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The renormalization group invariant charge and confinement
Volume 74B, number 1, 2
PHYSICS LETTERS
27 March 1978
THE RENORMALIZATION GROUP INVARIANT CHARGE AND CONFINEMENT
B.H. KELLETT 1
School of Physics, University of Melbourne, Parkville, Victoria, Australia 3052 Received 15 December 1977
The implications of exact renormalization group invariance for the structure of the invariant charge are examined. The most general invariant form is constructed, and it is shown that for finite q2, the invariant charge for both QED and QCD is non-singular. At q2 = 0 and q2 = _~, the invariant charge for QCD is either zero or infinite, and this forms the basis of a non-perturbative derivation of asymptotic freedom and confinement.
Renormalizability is an essential property of a realistic quantum field theory. Theories such as quantum electrodynamics (QED) and quantum chromodynamics (QCD) owe their success and current popularity to the fact that they are multiplicatively renormalizable. Physical amplitudes calculated in these theories are invariant under the multiplicative renormalization group [ 1 - 3 ] , which guarantees that physical quantities are finite, and independent of the ultraviolet cut-off A. Renormalization group (RG) invariance is widely used to extend the range of perturbation calculations, and in this letter we investigate the most general form that the vector propagator function and invariant charge in QED and QCD can take consistent with exact RG invariance. In the unrenormalized theory, we write the dimensionless dressed transverse vector propagator function d(q2, A 2, m 2, g2), where q2 is the momentum, A is the ultraviolet cut-off, m is the fermion mass, and g2 is the bare coupling constant, in terms of the sum over one-particle irreducible (1Pl)vacuum polarization Green's function g2H(q2, A 2, m 2, g2) as d(q 2, A2, m 2, g2) = [1 - g2II(q 2, A2, m 2, g 2 ) ] - 1
(1) Multiplicative renormalization implies that the renormalized propagator function is I Australian Research Grants' Committee Fellow.
D(q2,/a2, m 2, g2) = Z 3 I (A,/a)d(q 2, A 2, m 2, g~), (2) where p is the euclidean renormalization point
D(q2 = _/12, p2, m 2 , g 2 ) = 1,
(3)
and gu is the renormalized coupling constant g2 = Z3 (A,/~)g2.
(4)
Eq. (4) for the renormalized charge is well known in QED, and it is also true for QCD in a ghost-free gauge, where the gluon vertex renormalization function Z 1 = Z 3. Because of this simplification, we use the ghostfree gauge for QCD throughout this letter. From the multiplicative transformations (2) and (4) it follows that the invariant charge g2 (q2) = g2 D(q2 ' la2, m 2, g2)
(5)
is independent of the renormalization point ~. However, if we take the perturbation expansion of H as a finite power series in g2, explicit calculation readily demonstrates that RG invariance is exact only at the one-loop or g2 level. When higher-order terms are included, a finite-order expansion in one renormalized coupling constant corresponds to an infinite power series in any other renormalized coupling constant, so manifest RG invariance must be sacrificed if we work consistently to a particular order of perturbation 85
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theory. Looked at in another way, RG invariance implies a relation between different orders of perturbation theory, and it is this relationship that applications of the RG seek to exploit, to improve on straightforward perturbation theory. Thus, if we can find the most general form of the invariant charge (5) consistent with exact RG invariance, comparison of the general form with the perturbation results will enable us to sum at least part of the full perturbation series to all orders. This approach is rather different from the conventional applications of the differential equations of the RG in that we work with the integral forms, and thus obtain directly the general solution of the Callan-Symanzik equation [4] for instance. From eqs. ( 1 ) - (4) we obtain
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through a function f(q2/m2, g2 (q2)). The connection with the differential approach through the CallanSymanzik equations [4,5] is obvious. There the general solution is written in terms of an effective coupling with a q2-dependent renormalization prescription. From eqs. (3) and (5), the coupling g. renormalized at q2 = _/22 is given by g2 = g2 (q2 = ~'_/22), so the effective coupling with a sliding renormalization is just the invariant charge, ~(q2) = gd(q2). The most general form of the vacuum polarization function consistent with manifest RG invariance is, therefore, if(q2, A 2, m 2 ' g2) = c log ( - A 2 / q 2) + h(A2/m 2) (7)
_ f(q2/m2 ' g2 (q2)),
D - l ( q 2, U2, rn2,g2u) = 1
(6) _ gu2 [ii(q2, A 2, m 2, g2) _ i1 (_/22, A 2, m 2, g2)]. Multiplicative renormalization of D implies that I1 is renormalized by subtraction, as is well known. Since D is defined to be independent of A, we require the most general form of 11 such that the right-hand side of eq. (6) is independent of A. Since I1 is dimensionless, it is a function of A2/q 2, A2/m 2 and q2/m2. Any A2/m 2 dependence cancels exactly, and q2/m2 is automatically independent of A, so I1 can involve arbitrary functions F(qZ/m 2) and h(A2/rn2). Similarly, dependence on A2/q 2 is necessarily of the form const X log (-A2/q2), since the A dependence of any polynomial in AZ/q 2, or higher power of log ( - A 2 / q 2) does not cancel. These possibilities enable us trivially to write the order-g 2 contributions in manifestly invariant form, but this was already known. The problem arises when we include higher orders, since eq. (4) shows that go is an implicit function of A, and I1 cannot depend on go explicitly. Furthermore, it cannot depend on the renormalized coupling gu explicitly either, for that would violate invariance under changes in the renormalization point/2. The only possibility is that I1 depends on some function o f g 0 and A, that is independent of A, or on some function of/2 and gu' that is independent of/~. The only function that satisfies these conditions is the invariant charge gd (q2) itself, so the most general form of II consistent with renormalization group invariance involves higher orders only 86
where c is a constant. The renonnalized transverse propagator is then D-l(q2,/22, m2,g2) = 1 + g2 [c log(-q2//22)
{8) + f(q2/m2 ' g2) _ f(_/22/m2 ' g2)]. If the renormalization point is changed from/21 to /22, the invariant charge is unchanged:
g~D(q2,/21,m2,g~) = g2D(q 2 2 ,122 2 ,m 2 ,g22 ),
(9)
and the renormalized couplings are related by g2 = Z3(/22 '/21)g~'
(10)
Z3(/22'/21 ) = D(-/22'/'IT' rn2'
m:,d).
(11)
The check of the transformation properties of D is a trivial exercise that we leave to the reader. The invariant charge is g2(q2) : g2/[1 + g2(c log ( _ q 2 / , 2 )
(12) +f(q2/m2 ' g2) _ f(_/22/m2 ' g2)}] , and this form is exactly'invariant under changes in/2. The function f(q2/m2, g2) is completely arbitrary, and it is not difficult to convince oneself by explicit calculation that eq. (12) is indeed the most general in-
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variant form. The expression for the invariant charge therefore depends on itself through the function f Such non-linear situations are common in renormalization theory where, for instance, it is well known that the renormalization constant Z 3 is calculated as a series in gu, and hence depends on itself through eq. (4). The function f(q2/m2, g2(q2))can be determined from perturbation theory by expanding it as a power series in g2, and using eq. (12). Comparison of the resulting series in g2 for say D -1 , with the perturbation result for the same quantity, determines the coefficients of the series in g2. Writing the result in terms o f g 2 automatically includes and sums just that part of all higher orders in g2 necessary for RG invariance. The term gu2 l o g ( - q 2 / ~ 2) in eq. (12) is characteristic of a theory with self-coupled massless fields, and it is not present in QED with massive fermions. In that case, it is replaced by the function F(q2/m2), which is given by the usual one-loop perturbation approximation. In a purely massless theory, such as QCD without quarks, there is no external mass scale; terms in q2/m2 do not appear, a n d f i s a function o f g 2 alone. In QCD with massive quarks, both the log (_q2/~2) and F(q2/m q2) terms are present. The important properties of the invariant charge for QCD can be obtained by considering the case of massless quarks and gluons. The function f(g2) has no implicit q2 dependence in this case, and i f f ( g 2) is a simple polynomial, there is no term in g4 in the perturbation series for D l(q2,/~2, g2). Consequently, a term in log(g 2) must be present in f, and we can write f(g2) = c' log (gd2) + f(g2). Higher powers of log (g2d) generate terms in log (gu) , so these are ruled out. Using the known perturbation theory results to O(g 4) [ 5 - 7 ] , the invariant charge for QCD without quarks can be written in manifestly RG invariant form as gd2(q 2) = g2/[1
+ (g2/167r2)C2(G){~
log (-q2//l 2) (13)
17 log (gd/gli) 2 2
+~f(g2) _ f(gu2))]
,
where C2(G ) is the value of the quadratic Casimir operator for the gluon representation of the colour gauge group. For finite q2, f(g2) diverges as g2 _+ ,~, whatever the form o f f , so g2 must be finite. Furthermore, if
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gd2 vanishes for some finite q2, renormalizing at this point gives gu= 0 for some/l, and thus g2d(q2 ) = 0 for all q2, which is the trivial free-field solution that we shall ignore. We have, therefore, proved that the invariant charge is a finite, non-vanishing function o f q 2 for all f'mite, non-zero q2. This is an important result, because low-order perturbation theory indicates a pole for [q2l < / l 2 [5]. Our non-perturbative solution shows that this pole does not exist. The log (-q2//12) driving term in eq. (13) is exact for all q2, so the denominator diverges for q2 = 0 and q2 = +_~. Since f(g2)is finite for finite g2d, it follows that g2 is necessarily either zero or infinite at these end points. Which possibility obtains depends on the value of the function f ( g 2 ) as g2 _+ ,,~. ifj~(~) = 0, any finite value, or f ( ~ ) = -~,,, then gd2(0) necessarily vanishes, and g2(__~) may either vanish or diverge. On the other hand, ifY(~) = +,'% then gd2 must vanish as q2 - ~ , and g2(0) may vanish, or it may diverge. In every case, therefore, there is a solution that is asymptotically free, although it is not proved from these considerations alone that nature chooses this solution. Furthermore, there is also always a solution for which the invariant charge vanishes at q2 = 0, and it is this that we interpret as a signal for confinement. Presumably the one case for which g2 may diverge at q2 = 0 also signals confinement, although the divergence there is like log log (q2) or slower, which is nothing like the 1/q2 divergence associated with the popular linear confining potential. When g2d(0) vanishes, the divergence of II (q2) at q2 = 0 is only logarithmic, and it does not cancel the 1/q2 pole in the full propagator. Thus the vector propagator does not acquire a mass [8], and there is a genuine zero in the invariant charge associated with a massless gluon. This zero ofgd2 (q 2 = 0) is sufficient to establish the permanent confinement of gluons. In the renormalized field theory, the strength of the coupling of a gluon of momentum q is given by gd(q2). For external gluons on the mass shell, q2 = 0, the effective coupling strength is the residue of the pole of the propagator, which is again the invariant charge g2(q2 = 0). This is exactly the situation in QED, where the value of the invariant charge at q2 = 0 is e 2 = 47r/137; the physical charge that appears in low-energy theorems. Since g2(0) = 0, external gluons on their mass shell decouple, and ghions are permanently confined. 87
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The value of this mechanism for gluon confinement is that it is non-perturbative. All orders of perturbation theory are included exactly in the general form (13). Moreover, the result is explicitly independent of the renormalization point, and of the value of the renormalized couplinggu, so that it does not depend on whether we use a fixed or a sliding q2-dependent renormalization scheme. Confinement is seen to be a consequence of the log(-q2//l 2) term arising from massless field loops. This term survives in QCD with massive quarks. The quark mass merely introduces a function F(q2/m 2) that vanishes as q2 -+ 0, and does not affect the result. Notice further that this log (-q2//12) term appears in QED when the fermions are massless, so massless QED is also confined. Given gluon confinement, quark confinement is automatic. |t is well known from QED that if the infrared divergences associated with the fermion selfenergy and vertex functions are regulated by an infrared cut-off mass X, the electron production cross section vanishes as X -+ 0. In QED, the corresponding divergent probability for the emission of soft real photons cancels this zero, leaving a finite amplitude [9]. This mechanism is unavailable in QCD with confined gluons, so the amplitude zero survives, and quarks are also confined. The fact that quarks and ghions are not produced as asymptotic single-particle states solves the major problem of confinement in QCD, but this does not of itself forbid the production of colour non-singlet bound states of two or more quarks. An explanation of complete colour confinement lies outside the scope of the simple ideas presented here. Cornwall and Tiktopoulos [10] argue that the exponentiation and factorization of the leading infrared logarithmic singularities occurs for bound as well as elementary particle states, so that all colour non-singlet production amplitudes vanish as the infrared cut-off mass goes to zero. Whatever the status of this particular argument, it would appear that colour exchange reactions are forbidden, since colour non-singlet bound states are not observed. The confining solution of eq. (13) for which g2d(0) = 0 has some further interesting properties. The derivative of g2d(q2) with respect to q2 (which is essentially the Callan-Symanzik 13function) is
88
27 March 1978
q2 dg2 _ -11C2(G ) dq 2
487r2 (14)
x g 4 [ ' l +C2(G)
d[_-
2/ 2~-,~2~
17
16rr2gdlgdJtgd)---tl}l
-1
where T ( x ) = df/dx. At q2 = 0, therefore, the derivative is negative, and g2(q2) is negative for small _q2. Because of the log(g 2) term, g2 is in fact complex for small spacelike q2. However, g~] is real for sufficiently large spacelike q2, and the position of the branchpoint depends on the function f. This avoids any problem usually associated with a negative or complex tenorrealized coupling constant, such as a classical energy spectrum unbounded below, because we can always choose to renormalize at some point for which the coupling is real and positive, and the physical properties of the theory are independent of the renormalization point. I f g 2 vanishes for both q2 = 0 and q2 = _~o, the derivative (14) must vanish for some finite _q2. With gd2 complex, the real and imaginary parts of eq. (14) can vanish at different values o f q 2. In fact, the derivative can be exactly zero for finite q2 and g2 only if f,(g2) has a pole. Although we do not know f(g2d) in detail, it is known that it cannot diverge for finite g2, so its derivative can diverge only i f f is in fact nonpolynomial, which would conflict with its origins in perturbation theory. Like QCD, QED is invariant under its own RG, and the QED invariant charge e2(q 2) also has the form (12). With massive fermions, the explicit q2 dependence of the function f(q2/m2, e2) for QED means that the e 4 term does not necessarily cancel in a polynomial in e 2. However, the possible ambiguity between log(e 2) and F(q2/m2)e 2 terms in the fourth order is resolved by the inclusion of high orders, or by the requirement that the zero mass limit be smooth [11]. In the onshell normalization, 122 = 0, the known perturbation theory results [12, 13] lead to e2(q 2) = e2/[1 - (e2/48rr 2) {4F 1(q2/m2) + 9 log (e2/e 2)
+f(q2/m2, e 2) _ f ( 0 ,
e2)}] ,
(15)
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where e 2 = 47r/137 is the physical charge, and
F 1 (q2/m2) is a known function. For finite q2, e2(q2) must be finite as in QCD, and our non-perturbative approach solves the problem of the apparent divergence of e 2 for q2 >> m 2, which, it has been suggested, indicates a breakdown of perturbation theory in this region [2]. The original motivation for the RG of Gell-Mann and Low [1] was to solve the problems of the short-distance behaviour of QED occasioned by this apparent singularity, and we have shown that RG invariance naturally leads to a non-perturbative solution that does not have this singularity. At q2 = 0, F I ( 0 ) = 0 and e2(0) = e 2 as required. As q2 _+ _0% the possible solutions for e2(q 2) depend on f(oo, e 2 = oo), and as for QCD, we find that there is always a solution for which e2(oo) = 0. Thus there is a possibility that QED may be asymptotically free. In this letter we have developed an explicit (nonlinear) expression for the invariant charge in any multiplicatively renormalized field theory. Even though we cannot solve perturbation theory to all orders, this general form gives some general properties. In particular, the invariant charge is always a non-singular function for all finite (non-zero) q2 and for a theory with coupled massless fields, such as QCD, there is always a solution for which the invariant charge vanishes at q2 = 0. This solution permanently confines
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quarks and gluons. Similarly, there is always a solution for which the invariant charge vanishes as q2 _+ _0% and the theory is asymptotically free. The crucial input is the requirement that the results of low-order perturbation theory be manifestly RG invariant.
References [1] M. Gell-Mann and F.E. Low, Phys. Rev. 95 (1954) 1300. [2] N.N. Bogoliubov and D.V. Shirkov, Introduction to the theory of quantized fields (Interscience, New York, 1959). [3] K. Wilson, Phys. Rev. D3 (1971) 1818. [4] C.G. Callan, Phys. Rev. D2 (1970) 1541; K. Symanzik, Commun. Math. Phys. 18 (1970) 227. [5] D.J. Gross and F. Wilczek, Phys. Rev. D8 (1973) 3633; H.D. Politzer, Phys. Rept. 14C (1974) 129. [6] D.R.T. Jones, Nucl. Phys. B75 (1974) 531. [7] W.E. Caswell, Phys. Rev. Lett. 33 (1974) 244. [8] J. Schwinger, Phys. Rev. 125 (1962) 397; R. Jackiw and K. Johnson, Phys. Rev. D8 (1973) 2368; J.M. Cornwall and R.E. Norton, Phys. Rev. D8 (1973) 3338. [9] F. Bloch and Z. Nordsieck, Phys. Rev. 52 (1937) 54. [10] J.M. Cornwall and G. Tiktopoulos, Phys. Rev. D13 (1976) 3370. [11] B.H. Kellett, to be published. [12] R.P. Feynman, Phys. Rev. 76 (1949) 769. [13] R. Jost and J.M. Luttinger, Helv. Phys. Acta 23 (1950) 201; G. K~illen and A. Sabry, Kgl. Dan. Vid. Selsk. Mat. Fys. Medd. 29 (1955) No. 17.
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27 March 1978
THE RENORMALIZATION GROUP INVARIANT CHARGE AND CONFINEMENT
B.H. KELLETT 1
School of Physics, University of Melbourne, Parkville, Victoria, Australia 3052 Received 15 December 1977
The implications of exact renormalization group invariance for the structure of the invariant charge are examined. The most general invariant form is constructed, and it is shown that for finite q2, the invariant charge for both QED and QCD is non-singular. At q2 = 0 and q2 = _~, the invariant charge for QCD is either zero or infinite, and this forms the basis of a non-perturbative derivation of asymptotic freedom and confinement.
Renormalizability is an essential property of a realistic quantum field theory. Theories such as quantum electrodynamics (QED) and quantum chromodynamics (QCD) owe their success and current popularity to the fact that they are multiplicatively renormalizable. Physical amplitudes calculated in these theories are invariant under the multiplicative renormalization group [ 1 - 3 ] , which guarantees that physical quantities are finite, and independent of the ultraviolet cut-off A. Renormalization group (RG) invariance is widely used to extend the range of perturbation calculations, and in this letter we investigate the most general form that the vector propagator function and invariant charge in QED and QCD can take consistent with exact RG invariance. In the unrenormalized theory, we write the dimensionless dressed transverse vector propagator function d(q2, A 2, m 2, g2), where q2 is the momentum, A is the ultraviolet cut-off, m is the fermion mass, and g2 is the bare coupling constant, in terms of the sum over one-particle irreducible (1Pl)vacuum polarization Green's function g2H(q2, A 2, m 2, g2) as d(q 2, A2, m 2, g2) = [1 - g2II(q 2, A2, m 2, g 2 ) ] - 1
(1) Multiplicative renormalization implies that the renormalized propagator function is I Australian Research Grants' Committee Fellow.
D(q2,/a2, m 2, g2) = Z 3 I (A,/a)d(q 2, A 2, m 2, g~), (2) where p is the euclidean renormalization point
D(q2 = _/12, p2, m 2 , g 2 ) = 1,
(3)
and gu is the renormalized coupling constant g2 = Z3 (A,/~)g2.
(4)
Eq. (4) for the renormalized charge is well known in QED, and it is also true for QCD in a ghost-free gauge, where the gluon vertex renormalization function Z 1 = Z 3. Because of this simplification, we use the ghostfree gauge for QCD throughout this letter. From the multiplicative transformations (2) and (4) it follows that the invariant charge g2 (q2) = g2 D(q2 ' la2, m 2, g2)
(5)
is independent of the renormalization point ~. However, if we take the perturbation expansion of H as a finite power series in g2, explicit calculation readily demonstrates that RG invariance is exact only at the one-loop or g2 level. When higher-order terms are included, a finite-order expansion in one renormalized coupling constant corresponds to an infinite power series in any other renormalized coupling constant, so manifest RG invariance must be sacrificed if we work consistently to a particular order of perturbation 85
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theory. Looked at in another way, RG invariance implies a relation between different orders of perturbation theory, and it is this relationship that applications of the RG seek to exploit, to improve on straightforward perturbation theory. Thus, if we can find the most general form of the invariant charge (5) consistent with exact RG invariance, comparison of the general form with the perturbation results will enable us to sum at least part of the full perturbation series to all orders. This approach is rather different from the conventional applications of the differential equations of the RG in that we work with the integral forms, and thus obtain directly the general solution of the Callan-Symanzik equation [4] for instance. From eqs. ( 1 ) - (4) we obtain
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through a function f(q2/m2, g2 (q2)). The connection with the differential approach through the CallanSymanzik equations [4,5] is obvious. There the general solution is written in terms of an effective coupling with a q2-dependent renormalization prescription. From eqs. (3) and (5), the coupling g. renormalized at q2 = _/22 is given by g2 = g2 (q2 = ~'_/22), so the effective coupling with a sliding renormalization is just the invariant charge, ~(q2) = gd(q2). The most general form of the vacuum polarization function consistent with manifest RG invariance is, therefore, if(q2, A 2, m 2 ' g2) = c log ( - A 2 / q 2) + h(A2/m 2) (7)
_ f(q2/m2 ' g2 (q2)),
D - l ( q 2, U2, rn2,g2u) = 1
(6) _ gu2 [ii(q2, A 2, m 2, g2) _ i1 (_/22, A 2, m 2, g2)]. Multiplicative renormalization of D implies that I1 is renormalized by subtraction, as is well known. Since D is defined to be independent of A, we require the most general form of 11 such that the right-hand side of eq. (6) is independent of A. Since I1 is dimensionless, it is a function of A2/q 2, A2/m 2 and q2/m2. Any A2/m 2 dependence cancels exactly, and q2/m2 is automatically independent of A, so I1 can involve arbitrary functions F(qZ/m 2) and h(A2/rn2). Similarly, dependence on A2/q 2 is necessarily of the form const X log (-A2/q2), since the A dependence of any polynomial in AZ/q 2, or higher power of log ( - A 2 / q 2) does not cancel. These possibilities enable us trivially to write the order-g 2 contributions in manifestly invariant form, but this was already known. The problem arises when we include higher orders, since eq. (4) shows that go is an implicit function of A, and I1 cannot depend on go explicitly. Furthermore, it cannot depend on the renormalized coupling gu explicitly either, for that would violate invariance under changes in the renormalization point/2. The only possibility is that I1 depends on some function o f g 0 and A, that is independent of A, or on some function of/2 and gu' that is independent of/~. The only function that satisfies these conditions is the invariant charge gd (q2) itself, so the most general form of II consistent with renormalization group invariance involves higher orders only 86
where c is a constant. The renonnalized transverse propagator is then D-l(q2,/22, m2,g2) = 1 + g2 [c log(-q2//22)
{8) + f(q2/m2 ' g2) _ f(_/22/m2 ' g2)]. If the renormalization point is changed from/21 to /22, the invariant charge is unchanged:
g~D(q2,/21,m2,g~) = g2D(q 2 2 ,122 2 ,m 2 ,g22 ),
(9)
and the renormalized couplings are related by g2 = Z3(/22 '/21)g~'
(10)
Z3(/22'/21 ) = D(-/22'/'IT' rn2'
m:,d).
(11)
The check of the transformation properties of D is a trivial exercise that we leave to the reader. The invariant charge is g2(q2) : g2/[1 + g2(c log ( _ q 2 / , 2 )
(12) +f(q2/m2 ' g2) _ f(_/22/m2 ' g2)}] , and this form is exactly'invariant under changes in/2. The function f(q2/m2, g2) is completely arbitrary, and it is not difficult to convince oneself by explicit calculation that eq. (12) is indeed the most general in-
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variant form. The expression for the invariant charge therefore depends on itself through the function f Such non-linear situations are common in renormalization theory where, for instance, it is well known that the renormalization constant Z 3 is calculated as a series in gu, and hence depends on itself through eq. (4). The function f(q2/m2, g2(q2))can be determined from perturbation theory by expanding it as a power series in g2, and using eq. (12). Comparison of the resulting series in g2 for say D -1 , with the perturbation result for the same quantity, determines the coefficients of the series in g2. Writing the result in terms o f g 2 automatically includes and sums just that part of all higher orders in g2 necessary for RG invariance. The term gu2 l o g ( - q 2 / ~ 2) in eq. (12) is characteristic of a theory with self-coupled massless fields, and it is not present in QED with massive fermions. In that case, it is replaced by the function F(q2/m2), which is given by the usual one-loop perturbation approximation. In a purely massless theory, such as QCD without quarks, there is no external mass scale; terms in q2/m2 do not appear, a n d f i s a function o f g 2 alone. In QCD with massive quarks, both the log (_q2/~2) and F(q2/m q2) terms are present. The important properties of the invariant charge for QCD can be obtained by considering the case of massless quarks and gluons. The function f(g2) has no implicit q2 dependence in this case, and i f f ( g 2) is a simple polynomial, there is no term in g4 in the perturbation series for D l(q2,/~2, g2). Consequently, a term in log(g 2) must be present in f, and we can write f(g2) = c' log (gd2) + f(g2). Higher powers of log (g2d) generate terms in log (gu) , so these are ruled out. Using the known perturbation theory results to O(g 4) [ 5 - 7 ] , the invariant charge for QCD without quarks can be written in manifestly RG invariant form as gd2(q 2) = g2/[1
+ (g2/167r2)C2(G){~
log (-q2//l 2) (13)
17 log (gd/gli) 2 2
+~f(g2) _ f(gu2))]
,
where C2(G ) is the value of the quadratic Casimir operator for the gluon representation of the colour gauge group. For finite q2, f(g2) diverges as g2 _+ ,~, whatever the form o f f , so g2 must be finite. Furthermore, if
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gd2 vanishes for some finite q2, renormalizing at this point gives gu= 0 for some/l, and thus g2d(q2 ) = 0 for all q2, which is the trivial free-field solution that we shall ignore. We have, therefore, proved that the invariant charge is a finite, non-vanishing function o f q 2 for all f'mite, non-zero q2. This is an important result, because low-order perturbation theory indicates a pole for [q2l < / l 2 [5]. Our non-perturbative solution shows that this pole does not exist. The log (-q2//12) driving term in eq. (13) is exact for all q2, so the denominator diverges for q2 = 0 and q2 = +_~. Since f(g2)is finite for finite g2d, it follows that g2 is necessarily either zero or infinite at these end points. Which possibility obtains depends on the value of the function f ( g 2 ) as g2 _+ ,,~. ifj~(~) = 0, any finite value, or f ( ~ ) = -~,,, then gd2(0) necessarily vanishes, and g2(__~) may either vanish or diverge. On the other hand, ifY(~) = +,'% then gd2 must vanish as q2 - ~ , and g2(0) may vanish, or it may diverge. In every case, therefore, there is a solution that is asymptotically free, although it is not proved from these considerations alone that nature chooses this solution. Furthermore, there is also always a solution for which the invariant charge vanishes at q2 = 0, and it is this that we interpret as a signal for confinement. Presumably the one case for which g2 may diverge at q2 = 0 also signals confinement, although the divergence there is like log log (q2) or slower, which is nothing like the 1/q2 divergence associated with the popular linear confining potential. When g2d(0) vanishes, the divergence of II (q2) at q2 = 0 is only logarithmic, and it does not cancel the 1/q2 pole in the full propagator. Thus the vector propagator does not acquire a mass [8], and there is a genuine zero in the invariant charge associated with a massless gluon. This zero ofgd2 (q 2 = 0) is sufficient to establish the permanent confinement of gluons. In the renormalized field theory, the strength of the coupling of a gluon of momentum q is given by gd(q2). For external gluons on the mass shell, q2 = 0, the effective coupling strength is the residue of the pole of the propagator, which is again the invariant charge g2(q2 = 0). This is exactly the situation in QED, where the value of the invariant charge at q2 = 0 is e 2 = 47r/137; the physical charge that appears in low-energy theorems. Since g2(0) = 0, external gluons on their mass shell decouple, and ghions are permanently confined. 87
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The value of this mechanism for gluon confinement is that it is non-perturbative. All orders of perturbation theory are included exactly in the general form (13). Moreover, the result is explicitly independent of the renormalization point, and of the value of the renormalized couplinggu, so that it does not depend on whether we use a fixed or a sliding q2-dependent renormalization scheme. Confinement is seen to be a consequence of the log(-q2//l 2) term arising from massless field loops. This term survives in QCD with massive quarks. The quark mass merely introduces a function F(q2/m 2) that vanishes as q2 -+ 0, and does not affect the result. Notice further that this log (-q2//12) term appears in QED when the fermions are massless, so massless QED is also confined. Given gluon confinement, quark confinement is automatic. |t is well known from QED that if the infrared divergences associated with the fermion selfenergy and vertex functions are regulated by an infrared cut-off mass X, the electron production cross section vanishes as X -+ 0. In QED, the corresponding divergent probability for the emission of soft real photons cancels this zero, leaving a finite amplitude [9]. This mechanism is unavailable in QCD with confined gluons, so the amplitude zero survives, and quarks are also confined. The fact that quarks and ghions are not produced as asymptotic single-particle states solves the major problem of confinement in QCD, but this does not of itself forbid the production of colour non-singlet bound states of two or more quarks. An explanation of complete colour confinement lies outside the scope of the simple ideas presented here. Cornwall and Tiktopoulos [10] argue that the exponentiation and factorization of the leading infrared logarithmic singularities occurs for bound as well as elementary particle states, so that all colour non-singlet production amplitudes vanish as the infrared cut-off mass goes to zero. Whatever the status of this particular argument, it would appear that colour exchange reactions are forbidden, since colour non-singlet bound states are not observed. The confining solution of eq. (13) for which g2d(0) = 0 has some further interesting properties. The derivative of g2d(q2) with respect to q2 (which is essentially the Callan-Symanzik 13function) is
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q2 dg2 _ -11C2(G ) dq 2
487r2 (14)
x g 4 [ ' l +C2(G)
d[_-
2/ 2~-,~2~
17
16rr2gdlgdJtgd)---tl}l
-1
where T ( x ) = df/dx. At q2 = 0, therefore, the derivative is negative, and g2(q2) is negative for small _q2. Because of the log(g 2) term, g2 is in fact complex for small spacelike q2. However, g~] is real for sufficiently large spacelike q2, and the position of the branchpoint depends on the function f. This avoids any problem usually associated with a negative or complex tenorrealized coupling constant, such as a classical energy spectrum unbounded below, because we can always choose to renormalize at some point for which the coupling is real and positive, and the physical properties of the theory are independent of the renormalization point. I f g 2 vanishes for both q2 = 0 and q2 = _~o, the derivative (14) must vanish for some finite _q2. With gd2 complex, the real and imaginary parts of eq. (14) can vanish at different values o f q 2. In fact, the derivative can be exactly zero for finite q2 and g2 only if f,(g2) has a pole. Although we do not know f(g2d) in detail, it is known that it cannot diverge for finite g2, so its derivative can diverge only i f f is in fact nonpolynomial, which would conflict with its origins in perturbation theory. Like QCD, QED is invariant under its own RG, and the QED invariant charge e2(q 2) also has the form (12). With massive fermions, the explicit q2 dependence of the function f(q2/m2, e2) for QED means that the e 4 term does not necessarily cancel in a polynomial in e 2. However, the possible ambiguity between log(e 2) and F(q2/m2)e 2 terms in the fourth order is resolved by the inclusion of high orders, or by the requirement that the zero mass limit be smooth [11]. In the onshell normalization, 122 = 0, the known perturbation theory results [12, 13] lead to e2(q 2) = e2/[1 - (e2/48rr 2) {4F 1(q2/m2) + 9 log (e2/e 2)
+f(q2/m2, e 2) _ f ( 0 ,
e2)}] ,
(15)
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where e 2 = 47r/137 is the physical charge, and
F 1 (q2/m2) is a known function. For finite q2, e2(q2) must be finite as in QCD, and our non-perturbative approach solves the problem of the apparent divergence of e 2 for q2 >> m 2, which, it has been suggested, indicates a breakdown of perturbation theory in this region [2]. The original motivation for the RG of Gell-Mann and Low [1] was to solve the problems of the short-distance behaviour of QED occasioned by this apparent singularity, and we have shown that RG invariance naturally leads to a non-perturbative solution that does not have this singularity. At q2 = 0, F I ( 0 ) = 0 and e2(0) = e 2 as required. As q2 _+ _0% the possible solutions for e2(q 2) depend on f(oo, e 2 = oo), and as for QCD, we find that there is always a solution for which e2(oo) = 0. Thus there is a possibility that QED may be asymptotically free. In this letter we have developed an explicit (nonlinear) expression for the invariant charge in any multiplicatively renormalized field theory. Even though we cannot solve perturbation theory to all orders, this general form gives some general properties. In particular, the invariant charge is always a non-singular function for all finite (non-zero) q2 and for a theory with coupled massless fields, such as QCD, there is always a solution for which the invariant charge vanishes at q2 = 0. This solution permanently confines
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quarks and gluons. Similarly, there is always a solution for which the invariant charge vanishes as q2 _+ _0% and the theory is asymptotically free. The crucial input is the requirement that the results of low-order perturbation theory be manifestly RG invariant.
References [1] M. Gell-Mann and F.E. Low, Phys. Rev. 95 (1954) 1300. [2] N.N. Bogoliubov and D.V. Shirkov, Introduction to the theory of quantized fields (Interscience, New York, 1959). [3] K. Wilson, Phys. Rev. D3 (1971) 1818. [4] C.G. Callan, Phys. Rev. D2 (1970) 1541; K. Symanzik, Commun. Math. Phys. 18 (1970) 227. [5] D.J. Gross and F. Wilczek, Phys. Rev. D8 (1973) 3633; H.D. Politzer, Phys. Rept. 14C (1974) 129. [6] D.R.T. Jones, Nucl. Phys. B75 (1974) 531. [7] W.E. Caswell, Phys. Rev. Lett. 33 (1974) 244. [8] J. Schwinger, Phys. Rev. 125 (1962) 397; R. Jackiw and K. Johnson, Phys. Rev. D8 (1973) 2368; J.M. Cornwall and R.E. Norton, Phys. Rev. D8 (1973) 3338. [9] F. Bloch and Z. Nordsieck, Phys. Rev. 52 (1937) 54. [10] J.M. Cornwall and G. Tiktopoulos, Phys. Rev. D13 (1976) 3370. [11] B.H. Kellett, to be published. [12] R.P. Feynman, Phys. Rev. 76 (1949) 769. [13] R. Jost and J.M. Luttinger, Helv. Phys. Acta 23 (1950) 201; G. K~illen and A. Sabry, Kgl. Dan. Vid. Selsk. Mat. Fys. Medd. 29 (1955) No. 17.
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