................... ....... ......................... ......................... . 1 ......... 1 ... .. ... ......................................................................................... ..2 k 2 ..j..... ρ ν ........ . . . . . .. . . ........... . . . . . .... .... ..... .... .... ... ...
q
λ
p−P λ
γ
P −p
P
γ
p
λ
p0
(b)
Fig. 15.11a, b. δ1 from quark–gluon vertex function
530
15 Asymptotic Freedom in QCD
The expression of the vertex function (Fig. 15.11a) can be directly obtained from (14.2), with a single modification: the γ µ (1 − γ5 ) in (14.2) is now replaced by −i gs γ µ Ti , where Ti ≡ 21 λi . Because of this additional color matrix Ti , the Casimir operator Tj Tj in (14.3) is now replaced by Tj Tj −→ Tj Ti Tj = Tj Tj Ti + Tj [Ti , Tj ] = C2 (Nc ) Ti + i Tj fijk Tk 2 1 Nc − 1 1 − 2 Nc Ti , = C2 (Nc )Ti + i fijk 2 ifjkl Tl = 2Nc
such that, after including the factor −i gs , the overall substitution is 2 Nc2 − 1 Nc − 1 1 4 = −→ −i gs − 2 Nc Ti . (15.68) 3 2Nc 2Nc This substitution tells us that the Γµ (p2 , p3 ) obtained previously in (14.11) and (14.12), in particular its divergent part associated with Γ(2− n2 ) of F1 (q 2 ), can be immediately adapted to ΓµA (p0 , p) of Fig.15.11a by the substitution (68). We denote it by ΓµA (p0 , p)|div , where ( ) 2 2 2 Γ(2 − n ) g N − 1 (n − 2) µ 0 s c µ 1 2 ΓA (p , p) div = −igs γ Ti − 2 Nc n 2Nc 4 µ4−n (4π) 2 2 2 Γ(2 − n2 ) gs Nc − 1 1 − = −igs γ µ Ti N . (15.69) 2 c 16π 2 2Nc µ4−n n
Like (53), the denominator [−m2 (x + y)2 + q 2 xy]2− 2 in (14.12) is simply n taken as (µ2 )2− 2 = µ4−n , since only the divergent part is needed. For later use, we keep n 6= 4 in the factor Γ(2 − n2 )/µ4−n , because we will take its derivative with respect to µ to get the β-function. The diagram of Fig. 15.11b is given, always in the Feynman–’t Hooft gauge ξ = 1, by Z i(6P + m) −i −i d4 P µ 0 (−i gs γν Tk ) 2 (−i gs γρ Tj ) ΓB (p , p) = (2π)4 P − m2 (p − P )2 (P − p0 )2 (−gs fijk )[gµρ (q − p + P )ν + gρν (p + p0 − 2P )µ + gνµ (P − p0 − q)ρ ] . With fijk Tk Tj = −i 12 Nc Ti , we have Z −gs3 dn P γν (6P + m)γρ T µνρ (p0 , p, P ) ΓµB (p0 , p) = Nc Ti , (15.70) 2 (2π)n (P 2 − m2 )(p − P )2 (P − p0 )2 T µνρ (p0 , p, P ) ≡ [gµρ (q − p + P )ν + gρν (p + p0 − 2P )µ + gνµ (P − p0 − q)ρ ] . To compute the divergent part of ΓµB (p0 , p), i.e. the coefficient of Γ(2 − n2 ), we only need the quadratic power of the integration variable P in the numerator of (70). This is found to be (γν 6P γρ ) [gµρ P ν − 2gρν P µ + gνµ P ρ ] = 2P 2 γ µ + 2(n − 2) 6P P µ .(15.71)
15.3 One-Loop Computation of the QCD β -Function
531
After writing the product of the three terms in the denominator of (70) as an integration over the auxiliary Feynman variables x and R y and using (71) in the numerator, we finally get for the divergent part of dn P , Z 1 Z 1−x 2(n − 1)Γ(2 − n2 ) γ µ i 3i Γ(2 − n2 ) µ dx dy −→ γ . n n 16π 2 µ4−n 0 (4π) 2 0 [m2 (1 − x − y)2 − q 2 xy]2− 2 n
Like (53) or (69), the denominator [m2 (1 − x − y)2 − q 2 xy]2− 2 is taken as µ4−n for the divergent part. With (70), we obtain 2 gs 3Nc Γ(2 − n2 ) . (15.72) ΓµB (p0 , p) div = −igs γ µ Ti 16π 2 2 µ4−n
The QCD vertex function ΓµQCD (p0 , p) is the sum ΓµA (p0 , p) + ΓµB (p0 , p). By definition, the QCD counterterm (−igs γ µ Ti ) δ1 is used to cancel the divergence of the vertex function ΓµQCD (p0 , p), just as the QED counterterm ieγ µ δ1 of (22) was used to cancel the divergence of ieΓµ (p0 , p) in (23). Note that the factor (−igs γ µ Ti ) already appears in (69) and (72). Thus −igs γ µ Ti δ1 + ΓµA (p0 , p) div + ΓµB (p0 , p) div = 0 , from which we get Γ(2 − n2 ) −gs2 Nc2 − 1 δ1 (µ) = + N . c 16π 2 2Nc µ4−n
In the most general gauge parameter ξ, one finds Γ(2 − n2 ) 1−ξ −gs2 Nc2 − 1 ξ + 1 − N . δ1 (µ) = c 16π 2 2Nc 4 µ4−n
(15.73)
(15.74)
From (67) and (74), we notice that δ1 is not equal to δq as it is in QED.
15.3.3 Gluon Self-Energy Counterterm Zglu The self-energy of the gluon field, δglu , remains to be evaluated. Its calculation is more involved and the corresponding diagrams are shown in Fig. 15.12. The three diagrams Fig. 15.12a–c are evaluated from the Feynman rules given in Chap. 8 for QCD. These rules which only involve the ‘tree’ diagrams do not apply to Fig. 15.12d. We are already familiar with Fig. 15.12a. Except for the additional SU(3) color matrices Ti and Tj at each point where a gluon is attached to the fermionic loop, this diagram is identical to the photon self-energy computed in (2). We have Z 4 d P i i def µ ν i Πµν (q) = (−1) Tr (−ig γ T ) (−ig γ T ) . s i s j A (2π)4 6P − m 6P − 6 q − m (15.75)
532
15 Asymptotic Freedom in QCD
So Πµν A (q) and the corresponding QCD counterterm of Fig. 15.12a, called A δglu (µ), can be directly taken from (6) and (14) by including the trace factor Tr[Ti Tj ] = 12 δij . Since there are Nf flavors of quarks in the loop integral – all in the fundamental SUc (3) representation – their contributions must be summed up and it results in a multiplicative factor Nf . Below the threshold of heavy quarks, Nf = 3, if we only count the u, d, s light quarks. For higher energy, Nf = 6, which includes c, b, t quarks. From (6), the divergent part of Πµν A (q) is n 2 = Nf δij −gs Γ(2 − 2 ) (q 2 gµν − q µ q ν ) . Πµν (q) (15.76) A div 2 12π 2 (µ)4−n
From (14), we deduce Nf −gs2 Γ(2 − n2 ) A . δglu (µ) = 2 12π 2 (µ)4−n
............ ... . ................. ................... . .. .................. .................. . . . . . . . . . ................... .. ............. ...................... .... ....................... ...... ................ .......... . ...... ..... ........ .............. .... .. . .. ..... .......... .... .. ........... ........ . . . ....... ..... ... .. ............... ...................... ............ .... .... .... ................... ................. .........
................................. ........ .... j .......... ......... ......... ........ .... .... .. ..i ... ................................................................. .. .. . . .. .. .. . . . . . . .. ......... .......... ......... .......... . .. . . .... .... ........ . . ...............................
T
T
k
k
(b)
(a) `, β
... ............ ..... ................. ........ .... .... ......................... .............. ............. . .. ...... ............ .. .. .. .. .. ............. ......... ......... ........ ...................................................................... .. .. .. .. .. .......... .......... ...................... ......... ....... ... ... ... ....... . . . ................. .................. .... ....... . ............. .. ... .... .............. .... ................. .... .................. ................ ..........
i, µ q
(c)
q+k k
m, α
j, ν q
`
...... . . ................. . . .. ..... .... .. ... . . . .... .... ... ... . .......... ......... ......... .......... ........................................................... .. . .. .. .. . .. .. . .......... ......... ............ ......... .... .... .... .... . ..................... .................. . . . .. ... .... ..... .. . . ............... . . .. . . ..
i, µ q
(d)
j, ν
q+k k
q
m
Fig. 15.12a–d. δglu from gluon self-energy
The next three diagrams Fig. 15.12b, c, d are typically non-Abelian and absent in QED. The diagram of Fig. 15.12b does not contribute to δglu ; this can be seen by considering the degree of divergence of the corresponding integral which is quadratic and not logarithmic. Indeed, besides the fourgauge-boson coupling constant, the relevant loop integral of Fig. 15.12b is Z n Z n d k dn k n−2 Γ(2 − 2 ) = lim ∼ lim v −→ 0 for n > 2 . (15.77) v→0 v→0 k2 k 2 + v2 1 − n2 B To compute δglu (µ), we need the coefficient of Γ(2 − B vanishes as n → 4, then δglu (µ) = 0.
n ). 2
It is vn−2 which
15.3 One-Loop Computation of the QCD β -Function
533
With the same convention as in (75) for +i Πµν A (q), the gluon self-energy +iΠµν (q) of Fig. 15.12c is obtained using Feynman rules: C def
+iΠµν C (q) =
1 2
Z
dn k −i −i (−gs fiml ) (−gs fjml ) N µν (k, q) , (2π)n k 2 (q + k)2
where N µν (k, q) = gµα (q − k)β + gαβ (2k + q)µ + gβµ (−k − 2q)α × δαν (k − q)β + gαβ (−2k − q)ν + δβν (k + 2q)α , (15.78)
the factor 12 in front of the right-hand side of the above equation is due to two identical internal bosons in the loop. We find N µν (k, q) = − gµν (2k 2 + 2k · q + 5q 2 ) + (6 − 4n)k µ k ν + (3 − 2n)(k µ q ν + k ν q µ ) + (6 − n)q µ q ν .
−2 −2 After introducing the auxiliary variable x for the R nproduct k (q +k) in the denominator of (78) following Feynman, the d k integration can be done, using formulas in the Appendix. Together with fiml fjml = Nc δij , we get
Πµν C (q)
=
Z
n 2(4π) 2
1
dx
Γ(2 − n2 )
(
Γ(1 − n2 ) Γ(2 − n2 ) 0 [q 2 x(1 ) 2 µν µ ν 2 2 +q g [5 − 2x(1 − x)] − q q 6(1 − x + x ) − n(1 − 2x) . gs2 Nc δij
n − x)]2− 2
3 q 2 gµν (n − 1)x(1 − x)
Using (1 − n2 )Γ(1 − n2 ) = Γ(2 − n2 ), we finally obtain Πµν C (q)
=
gs2 Nc δij n
Z
1
dx
Γ(2 − n2 )
n
(
q 2 gµν
5 (4n − 5) x(1 − x) − 2 n−2
0 (4π) 2 [q 2 x(1 − x)]2− 2 ) h i 2 2 µ ν n (1 − 2x) − 3(1 − x + x ) , +q q 2 gs2 Nc δij 19 2 µν 11 µ ν Γ(2 − n2 ) µν ΠC (q) div = q g − q q . 16π 2 12 6 µ4−n
(15.79)
For the most general ξ gauge, we have for Πµν C (q)|div , gs2 Nc δij 16π 2
Γ(2 − n2 ) 19 2 µν 11 µ ν 1 − ξ 2 µν q g − q q + (q g − q µ q ν ) . (15.80) 12 6 2 µ4−n
Gauge invariance of QCD implies that the gluon self-energy, like the photon self-energy, must satisfy qµ Πµν (q) = qν Πµν (q) = 0, i.e. it must have the structure Πµν (q) = (q 2 gµν − q µ q ν )Π(q 2 ). However, in contrast to (76), this structure is not found in (80) for Πµν C (q). There must exist something new
534
15 Asymptotic Freedom in QCD
in the Feynman rules for non-Abelian theories when pure gluons in loops are connected to external gluons, as in Fig. 15.12c. A systematic solution of this problem is referred to as the Faddeev– Popov (FP) prescription, according to which contributions of fictitious scalar particles obeying Fermi statistics, called FP ‘ghosts’, must be included. They only appear in closed loops with a minus sign. These color-octet ghosts couple only to gauge bosons and not to fermions. Their Feynman rules (propagator and coupling) are indicated in Fig. 15.13. The two-ghost–two-gluon fourpoint vertex does not exist. We will come back to ghosts later. i
k
......... .. . . . . . . . . . .
. . ......... . . . . . . µ ... ... ... ... ... . . ............................................................................. ... .... .... ... ... . . . . . ........ . . . . .
j
µ
P
i
k
i δik /P 2
gs fijk Pµ
Fig. 15.13. Feynman rules for FP ghosts (dotted line) coupled to gauge boson
According to this rule, the FP ghost contribution (Fig. 15.12d) to the gluon self-energy is Z dn k i i def µν +iΠD (q) = (−1) (gs flim ) (gs fmjl ) (q + k)µ k ν . n 2 (2π) k (k + q)2
With flim fmjl = −fiml fjml = −Nc δij , and using formulas from the Appendix, Z −gs2 Nc δij 1 x(1 − x) Γ(2 − n2 ) 1 2 µν Γ(1 − n2 ) µν µ ν ΠD (q) = dx −q q , n n 2q g Γ(2 − n2 ) 0 (4π) 2 [q 2 x(1 − x)]2− 2 2 µν n µ ν 2 = gs Nc δij q g + q q Γ(2 − 2 ) . (15.81) Πµν (q) D div 2 4−n 16π 12 6 µ
µν Adding (80) to (81), gauge invariance is restored in the sum Πµν C (q)+ΠD (q): gs2 Nc δij 5 1 − ξ Γ(2 − n2 ) 2 µν µν µν ΠC (q) div + ΠD (q) div = + (q g − q µ q ν ) . 16π 2 3 2 µ4−n
When the above equation is added to Πµν A (q) in (76), we obtain the divergent part of the gluon self-energy described by the four diagrams of Fig. 15.12 in its totality. Thus, 2 Γ(2 − n2 ) 2 µν 5 1−ξ 2 = gs δij Πµν (q) + N − N (q g − q µ q ν ) , c f glu div 16π 2 3 2 3 µ4−n from which we get
g2 Γ(2 − n2 ) δglu (µ) = s 2 16π µ4−n
5 1−ξ + 3 2
2 Nc − Nf 3
.
(15.82)
15.3 One-Loop Computation of the QCD β -Function
535
15.3.4 The Running QCD Coupling Plugging (67), (74), and (82) into (65), we obtain β(gs ) =
−gs3 16π 2
11 2 Nf Nc − 3 3
.
(15.83)
As announced, the gauge-dependence ξ disappears in β(gs ) when the various contributions of (67), (74) and (82) are added. Furthermore, for Nf < ( 11/2) Nc , i.e. for the number of quark species less than 16, the Callan– Symanzik β-function is negative, then according to (43) the running coupling decreases for increasing energy, and so QCD is asymptotically free. The counterterms Zj that we have computed are summarized in Table 15.1. Table 15.1. Coefficients of the counterterms at the one-loop level QCD :
Coefficients δ1 (vertex)
−
Nc −1 ξ 2Nc
δ2 (fermion)
−
Nc −1 ξ 2Nc
δ3 (gauge boson) †
1 from e− ,
5 9
5 3
+
gs 16π
1−ξ 2
log
− 1−
Λ µ 1−ξ 4
QED :
Nc
Nc − 23 Nf
e 16π
log
Λ µ
−ξ −ξ − 23 Nf (1 + 59 Nc )†
= ( 32 )2 + ( −1 )2 from u, d quarks 3
From (48) or (65), we infer a general rule: a negative δ1 just as a positive δext (δ2 and δ3 ) would render the theory asymptotically free. Their roles can be seen in Table 15.1. A glance at the various coefficients δj reveals that the negative sign (−1)Nc in δ1 and the positive sign (+ 35 )Nc in δ3 are mainly responsible of the asymptotic freedom. This is due to the gluonic non-Abelian character of Fig. 15.11b and Fig. 15.12c. On the other hand, due to the fermionic loop in Fig. 15.12a, the minus sign in −( 23 )Nf of δ3 is the main obstacle for QCD or the Abelian QED to be asymptotically free. Finally, we remark that for the pure Yang-Mills gauge fields, e.g. QCD without quarks, the corresponding β(gs ) can be obtained from Fig. 15.8c. This has no equivalent in QED, and is computed according to β(gs ) = gs µ
∂ 3 ∗ −∆glu (µ) + δ (µ) , 1 ∂µ 2 glu
(15.84)
where Z1glu = 1+∆glu 1 is the three-gluon loop vertex counterterm derived from the diagrams of Fig. 15.14. Only pure gluon fields, and not quarks, come in ∗ Z1glu . The δglu (µ) is taken from (82) but without the Nf term, i.e. without Fig. 15.12a .
536
15 Asymptotic Freedom in QCD
...... .... ................. .. ...... ................. . ....... ................. .. ....... ................. . .... ... .... .... ....... ............... ...... ...... ............ .... ............. .... ...... ......... ..... .... ............ .. ........ ........ . ..... .. .......... ......... . . . . . . . . . ......... . ... ...... .... ...... ....................................................... ............. ... .. ... ............. ................... .................. .................. .. ............. .................. .
...... .... ............... ...... . . .............. ....... . . . . . . . ......... ....... ............... . . . . .. . . ..
.. . .. .. . . . . . . . . . . . . .......... .. . . ............. ... ................. . . . . . . . ..... ........ .. ............. .................... ................. . . .................. .. ........... ................... .
(a)
...... .... ................ ...... . .................. ...... .................. ....... .................. ............. .............. .................. ............. ...... . . . . ........... .... ...... .......... ...... .... ............ .. ........ ....... . ..... .. . ........ . . . . . . . . . . . ........... .. ...... .............................................. ........................... .................... ................. .. ............... ..................... .. ................. . ...................... .. ..........
(b)
(c)
Fig. 15.14a–c. Loop diagrams of the three-gluon vertex ∆glu 1
The easiest way to understand asymptotic freedom is by considering the magnetic screening properties of the vacuum.2 In a relativistic theory (with natural units c = 1), the dielectric constant ε is equal to 1/µ where µ is the magnetic permeability. In QED, all media are diamagnetic, i.e. µ < 1. The origin of the diamagnetism may be understood by the Lenz law: The electrically charged particles in response to an external magnetic field will set up a current which itself induces a magnetic field that acts to decrease the external magnetic field. Then ε > 1 which implies the screening effect. However in non-Abelian QCD, paramagnetism is possible. Gluons are color charged particles of spin one, they behave as color magnetic dipoles that align themselves parallel to the external chromomagnetic field, thus increasing its magnitude and producing µcolor > 1, hence the color charge is antiscreened with εcolor < 1. We can therefore regard the QCD vacuum as a color paramagnetic medium, and asymptotic freedom arises if the antiscreening due to gluons overcomes the screening due to quarks. On the other hand, as one goes toward the infrared low energy regime, εcolor → 0, the effective coupling strength would diverge. This would cause an effect similar to the QED Meissner phenomenon in which the magnetic field is repelled by the superconducting material. In QCD, it would be the chromoelectric field that exhibits the Meissner effect. This in turn would squeeze the chromoelectric lines of force between a quark and an antiquark into a narrow tube, and furthermore, the energy of the tube, i.e. the quark potential V (r) would increase proportionally to the distance r separating q and q. This suggests confinement of quarks and gluons. Now the QCD running coupling can be obtained by putting (83) into (56). Exactly as in (64), the solution for gs (q 2 ) and αs (q 2 ) = gs2 /4π is αs (q 2 ) = 2
αs(µ2 ) 1 + b0
αs (µ2 ) 4π
2
log µq 2
, with b0 =
2 11 Nc − Nf . 3 3
(15.85)
Nielsen, N. K., Am. J. Phys. 49 (1981) 1171; Hughes, R. J., Nucl. Phys. B186 (1981) 376.
15.3 One-Loop Computation of the QCD β -Function
537
Since b0 is positive with Nc = 3 and Nf = 6, the strong coupling αs (q 2 ) decreases with growing q 2 , in opposite direction to the electromagnetic coupling αem (q 2 ) as given in (64). Also in contrast to QED where αem(0) can be directly measured, for QCD the corresponding Thomson limit is not accessible: at q 2 = 0, the strong coupling gs is very strong. The solution (85) only gives us the evolution of the running coupling as a function of q 2 , but the absolute value of αs at a certain fixed value of q 2 , i.e. the constant of integration of the differential equation (56), must be determined by experiment. A conventional choice is to introduce a mass parameter ΛMS which provides a parameterization of αs (q 2 ). By defining ΛMS as follows, one can rearrange (85) in a simple form: 1 = gs2
b0 µ 2π log =⇒ αs (q) = for q ΛMS . 2 8π ΛMS b0 log(q/ΛMS )
(15.86)
Experiments (Fig. 15.15) yield a value of ΛMS ranging from 150 to 250 MeV, and perturbative QCD theory is valid when q 2 is much larger than Λ2MS . Depending on q 2 , the number of the quark flavor Nf in b0 is taken between 3 and 6. For instance, at the τ lepton mass (Nf = 3), then αs (Mτ ) ≈ 0.37, and at the Z 0 weak boson mass (Nf = 5), αs (MZ 0 ) ≈ 0.12. On the other hand, the strong interaction becomes strong at distance larger than 1/ΛMS which is the size of the light hadrons.
Fig. 15.15. Various measurements of αs (Q) from Yamauchi M., Proc. 17th Int. Symp. on Lepton–Photon Interactions, 1995 Beijing (ed. Zheng Zhi-Peng and Chen He-Sheng). World Scientific, Singapore 1996 (Courtesy Masanori Yamauchi)
538
15 Asymptotic Freedom in QCD
As an immediate application of (86), let us illustrate the implication of the CS equation for the QCD corrected decay rate Γ(τ − → ντ + hadrons). In (14.84), we have computed this quantity which depends on αs , the latter must be defined at some scale µ. This is in contrast to αem naturally defined by on-mass-shell electron and photon, which is impossible for quarks and gluons. CS equation allows us to avoid this problem. Since Γ(Mτ , µ, αs) ≡ Γ(τ − → ντ + hadrons) is a physical measurable quantity, it is independent of any scale µ and therefore obeys ∂ ∂ µ + β(gs ) Γ(Mτ , µ, αs) = 0 . (15.87) ∂µ ∂gs Such physical quantities change only because of their dependence on the coupling constant and the renormalization point, and so satisfy (87). The corresponding CS functions γ are not present for physical quantities whose normalizations Z follow from their definitions. On dimensional grounds, we can write Γ(Mτ , µ, αs) as Γ0 F ( Mµτ , αs). The above equation tells us that the dimensionless F depends on its arguments only through the running coupling constant αs(µ) evaluated at µ = Mτ . This allows us to replace in (14.84)the fixed αs with the running coupling constant αs (Mτ ): Γ0 Nc 1 + π1 αs (Mτ ) .
15.4 Ghosts
The appearance of ghosts in Fig. 15.12d and the corresponding Feynman rules in Fig. 15.13 at first sight seem to be entirely ad hoc. It is not so. At the origin, ghosts represent a prescription proposed by Faddeev and Popov to cure some difficulties encountered in the quantization of massless gauge fields (see Chaps. 2 and 8), whether the gauge symmetry is spontaneously broken (electroweak) or not (QCD). The problem has to do with the redundant degrees of freedom of the gauge field Aµ , its polarization vector εµ has only two transverse physical components (see Sect. 2.5), whereas all of its four components are treated covariantly in the quantization. This redundancy is common to both massless photon and gluon fields; it must be removed.
15.4.1 The Faddeev–Popov Gauge-Fixing Method To quantize these fields and to single out their essential difference, one could adopt the functional path integral formulation which is not used in this book, but some rudimentary notions would be useful. In this formalism, the key ingredients for the quantization are the functional integrals Z Z DAµ exp (i S[Aµ ]) for QED , DAjµ exp i S[Ajµ ] for QCD , (15.88) where S[Aµ ] and (S[Ajµ ]) are the actions of the photon and gluon fields Z Z j 2 S[Aµ ] = d4 x − 41 (Fµν )2 , S[Ajµ ] = d4 x − 41 (Fµν ) .
15.4 Ghosts
539
Ajµ , j = 1 · · · 8 are the gluon fields in the adjoint representation of the SUc (3) color group. As explained in Chap. 8, the transformations, parameterized by an arbitrary function ω(x) [ωi (x)] applied to the field Aµ (x) [Aiµ (x)]: Aµ (x) −→ Aµ (x) + ∂µ ω(x) ≡ A[ω](x) , or Aiµ (x) −→ Aiµ (x) + ∂µ ωi (x) − gs f ijk Ajµ (x)ωk (x) ≡ Ai [ω](x) ,
(15.89)
i leave Fµν (Fµν ) and the actions unchanged. We can use the freedom of adding an arbitrary ω(x) [ωi (x)] to Aµ [Aiµ ] to make the latter satisfy some relations. Choosing Aµ (Aiµ ) to satisfy some conditions is called fixing the gauge. In the functional integrals (88), we redundantly integrate over an infinite number of physically equivalent gauge field configurations Aµ (Aiµ ). Without fixing the gauge, we overquantize unphysical variables. We must remove the redundancy by some gauge-fixing conditions in the course of quantization. To count each configuration only once, Faddeev and Popov use a constraint G(A[ω]) = 0 as a gauge-fixing condition. For instance G(A[ω]) = ∂ µ Aµ − h(x), where h(x) is an arbitrary function and h(x) = 0 is the familiar covariant Lorentz gauge condition [see (2.140) for QED]. A similar condition applies to the nonAbelian Ai [ω] field, cf (8.61), (8.118). We could force the integral to cover only the configurations satisfying G(A[ω]) = 0 by inserting a delta functional δ{G(A[ω])} in (88). This functional change implies that a Jacobian factor (which is the determinant det ∂G(A[ω]) ) enters into the integrand. We omit ∂ω the Lorentz index µ and the color index i to simplify the notations. Since Z ∂G(A[ω]) 1 = Dω(x) δ{G(A[ω])} det , (15.90) ∂ω
equation (88) can be rewritten as Z Z ∂G(A[ω]) Dω DA exp(i S[A]) δ{G(A[ω])} det . ∂ω
(15.91)
Now in the above functional integral, the only difference between QED and QCD is that in the former, the ∂G(A[ω])/∂ω is independent of the field A, whereas the corresponding ∂G(Ai [ω])/∂ω of the non-Abelian case depends on the field Ai because of the third term f ijk Aj ωk in (89). In the Abelian case, we R can factor the determinant of ∂G(A[ω])/∂ω out of the functional integral DA and use a Gaussian function (weighted by an arbitrary finite number ξ) to integrate exp(i S[A]) δ{G(A[ω])}, the result of the integration yields a term (1/2ξ)[∂ µAµ ]2 to be added to the original Lagrangian − 41 (Fµν )2 . This (1/2ξ)[∂ µ Aµ ]2 term coming from δ{G(A[ω])} gives rise to the propagator of massless gauge fields obtained in Chap. 8. This propagator, which depends on an arbitrary gauge parameter ξ, is common to the photon and gluon: −i (1 − ξ) q µ q ν µν g − . q 2 + i q2
540
15 Asymptotic Freedom in QCD
Now comes the crucial difference between the Abelian and the non-Abelian i cases. For the R latter, because the determinant depends on Aµ , it cannot be put outside DA, therefore, after the integration has been done, we obtain a new term in addition to (1/2ξ) [∂ µAiµ ]2 . This new term is si (−∂ 2 δ ik + gs f ijk δ µ Ajµ )sk ,
(15.92)
where si is a scalar field obeying the anticommutation rule, like a fermionic field, hence the name of ghost. This object is a representation of det
∂ G(Ai [ω]) = det (∂ µ Dµ ) where Dµ ωi = ∂µ ωi − gs f ijk Ajµ ωk ∂ω Z Z = Ds D s exp i d4 x s (−∂ µ Dµ )s ,
using the general Gaussian integral formula involving a Hermitian matrix X with eigenvalues xj : ! Y YZ det X = xi = dθi∗ dθi exp [−Σi θi∗ xi θi ] i
i
=
YZ i
dθi∗ dθi
!
exp [−θi∗ Xij θj ] .
The functional integration variable θ must be a Grassmann (anticommuting) function; if we use an ordinary commuting function, we would get the inverse of the determinant, (det X)−1 , and not det X. This may be seen by pobserving R the generalization of the one variable integral dx exp[−Ax2 ] = π/A: Z (det X)−N = C DΦDΦ∗ exp[−Φ∗ XΦ]
where Φ is a complex scalar field with N components. Feynman rules are derived for closed loops of Φ fields, each loop goes with a factor N . Since we want N to be −1, our closed loops will go with a factor −1, just like the rules for fermions. The kinetic term ∂ 2 δ ik si sk in (92) gives rise to the ghost propagator, whereas the second term gs f ijk si δ µ Ajµ sk represents the gluon-ghost-ghost vertex; both are displayed through the Feynman rules of Fig. 15.13. Ghosts, derived from non-Abelian gauge fields, only couple to the latter and only appear in loops; they do not exist as external free particles. Because of the self-interacting three-boson coupling which is typically non-Abelian, the ghost term (92) occurs not only in QCD but also in electroweak interactions. In the above discussion we considered only massless gauge fields. When these fields become massive through the Higgs mechanism, their quantization is similar to the massless case, and ghosts are also present. Absent at the tree level, the elastic photon–photon scattering and the decay Z0 → 3 photons are pure loop effects (Fig. 15.16). Because of the non-Abelian γW + W − and Z0 W + W − couplings, ghosts appear in the loops of γ + γ → γ + γ and Z0 → 3γ processes. They are massive and charged.
15.4 Ghosts . ........ ........... ............. .......... . ..................................................... ... ... .... ..... ... ... ... .... ... ... ... ... .. .................................................. . ............ ............... .............. ........ .
. ........ ........... .............. . . . . ..... .... . . ................................................ . . . . . . ....... ......... .... ... . ........ ......... . . . . ... ........ ........ . . ... ... ............ ..... ..... .................. ............. .... .... ..... ..... ......... .... .. .............. ....... .
W
. ........ ........... . ............ .......... . ....... . . . . . . . . . .... .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . ....... . ............ ............... .............. ........ .
(b)
(c)
W
W
(a)
W
541
Fig. 15.16. Contributions to γγ → γγ scattering: (a) fermion box; (b) W-boson box; and (c) ghost box
15.4.2 Ghosts and Unitarity We have seen that ghosts restore the gauge-invariant structure −q 2 gµν +q µ q ν of the gluon self-energy Πµν glu (q). Next, ghosts arise from the Faddeev–Popov gauge-fixing quantization by path integrals. Now we show that ghosts are even more indispensable. Without them, unitarity would not be satisfied, which would be as disastrous as a nonconservation of probability. The amplitude of any physical process computed to some perturbative order must satisfy unitarity to that order. The of unitarity for P requirement ∗ the S-matrix operator S S † = S † S = 1 or n San Sbn = δab , implies that the matrix element of the operator T , defined by S = 1 + i T , satisfies XZ
2 Im hb | T | ai = dρj hb | T | ji j T † a , j
where dρj is the phase space of the intermediate state |ji. The imaginary part of the scattering amplitude ha | T | bi is directly related to the sum over the products of physical amplitudes connecting any intermediate state |ji to the initial state |ai and the final state |bi. For a two-body → two-body amplitude, the unitarity constraint is expressed as n XZ Y d3 [qj ] (2π)4 δ 4 ([qj ] − k − k 0 ) 2 Im A(k + k 0 → p + p0 ) = 3 2E (2π) j n j=2
× A(k + k 0 → [qj ]) A∗ (p + p0 → [qj ]) (15.93)
where [qj ] collectively denotes momenta of the intermediate state |ji. In the particular case of two-body forward elastic scattering k = p and k 0 = p0 , the right-hand side of (93) contains a squared amplitude |A|2. To calculate a cross-section from |A|2 , we must include the kinematic factor S/4F according to (4.54). When we multiply this kinematic factor by both sides of (93), we obtain the optical theorem relating the imaginary part of the two-body forward elastic scattering amplitude to the total cross-section σtot for the production of all states. Thus, Im Aelas. (k + k 0 → k + k 0 ) = [2Ecm Kcm ] σtot(k + k 0 → anything) q Im Aelas. (s, t = 0) = λ(s, m21 , m22 ) σtot(s) , (15.94)
542
15 Asymptotic Freedom in QCD
where Ecm is the total energy of the two-body system and Kcm is the momentum of either particle in the center-of-mass frame. The variables Ecm and Kcm are written in terms of the Mandelstam variables s and t in (94). As an application of (93), we consider the quark–antiquark scattering where the intermediate state are two gluons, as described by Fig. 15.17. Unitarity relates the imaginary part of the q(p1 ) + q(p2 ) → q(p3 ) + q(p4 ) amplitude at order gs4 to the q(p1 )+q(p2 ) → g(k1 )+g(k2 ) squared amplitude: Z 2 where 2 ImA(q + q → q + q) = 21 dρ2 |A(q + q → 2 gluons)| , Z Z d3 k1 d3 k2 dρ2 ≡ (2π)4 δ 4 (k1 + k2 − p1 − p2 ) (15.95) 3 (2π) 2E1 (2π)3 2E2 is the two-gluon phase space integral, and the symmetry factor 21 on the right-hand side of (95) is due to two identical bosons in the final state. ... 0 . 3 1 .. .. ...... .... ............................................................................................................................................. ... ... ... .. ..... .. ... ... . . ... .. ... ... ... ... ... ... ... .... . . . . . . . . . . . . .. . .. .. ... ............................................................................................................................................. ... ... .. .. ... 0 2 4
p
µ
µ
p
p
ν
ν
p
... 0 ... 1 3 .................................................... ... ........................................... .................. ............. ... ................. ................... ....... ... .............................. ... ... ..... ..... .... .... .... ....................................... .... ... ............ ..................... ... . ................ . . ....................................................... ........................................... ... ... ... ... 0 2 4
p
µ
ν
p
p
ν
µ
p
(b)
(a) .
.. p..1.................................µ . ................ ..µ0 ..... ... .... .. . ........
p3
.. ... ... .................. ............. ... ............... ... ............... ........ ....... ....... ... . ................... ........................ ... ... .............. .... ... ..... .... ... ...................... .... ........... ........... ........... . . ................................................ ..... 0 ... 4 2
p
... 0 3 ... . 1 . ......................................... .... . ............................. ....... . . ...... ........................ .... . .. . .... .... ....... ....... ....................... .. .... ... .. .. .. ..... . ... . ... ...... .............. . ......... ... .. .......................... .... . . . . . ...................... .... .... .... .... .. .... . . . ..................................................... ... 0 2
ν ν
p
(d)
p
µ µ
p
p
ν
p4
ν
(c)
... 0 ....... .... ..... ................. 3 ... ..... ........ ... ........ .... 1 . ........... ... .. .............. .. ............. ........... .. .... .......... ... .... .... ..... ..... ..... ..... ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............. ....... ....... ........ .... .......... ....... ........ ............... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... .. ... ............ ........ ... .... ... ... ....... . ... . .... ....... .... ........ .......... ....... ..... . ........... ........... ... .... ..... ... .. ................. ........ .............. .. 2 4 ..... 0 ..
p
p
µ
µ
ν
ν
p
p
(e)
Fig. 15.17a–e. The cuts (dashed lines) give the imaginary part of the amplitude (to order of gs4 ) for q + q ¯ → q+q ¯ via two-gluon exchange
The crucial point is the following: The two real gluons in the final state (Fig. 15.18) have only physical transverse polarizations; while the virtual gluons in Fig. 15.17, via their propagators with the tensor gµν , can have unphysical nontransverse polarization states [see (99) below where gµν contains nontransverse polarizations Gµν L (k)]. How can we be sure a priori that in (95), the unphysical polarizations of the left-hand side would disappear on the right-hand side? If they are present, something must exist to cancel them. Historically, by inspecting (95), Feynman recognized that unitarity hints at a missing piece in the non-Abelian theory at one-loop level. To verify explicitly the unitarity relation (95), let us proceed in three steps. First, the polarizations of massless gauge particles (see also Sect. 2.5) are discussed, next we go to the q + q → g + g amplitude, i.e. the right-hand side of (95), then its left-hand side is computed lastly.
15.4 Ghosts
543
Massless Gauge-Particle Polarizations. A massive spin-1 particle has three polarization states, hence its polarization four-vector eµ (k, σ), σ = 1, 2, 3 subject to the constraint kµ eµ (k, σ) = 0 is well defined. On the other hand, a massless gauge particle has only two physical polarization states eµ (k, λ), λ = 1, 2. They are transverse k · e(k, λ) = 0. These two spacelike transverse polarization vectors satisfy eµ (k, λ)e∗µ (k, λ0 ) = −δλλ0 , kµ eµ (k, λ) = 0 .
(15.96)
The first one is simply the relativistic version of e(k, λ) · e∗ (k, λ0 ) = δλλ0 . The three independent vectors k µ , eµ (k, 1), and eµ (k, 2) cannot completely span the four-dimensional space-time, i.e. the conditions (96) do not uniquely specify the transverse polarization states. Indeed, any eµ (k, λ) + ωk µ (where ω is an arbitrary parameter) also satisfies the conditions (96). To uniquely define the two transverse states of a massless gauge particle, one must impose one more condition to determine ω, i.e. one must fix the gauge. To achieve an orthogonal basis, we introduce an independent lightlike four-vector η µ orthogonal to the transverse polarization vectors eµ (k, λ). Together with k µ , eµ (k, 1), and eµ (k, 2), the η µ closes the basis. Since k 2 = 0 and η µ is independent of k µ , we must have ηµ k µ 6= 0. So η µ satisfies η 2 = 0 with ηµ eµ (k, λ) = 0 , ηµ k µ 6= 0 .
(15.97)
The two unphysical nontransverse polarization states can now be built up from k µ and η µ . When we compute the right-hand side of (95), we have to sum over the transverse polarization states. Thus, we define X µν gT (k) ≡ eµ (k, λ) eν∗ (k, λ) . (15.98) λ=1,2
µν The tensor gT (k) which must have the form agµν + bη µ η ν + ck µ k ν + dk µ η ν + µ ν eη k is reduced to µν gT (k) = −gµν + Gµν L (k) ,
where Gµν L (k) =
kµ ην + ηµ kν k·η
(15.99)
µν using (96) and (97). The extra term Gµν the lonL (k) subtracts out of g µ gitudinal and time components of e , i.e. the two unphysical nontransverse polarizations, called eµL (k), with kµeµL (k) 6= 0. The important thing is µν µν µν ν µ k µ gT (k) = kν gT (k) = 0 while kµ Gµν L (k) = k , kν GL (k) = k . (15.100)
For the problem on hand, with two gluons, a convenient choice of the nontransverse polarizations of the first (second) gluon k1 (k2 ) is the one in which the vector η µ could be taken as the momentum k2µ (k1µ ) of the other gluon in their center-of-mass.
544
15 Asymptotic Freedom in QCD
This frame is required, since according to (97), ηµ must be such that ηµ eµ (k, λ) = 0. In this frame k1 = (E0 = |k|, k) , k2 = (E0 , −k), such that k2µ eµ (k1 , λ) = 0 = k1µ eµ (k2 , λ) for transverse eµ (k, λ) having zero time component. With this choice, we have Gµν L (k1 ) =
k µ k ν + k1ν k2µ k1µ k2ν + k1ν k2µ = 1 2 = Gµν L (k2 ) . 2 2E0 k1 · k2
(15.101)
This explicit form of Gµν L is convenient for subsequent calculations. The Amplitude for q(p1 ) + q(p2 ) → g(k1 ) + g(k2 ) . To order gs2 , there are three diagrams drawn in Fig. 15.18 that contribute to the amplitude. Let us write it as A = e∗µ (k1 )e∗ν (k2 )v(p2 )Mµν u(p1 ), where Feynman rules give
i i γµ T i + γµ T i γν T j Mµν = (−igs )2 γν T j 6 p1 − 6 k 1 − m 6 k 1 − 6 p2 − m 2 ijk ρ k g f γ T + s g (k − k ) − g (2k + k ) + g (2k + k ) µν 2 1 ρ νρ 2 1 µ ρµ 1 2 ν .(15.102) (k1 + k2 )2
Note that Mµν is symmetric under the interchange of the two final state gluons (k1 , µ, T i ) ↔ (k2 , ν, T j ). The two terms inside the curly brackets { } of (102), corresponding to the two diagrams of Fig. 15.18a–b, are similar to the e+ + e− → γ + γ amplitude in QED (except for the color matrices T i and the coupling gs ). We call this amplitude M1µν . When we multiply M1µν by k1µ and use v(p2 )[6 p2 + m] = 0 = [6 p1 − m]u(p1 ), we obtain v(p2 )k1µ M1µν u(p1 ) = −igs2 v(p2 )γν [T i , T j ]u(p1 ) = +gs2 f ijk T k v(p2 )γν u(p1 ) which vanishes in Abelian QED. The last term associated with the brackets of (102) corresponds to Fig. 15.18c. Absent in QED, this amplitude is denoted by M2µν . When it is multiplied by k1µ , using v(p2 )[6 k1 + 6 k2 ]u(p1 ) = v(p2 )[6 p2 + 6 p1 ]u(p1 ) = 0, we get v(p2 )k1µ M2µν u(p1 ) = +gs2 f ijk T k v(p2 )
k2ν 6 k1 − γ u(p1 ) . ν (k1 + k2 )2
The sum k1µ [M1µν + M2µν ] = k1µ Mµν is v(p2 )k1µ Mµν u(p1 ) = gs2 f ijk T k v(p2 )
k2ν 6 k1 u(p1 ) ≡ k2ν P , (15.103) (k1 + k2 )2
where P is defined by (103), i.e. P = gs2 f ijk T k v(p2 )
6 k1 − 6 k2 u(p1 ) = gs2 f ijk T k v(p2 ) u(p1 ).(15.104) (k1 + k2 )2 (k1 + k2 )2
545
15.4 Ghosts
Note that P is symmetric under the interchange of the two gluons, therefore v(p2 )k2ν Mµν u(p1 ) = k1µ P.
(15.105)
From now on, to simplify the notations, we will write k1µ Mµν and k2ν Mµν without v(p2 ) and u(p1 ) although the insertion is implicitly understood. We learn from this analysis that a two-photon (or two-gluon) amplitude can always be written as eµ (k1 , λ) eν (k2 , λ)Mµν . When we replace eµ (k1 , λ) by k1µ or eν (k2 , λ) by k2ν , we get QED : k1µ Mµν = 0 = k2ν Mµν , QCD : k1µ Mµν 6= 0 , k2ν Mµν 6= 0 .(15.106) However, for QCD, from (96), (103), and (105), we have a weaker constraint k1µ k2ν Mµν = k1µ eν (k2 , λ)Mµν = k2ν eµ (k1 , λ)Mµν = 0 .
(15.107)
Using (103) and (105), most important is the following nonzero term for QCD k1µ k1ν Mµν = k2µ k2ν Mµν = k1 · k2 P . p
k
p2
k2
1 ... ... 1 .... .... ........................................................................................................................... ... .... .... .... ... ... ... ... ... ... ..... ... .... ... . . .................................................................................................................. .... .... .... ....
(15.108)
p1
k1
p2
k2
p
k
p2
k2
1 ... 1 ........ ........................ .......... . ........................ ............... . ........ . . . . .................. .......... .................................. .. . .. . . . ...... . ......... ....... ......... ..................... ................. .............. ....... . . . .................... . . . .. .. .... ........ .............
........................................................................... ... ....... .... ........................ ....................... ....................... .. ......................... ..... .............. ... ........................................... ................. . .. . ......................... . . . ............... . .......................................................
(c)
(b)
(a)
q → g+g Fig. 15.18a–c. Amplitude (to order gs2 ) of q + ¯
The Imaginary Part of A(q + q → q + q). The amplitude of Fig. 15.17 – which is constructed from the two vertices Mµν and M∗µ0 ν 0 connected by two gluons exchanged in the loop – can be written as [with P = 21 (p1 + p2 )] A=
1 2
Z
0
0
d4 K −i gµµ −i gνν M M∗ 0 0 , µν (2π)4 (K + P )2 + i (P − K)2 + i µ ν
(15.109)
the factor 12 in front takes into account the two identical gluons in the loop. The imaginary part of the amplitude is obtained – according to the Cutkosky rules – by putting the two virtual gluons on the mass-shell, i.e. by replacing their propagators with the delta functions 1 −→ (−2π i) δ(p2 ) . p2 + i
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Z
Z 4 Z 4 d4 K d k1 d k2 = (2π)4 δ 4 (k1 + k2 − 2P ) , and (2π)4 (2π)4 (2π)4 Z d3 kj = d4 kj δ(kj2 )θ(Ej ) , j = 1, 2 , 2Ej R it can be shown that when the four-dimensional integral d4 K/(2π)4 in (109) is multiplied by the two delta R functions, it becomes the dimensionless twogluon phase space integral dρ2 of (95). We find that the imaginary part of A(q + q → q + q), i.e. the left-hand side of (95) is Z h 0 i 0 2 Im A(q + q → q + q) = 21 dρ2 Mµν gµµ gνν M∗µ0 ν 0 Z h i 0 0 µµ0 νν 0 = 21 dρ2 Mµν −gT (k1 ) + Gµµ −gT (k2 ) + Gνν M∗µ0 ν 0 . L (k2 ) L (k1 ) Using Z
(15.110)
Equation (110) clearly shows that unphysical polarizations appear in Im A. On the other hand, the right-hand side of (95) with real gluons, i.e. the R quantity 21 dρ2 |A(q + q → 2 gluons)|2 , is equal to Z h 0 i µµ νν 0 1 dρ M g (k )g (k ) M∗µ0 ν 0 . (15.111) 2 µν 1 2 T T 2
As stated by (95), if unitarity is satisfied, (110) and (111) must be identical. Actually this is the case of e+ + e− → γ + γ in Abelian QED. Indeed, be0 ∗ cause of (106), the contraction of Gµµ L (k1 ) with Mµν Mµ0 ν 0 yields vanishing 0 result, and the same thing happens with Gνν L (k2 ). More generally, in Abelian µν processes, we can safely take gT (k) = −gµν . 0 µµ0 µµ0 νν 0 For QCD in (110), the cross terms gT (k1 )Gνν L (k2 ) + gT (k2 )GL (k1 ) also vanish when they are contracted with Mµν and Mµ0 ν 0 due to (107). 0 νν 0 The difference between (110) and (111) which comes from Gµµ L (k1 )GL (k2 ) would violate unitarity if it did not vanish. Using (101) together with (107) and (108), this difference actually is nonzero and equals to Z Z i Mµν h µ ν µ0 ν 0 µ ν µ0 ν 0 ∗ 1 0ν0 = dρ k k k k + k k k k M dρ2 PP ∗ . (15.112) 2 µ 2 2 1 1 2 (k1 · k2 )2 1 1 2 2 To cancel the right-hand side of (112), the ghost contribution coming from the imaginary part of the amplitude Aghost (q + q → q + q) in Fig. 15.19 is added to (110). ... .. ... ..
p
k ,a
p
k ,b
p
. 3 1 .... 1 . . .... . ..... . . ....... ... . .... . . . .......... . . . .. ......... .... ....... ....... ....... . . ... ......... .......... .......... ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... . . .. .. .. .. . . .. .............. ............... ..................... ....... . . .. . . ..... ......... ... .......... .... ..... .. .... ... 2 . ... 2 . 4 . ...
p
Fig. 15.19. Aghost from ghost loop. The dashed line cut gives the imaginary part of Aghost
Problems
547
The imaginary part of Aghost (q + q → q + q) is calculated similarly to (109) and (110) by the following operations: (i) as in the case of gluons, we replace the ghost propagators with the delta functions. After this replacement, the ghost loop integral becomes the phase space dρ2 of (95). (ii) substitute the q + q → g + g amplitude by the q + q → si + sj amplitude. The latter is obtained from Feynman rules, (−igs )v(p2 )γµ T k u(p1 )
−i [gs f ikj k1µ ] = P . (k1 + k2 )2
(15.113)
Because of the fermionic character of ghosts, there is a minus sign in the loop integral, but there is no bosonic symmetry factor of 12 . From all of these operations, we get Z 2 Im Aghost = − dρ2 P P ∗ . (15.114) So (112) and (114) cancel each other. Without the ghost contribution as given by (114), the difference between (110) and (111), i.e. (112), would not vanish, and unitarity would be violated. This example illustrates a general interpretation of ghosts as agents that neutralize the unphysical polarizations of the gauge bosons. Their presence was anticipated by Feynman well before the Faddeev–Popov quantization method was proposed.
Problems 15.1 Computation of Σ(p) by the Pauli–Villars regularization. Use the Pauli–Villars procedure (14.89) to compute the fermionic self-energy Σ(p) defined in (14.21) and (14.23). As Λ → ∞, one recovers the original theory. Compute Z2 and check that the pole Γ(2 − n2 ) can be identified with log(Λ2 /µ2 ). 15.2 Massive photon in space-time two dimensions. In one-space and one-time dimension (two-dimensional QED), derive the dimensions of the photon field, the fermion field, and √ the coupling e. Using (5) with n = 2, show that the photon has a mass e/ π when it couples to massless fermions. In two dimensions, the Π(q 2 ) can have a pole at q 2 = 0 which arises from the massless fermion-antifermion intermediate state. As discussed after (11), this is the reason why the photon can be massive (Schwinger’s mechanism). 15.3 Correction to the Coulomb potential. The Uehling effect. From (37), the running charge e is q 2 dependent. Derive the Fourier transform of the Coulomb potential e2 (q 2 )/q 2 , first in the nonrelativistic case q 2 m2 , then in the q 2 m2 case. The modified Coulomb potential shifts the atomic electron energy levels (Uehling effect).
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15 Asymptotic Freedom in QCD
15.4 The β- and γ-functions in λφ4 at one-loop. Draw to order O(λ2 ) the vertex of four interacting scalar fields. There are three loop diagrams, corresponding to the three Mandelstam variables s, t, u. Show that β(λ) = 3λ2 /(16π 2 ). The one-loop O(λ) correction to the two-point function gives the field-strength counterterm Zφ from which is derived the CS-function γ(λ). Show that γ(λ) vanishes. 15.5 The pure Yang–Mills β-function by (84). Consider QCD without quarks but only gluons with their three-gluon and four-gluon interactions. Compute the Z1glu = 1 + ∆glu from Fig. 15.14. Then show that the corre1 sponding β-function is β(gs ) = (−gs3 /16π 2 ) 11 3 Nc .
Suggestions for Further Reading Field Theory, Renormalization: Collins, J. C., Renormalization. Cambridge U. Press, Cambridge 1984 Hatfield, B., Quantum Field Theory of Point Particles and Strings. Addison-Wesley, Redwood, CA 1992 Itzykson, C. and Zuber, J. B., Quantum Field Theory. McGraw-Hill, New York 1980 Kaku, M., Quantum Field Theory. Oxford U. Press, New York 1993 Ramond, P., Field Theory: A Modern Primer (Second edition). Addison-Wesley, Redwood, CA 1989 Quantization of Yang–Mills fields: Faddeev, L. D. and Slavnov, A. A., Gauge Fields: Introduction to Quantum Theory. Benjamin, Reading, MA 1980 ’t Hooft, G., Under the Spell of the Gauge Principle. World Scientific, Singapore 1994 Vacuum polarization Πµν (q), vertex function Γµ (p0 , p), fermionic self-energy Σ(p): De Wit, B. and Smith, J., Field Theory in Particle Physics (Vol. I). NorthHolland, Amsterdam 1986 Peskin, M. E. and Schroeder, D. V., An Introduction to Quantum Field Theory. Addison-Wesley, Reading, MA 1995 Renormalization group method and β-functions: Cheng, Ta-Pei and Li, Ling-Fong, Gauge Theory of Elementary Particle Physics. Oxford U. Press, New York 1984 Peskin, M. E. and Schroeder, D. V., (op. cit.) Weinberg, S., The Quantum Theory of Fields (Vol. II.) Cambridge U. Press, Cambridge 1996 Ghosts and unitarity, Cancellation of gauge-dependent ξ terms: Aitchison, I. J. R. and Hey, A. J. G., Gauge Theories in Particle Physics (Second edition). Adam Hilger, Bristol 1989 Feng, Y. J. and Lam, C. S., Phys. Rev. D53 (1996) 2115 Gross, F., Relativistic Quantum Mechanics and Field Theory. Wiley-Interscience, New York 1993 Perturbative QCD, GLAP equations: Field, R. D., Applications of Perturbative QCD. Addison-Wesley, Redwood, CA 1989
