Keference International Centre for Theoretical

KEFERENCE

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

IC/86/342

INTERNATIONAL THE REGULAR EFFECTIVE ACTION OF GAUGE FIELD THEORY AND QUANTUM GRAVITY ATOMIC ENERGY AGENCY

Nazir S. Baaklini

UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION IC/86/342

International Atomic Energy Agency 0. INTRODUCTION and The effective action of quantum field theory, defined by the functional United Nations Educational Scientific and Cultural Organization integral , is an elegant and potentially, very powerful framework for compu- INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ting quantum effects, in a manner that preserves underlying fundamental symmetries. However, this framework had been utilized mainly in computing effective potentials, and in certain discussions of the divergent counter-terms needed in gauge field THE REGULAR EFFECTIVE ACTION OF GAUGE FIELD THEORY theory and quantum gravity. Whereas it should ~ae very fruitful fo develop expre- AND QUANTUM GRAVITY ssions for the effective action which could have general applicability, computations are still performed using conventional Feynman rules. What is needed is a perturbative formalism for the effective action vhicii applies to a general field Nazir S. Baaklini theory containing Bosonic as well as Fermionic fields, and vhich could be used to International Centre for Theoretical Physics, Trieste, Italy address fundamental issues of quantum field theory and quantum gravity. and Dahr el Chir Science Centre, Dhour el Choueir, Lebanon. On the other hand, our study of the effective action makes impact on two fundamental issues. One of these concerns the treatment of gauge field theory and other singular systems. The other concerns the ultraviolet divergences of quantum field theory and quantum gravity. Quite remarkably, both issues may be ABSTRACT tackled by the same concept, that of a properly metricizea functional space. This concept is realized, in one hand, by using the correct functional space metric We present a perturbative formalism for the effective quantum action which is determined by the Poisson brackets of the canonical field theory. For of a general Femi-Bose field theory. We propose a unitary alternative to the singular systems, the correct functional metric is determined by the modified 11) conventional virtual ghost prescription for handling the functional integral brackets in Dirac's treatment of constraints. This implements unitarity in of gauge fields, "based on the functional space metric that is determined by the perturbative expansion of the effective action. Our approach provides an the Dirac 'brackets of the Canonical theory, A gauge-invariant Gaussian cutoff elegant alternative to the conventional virtual ghost prescription * . is introduced by extending the functional space metric into a regular counter- Moreover, a fundamental gauge-invariant Gaussian cutoff may be implemented part. The regular one-loop contributions of a scalar field to the photon and through a regulating functional metric. The latter replaces the singular Dirae the graviton self-energies are computed. The otherwise logarithmically diver- delta-function which metricizes the virtual functional space by a regular, and gent contributions are found, in our scheme, to be independent of cutoff. gauge-covariant, counterpart.

In See.l, we begin by reviewing the definition of the effective action in terms of the functional integral. The perturbative formalism, for purely Bosonic fields, begins in Sec.2. In Sec.3, we derive expressions for expectation values on the basis of the Gaussian functional integral. These are utilized in HIRAMAHE - TRIESTE deriving the perturbative expressions of the effective action. The latter is October 1986 given to two-loop order In section t. We also prescribe graphical rules for constructing the higher orders. In Sec.5> we extend these results to Fenr.i-Bose theories, through a unified treatment of Fermionic and Bosonic fields. * To be submitted for publication.

-2- We begin our discussion of vector gauge field theory in Sec.6. Our This gives upon functional differetiation, and using Eq,(2), novel scheme of treating the virtual gauge invariance of the functional integral, by utilizing the correct functional metric, is presented in Sec.7. In Sec.8, ve introduce the regulating functional metric.

In Sec,9> we apply our functional metric regularization method to the Eqs.(l), (3), and

In Appendix A, ve develop a formalism for constructing the renormalized Making the shift * •+ f + + in Eq. (5), obtain effective action. Appendices B, C, and D provide supporting formalism and pedagogic applications to the systems of a self-interacting scalar, and quantum electrodyna- (6) mics. In Appendix E, we apply Dirac's Hamiltonian treatment to gauge fields, and provide pointers needed for computing the effective action of vector gauge field theory. Appendices (F, G, H, I, J) provide prerequisite mathematical material 2. ITERATIVE EXPANSION FOR EOSONIC THEORY especially developed for handling the computations of sections 9 and 10. We shall be concerned here with Bosonic field theory. The extension to Ferai-Bose systems will he given in a later section. The effective action will be expanded iteratively in Planck's constant h, 1. THE EFFECTIVE ACTION OF QUANTUM FIELD THEORY: DEFINITION

The generating functional for connected Green'n functions Z(J) is r(*} = rU) + hrU) + « r(+) + ... tT) defined by the functional integral Reinstating h into Eq.(6), we have eiZ(J> = (1) erf*) = /d,e t8) where f represents a quantum field, J. a corresponding external source, and W($) is the classical action functional. Here, we use a compact notation where We shall obtain an expression for rt*) to order h , and then prescribe graphic the index i represents all labels and spacetime arguments, and la = 1. The field rules for higher orders. To this end, we write the Taylor expansion $., being functionally integrated over, will be termed the virtual field. On the other hand, the effective field is defined by

(2) 19) 3J.

The generating functional for proper vertices or the effective action Here (VT, W., W , V , ...) are the classical action functional W($) and its 1 1J 1J id is defined by the Legendre transformation totally symmetric functional derivatives,

(10) rU) = z(J) - (3)

-3- -U-

T This expression is valid provided W is a nonsingular matrix. Me shall return From Eq. (8), obtain immediately that f( if) = W( gauge theory.

(11) By taking successive derivatives of Eqs.(lfi) and (17) with respect to W , obtain

Hence, the term proportional to W. cancels in Eq.(8). The term proportional to T. contributes to r, and so on. (16) I1 2 Utilizing Eos.(9) and (11) in Eq.(8), obtain w w i J (19) (i/h)r e (12)

Here,

[(1/3!) v ] + (13) " Vj*k ijk-M-i . (20) and we have defined the expectation value Here, W~ is the symmetric inverse of W

(21)

From Eq.(12), obtain to order and we have used

= W - (15) fi(TrlnW) = (22)

To evaluate this expression, we need to compute the expectation values of (23) polynomials in $,.

The expectation values vanish for odd powers of . This is shown by taking the expectation value iW < > which vanishes due to the 3. GAUSSIAN INTEGRAL AND EXPECTATION VALUES J-J convergence of the Gaussian integrand. But since W is nonsingular <$ > = 0 The Gaussian integral follows. Then taking successive derivatives of the latter with respect to W , obtain that all odd powers of $. give vanishing expectation values, (16)

<*i»,?k ..-> = 0, (odd power) (2lt) may be expressed by a functional determinant

1/2 <1> = |detw|" = (17)

-5- -6- 5. EXTENSION TO FERMI BOSE THEORY THE TWO_L00P EFFECTIVE ACTION AND HIGHER ORDERS In this section, we extend the preceeding results to theories Now back to the computation of Eq.(15), dropping all odd powers of containing Bosonic (commuting) fields + , and fermionic (anticommiting) fields and using Eqs.(18)-(2O), obtain * . Without loss of generality, both types are taken to be real. We shall give

a unified treatment. Consider the hyperfield $A = $ ). This obeys the gene- r = U + h(l/2)TrlnW - ralized commutation and differentiation rules

(2 ) (28) 5 +A*B

At thi3 point, use the first order result (29)

r = (i/2)Trmw ; C26) (30)

in Eq. (25), and o*btain Here, a(AB) is a sign factor defined by

1 if AB = ij or ig f = C27) -1 if AS = a6. (31) The resulting first and second order quantum contributions in Eq.(27) may be represented, respectively, by irreducible one and two-loop graphs. The We shall utilize the following definitions. three and four-leg vertices correspond to V.,. and W , respectively, while ijk ijkl = o(A). the Joining lines (effective propagators) correspond to W,-1 . Notice that the a(AB + CD) = o(AB)o(CD); (32) —1 —1_ —1 term W..(WT;¥, .,.)(W~ W .) which can be represented by a reducible graph (two IJ K± Kli mn Binj Hote that d(2A) = o (A) = 1. closed loops joined by a line) cancels in the derivation of Eq.(2T). This corres- ponds to the important fact that the effective action should only generate We shall employ totally hypersymmetric tensors T^j, ,. These have one-particle-irreducible vertices. the property of being multiplied by a(AB) under a permutation of any two indices The extension of the preceeding results to higher orders is straight- A and B. To construct a totally hypersymmetric tensor 'r^gf. \ from a general forward. However, the intermediate expressions becoae very lengthy. As a short- tensor 1HOC , construct the totally symmetric tensor and multiply each term by cut, we may turn to graphical rules. For instance, the third order quantum the proper sign factor which results from all permutations leading to the order contributions are represented by irreducible 3-loop graphs {these are depicted by (ABC...). For instance, the 3-rank hypersymmetric tensor Tr.=m is constructed Jackiw in his computational scheme of the effective potential ). The corres- as follows, ponding expressions are obtained by attributing (iW ) to each vertex, (iW~) ljit.. ij T = (l/^SlT +T + T + T +T to each connecting line, an overall factor of (-i), and combinatoric factors. {ABC} v Vl ABC ACB BAC BCA CAB

(1/3! : + "^^BAC '

. + c(AB + AC •

-7- -6- Note the useful notation [...]a in Eq.(33), which means that each term enclosed Hotice that consistently with Eq. (£8), Tr(MN) = Tr(JIM). Eqs. (39)-C*0) agree — 1/2 "by the 'brackets must be multiplied, as in Eq.(3l+), by a sign factor corresponding with the conventional result that the Gaussian integral gives |detw|~ for to permuting indices to the order specified by the first term. purely Bosonic fields, and |detw| for purely Fermionic. Notice that whereas W is hypersymmetric How, corresponding to Eq.(9), write

W (1*1) = W *I3/2(1/3! ) AB

its inverse W satisfies (35)

W W (1.2) where {W, Wft, W^, W^^,, ...) are the classical Ferai-Bose action functional AB BC W( $ .. -T r , then V1* (kk) Proceeding as in the case of purely Bosonic fields, obtain Eq.(15) The solution to Eq. (!*M is with the expression

(1*5) [] = h

where we have defined The Gaussian integral

(38)

Replacing *B in Eq. (l>3) by ?B*C?D> *B'c'D*E*p' etc...; obtain will be expressed by a generalized functional determinant (superdet),

<1> = -Cl/2)TrlnW e (39) where we have used the notation defined after Eqs. {33)-(3M. Also obtain where we have the generalized trace operation (supertrace)

TrM = o(A)M AA"

-9- -10- Now that we have an explicit two-loop expression for the effective AE FB CD AB CE FD action, that is applicable to a general Fenai-Bose theory, we should discuss the construction of the renormalized effective action. Such a construction is AEWFC DB AC DE FB essential for practical applications. In Appendix A, we construct the renonna- laized effective action for a general Fermi-Bose theory. In Appendix B, we apply the renormalized effective action formalism to the simple case of a self-

Again, the expectation values of odd powers are vanishing, by the argument interacting scalar. This application should serve to indicate how to translate used in the purely Bosonie case. the compact index notation into momentusi space. Appendix C supports the computations of Appendices A and B. In Appendix D, we consider quantum electrodynamics, Eq.(l5) may be expressed, using Eqs.(37), <1*5), C»7), and {1*8), and give brief pointers as how to apply the unified Fermi-Bose expressions to a such as theory with Dirac Fermions.

6. THE EFFECTIVE ACTION FOR GAUGE FIELD THEORY

-(1/2X1/3! Consider the classical Lagrangian density for non-Abelian gauge fields,

Now, using the first order result JL = -(l/li)Tr(F F }; (53)

= (i/2)TrlnW; (50) F = 3 A - 3 A - i[A , A ]; A = A J , in Eq.(49), obtain where A belongs to a noneommutative algebra, with generators J , totally antisymmetric structure constants f , and a symmetric metric g , = W + h(i/2)TrlnU + h2[(l/12)o[(B+C+D)

With respect to these transformations, the virtual gauge field transforms 6A = -7 n. (60) like a matter field,

The curvature tensor F transforms like iA = -I[fi, A ]. (69)

F - ei£V e"ifi; <5F = i[[i, F 1. (6l) u\i \i\l 11V y\l The effective gauge invariance will be kept manifest at all stages of computing the effective action. Hence, Eq. (53) is gauge-invariant. We also have virtual gauge invariance with the following transfor- Now making the replacement A •+• A + A , where A is the effective - y y y y mations, parametrized by Si, gauge field and A is the virtual counterpart, obtain

A = -7 (A + A)i) -v (A)SI + i[A , Si], (70) F ->-F + 7 A - v A - 1[A , A ]; (62) yv y\> y V v 11 u v U U

a abC b C a With respect to the virtual gauge transformations, the effective gauge field V A = V AV = (3 l + f A A )J . (63) is neutral, SA = 0.

Using Eq.(62), obtain for Eq.(53) Our iterative expansion of the effective action begins with the Gaussian integral Eqs.(l€)-(17). However, the bilinear kernel W corresponding L •+£ + Tr{(l/2)A 72A - (l/2)7 A 7 A - iA [F , A 1 to Eq. (6I1) is singular, and could not have a perturbatlvely well-defined y y uU^v iiyvv determinant, since in the free-field limit (v •* 3 ), it is proportional to Av"V (61.) (3 n -33). The conventional treatment of this problem is to add a gauge- fixing term to Eq.(6U) proportional to (7 A ) , thus modifying W into a non- where we have dropped the terms linear in the virtual gauge field A , since singular counterpart W. This is accompanied by a modification of the functional they cancel in the effective action Eq. (8), and we have used the follovings, integration measure with a Fade'ev-Popov Jacobian factor |detV (A)7 (A+A)j. The latter is interpreted, in the Feynman graph language, as corresponding to

Tr(vXY) = -Tr(XVY), (65) a virtual (complex and Fermionic) ghost field (u, u), with a Lagrangian density u7 (A)V (A+A)u. Unitarity is supposed to be satisfied by the fact that contri- Tr[A, B]C = TrA[B, C], (66) butions from the extra (longitudinal) degrees of freedom propagated by the nonsingular kernel W are cancelled by the fermionic ghost contributions. [ 7J* = -itF^. •]. (67) V The effects of the above gauge-fixing and measure-modifying prescription could easily be incorporated into the iterative expansion of the effective Notice that the effective action corresponding to Eq.(61») is action. However, we vould like to suggest an alternative scheme which would invariant with respect to two independent gauge transformations. First, we correspond to the canonical quantization of gauge field theory, with Dirac'e have the effective gauge invariance, with parameter multiplet il, modified brackets. This would implement offshell unitarity. To indicate possible

-13- -11*- shortcomings of the conventional approach, and to initiate our alternative The delta-functional constraint in Eq.(73) simply eliminates the scheme, we shall examine the quantization of gauge field theory by Dirac's longitudinal component corresponding to the decomposition method of treating constrained systems, and its significance in defining the correct functional space. A •* AT + AL; AT = AT A , AL = AL A ,

AT = (n - 3 3 /32) AL = 3 3 /32. 7. DIRAC BRACKETS AMD THE FUNCTIONAL SPACE METRIC uv pv )iv ;' uv u v (75)

For a system described, in the Hamiltonian formalism, by dynamical Hence, write variables $- and conjugate momenta TT , and respecting a set of constraints Ka, the functional integral in phase space is defined by W)

(71) Notice that the bilinear kernel in Eq.(76) becomes nonsingular {in the subspace of transverse fields), and that the external source is bound to be Here, i are the derivatives normal to the spacelike surface of quantization, transverse itself. Hence, obtain H is the Hamiltonian, and [Ka, K ] are the Poisson brackets among the const- T raints. To obtain the functional integral in field configuration space, one eiZ(J ) (77) must perform the formidable task of integrating over conjugate ssomenta in the presence of the delta-fun etionals. In conventional theory, one defines the To obtain the propagator functional integral directly in field space, with an integration measure consis- ting of delta-functionals describing the required constraints on the gauge field (78) components, and a corresponding Fade'ev-Popov Jacobian factor. Also, one employs the 't Hooft trick of converting the delta-functionals Into gauge-fixing terms notice that we must replace the usual functional differentiation rule in the argument of the exponential,

3J (x)/3J (y) = n «{x - y), (79) (72)

for ordinary sources, by that corresponding to transverse sources, where f are arbitrary functions. The above treatment of the delta-functionals could perhaps be naive, 3J (x)/3J (y) = A {x - y); A (x - y) = (n -33/3 )4(x - y). (80) and could violate unitarity by missing essential elements of the constrained T system. To illustrate this point, we shall discuss the functional integral of an In the functional space of transverse vector fields, A {x - y) plays the Abelian gauge system, with a simple constraint 3 A = 0, role of a metric. Hence obtain for the propagator, iZ(J) . ,„ , , i[(l/2}A 4 A + A J ] e /drjA (8 A )e u uv V p u , (73) = (i/3 )A (x - y). (81) where Z(J) is a generating functional, A = n 3 - 3 J , and J is an The transversal form of Eq.(8l) ensures the propagation of only transversal external source. The Jacobian factor, here, is irrelevant.

-15- -16-

T degrees of freedom. This point is missed in conventional theory. Hence, in our iterative expansion of the effective action for Let us now consider the same system from an effective action view- gauge field theory, we shall have for the basic Gaussian integral point, and write (86) A \) eU/2) Au [f l uv+ uY (82) where A. are the virtual gauge fields which are projected by the Dirac where Y ($) is a functional of the coupled effective fields. Again, conver- bracket operator. Notice that W is converted into a nonsingular counterpart W ting £q,(82) into an integral over transverse components, obtain in the projected subspace, since the parts which annihilate A drop out. The Gaussian functional integral may then be expressed by the functional (83) determinant of tf in the projected subspace,

Wow this Gaussian functional integral, being defined in the subspace of -l/2)Tr(AlnW) (87) transverse components, may be expressed as a functional determinant in that subspace. Hence, write where the Dirac bracket operator, A^, effects the trace summation.

The argument of the exponential in Eq.(87) gives the one-loop contribution. Higher-order contributions are still given by Eq.(27), however, with the general rule of replacing W by its nonsingular counterpart W, and (8b) each index summation is effected with the projecting metric k. The non-singular kernel W , its inverse W~, and the effective vertices of non-Abelian gauge It is important to notice that the trace summation in Eq. (81*) is properly theory, which constitute the rules of our iterative computation of the effected with the transversal (projecting) metric of the functional subspace. effective action, are given in Appendix E. From these remarks, it should be clear that the proper treatment of the functional integral must respect the metrical structure of the constrained functional space. 8. FUNCTIONAL METRIC REGULAEIZATION OF THE FFECTIVE ACTION On the basis of Dirac's treatment of constrained systems, it is Various regular!zation methods are often employed for handling possible to dispense with the constraints, provided one defines modified the divergences of renormalizable field theory, and eliminating them through Poisson brackets (see Appendix E), which determine the proper functional space the process of renormalization. Conventional regularization techniques (for metric. Hence, the functional integral could be relieved from measure factors instance, higher derivative kernels, dimensional continuation, etc..) do not (hence no ghosts!) and delta-functionals by working directly with the projected introduce a fundamental physical and unitary cutoff into quantum field theory, virtual components A * A. A , vhere A. , is determined by the Dirac brackets. i Ij J lj hence being unsuitable for treating nonrenormalizable theories like quantum These projected components would satisfy the functional differentiation rule gravity. We shall present here our technique of introducing a gauge-invariant and unitary cutoff into the structure of the iterative expansion of the effective (85) = V action. The cutoff is termed unitary in the sense that it does not modify the classical Lagrangian by introducing extra unphysical degrees of freedom. It is The structure of A for non-Abelian gauge fields is given in Appendix E. i j rather part and parcel of the quantization process.

-17- -18- Our scheme "begins with the thought that the ultraviolet divergences 9. THE PHOTON SELF_ENERGY in the effective action are due to the fact that the virtual Hilbert apace, over Consider the U(l) gauge-invariant Lagranglan density for a complex which the functional integral is defined, is metricized by a Dirac delta-function. scalar field, This is reminiscent of the fact that the Poisson brackets of the canonical theory

are defined by a Dirac delta-function. We shall replace the latter by a non- • 2 • 1 = Hi-iU; 7 =3 - ieA . (92) singular counterpart of the form U V V V V

For any operator U(3) acting between $ and + , we shall define its matrix BCx - y) • e~a 5(x - y), (86) elements in momentum space by

vhere a is a fundamental length squared, and V is the Laplacian operator ,q) = fax elpxn(3)e"iqX. (93) which is covariant with respect to the effective gauge fields of the theory. Note that B(x - y) is nonsingular. 7or instance, in the free-field limit, From Eq. (92), we obtain the Icernel needed in computing the one-loop contri-

h -i(x-y)p + op . 2, -(x-yrAa bution to the photon effective action, B(x - y) = /dpe = I(IT /a (69)

W(p,q) = (-n2 + m2)(p,q); Translating the above to the functional integral formalism, ve impose the gauge-covariant two-point function B(x - y) as a metric in the i = i i ; ir=i3+eA (x). space of virtual quantum fields, To express W(p,q) in terms of the photon field in aomentum space, proceed = B(x - y). (90) with the following expressions,

The momentum-space counterpart B(p, q) is simply the matrix element of the A(x) = fir e"lrxA(r); e"irx operator exp(-aV ) between relevant momentum eigenstates.

Since our functional integral is defined over the virtual space metriciaed ty B(x - y), the Gaussian contribution takes the form Sir = in TT + IT fin ; 6ir /efiA (r) • «~ it + if e , V u y V V u V (91) A (r)«A ( (95) Hence, the iterative expansion of the regular effective action would take the same fora as Eq.(27), however, with the only modification that each How writing the Taylor expansion index summation must be effected with the metric S... For gauge field theory, we must effect each index Buntmation with the combination (BAB) of the regula- W(p,q) = W(P.aJo + /drA^rHSW/SA^ rising operator A and the projecting operator B_. In the following sections, we r shall illustrate how to compute the effective action with the regularizing + (l/2)/drdsA)j(r)Av(s)[fi' W/6Aij(r)«Ay(s)](p,q)o, (96) functional metric.