Renormalized Effective Actions in Radially Symmetric Backgrounds: Exact Calculations Versus Approximation Methods

Renormalized Effective Actions in Radially Symmetric Backgrounds: Exact Calculations Versus Approximation Methods

Gerald V. Dunne Department of Physics, University of Connecticut, Storrs, CT 06269, U.S.A.    Jin Hur School of Computational Sciences, Korea Institute for Advanced Study, Seoul 130-012, Korea    Choonkyu Lee Department of Physics and Astronomy and Center for Theoretical Physics
Seoul National University, Seoul 151-742, Korea
Hyunsoo Min Department of Physics, University of Seoul , Seoul 130-743, Korea
Abstract

Our previously-developed calculational method (the partial wave cutoff method) is employed to evaluate explicitly scalar one-loop effective actions in a class of radially symmetric background gauge fields. Our method proves to be particularly effective when it is used in conjunction with a systematic WKB series for the large partial wave contribution to the effective action. By comparing these numerically exact calculations against the predictions based on the large mass expansion and derivative expansion, we discuss the validity ranges of the latter approximation methods.

pacs:
12.38.-t, 11.15.-q, 11.15.Ha

I Introduction

In field theoretic investigations one is often confronted by the rather formidable task to evaluate the one-loop effective action in some nontrivial background field. Until recently, it has not been possible in four spacetime dimensions to evaluate explicitly this renormalized quantity (including its full finite part), unless some very special background is chosen or a priori arbitrary parameters (e.g., mass values) are set to zero. In our recent publications idet (); radial () we made some headway to this old problem by developing an efficient calculational method – a combination of analytic and numerical schemes – for the exact computation of fully renormalized one-loop effective actions in radially symmetric backgrounds. For example, this method was first applied to the accurate determination of QCD single-instanton determinants for arbitrary quark mass values idet (), producing a result that interpolates smoothly between the known analytical massless and heavy quark limits. In Ref.radial () we generalized the calculational procedure to calculate the one-loop effective action in any radially symmetric background, not just an instanton. In the present paper, which is a sequel to Ref.radial (), we present some explicit examples and results (including the numerical contributions) to establish the efficiency and generality of our method. We also examine the validity of often-used approximation methods, such as the large mass expansion and the derivative expansion, compared to numerically exact calculations.

In Ref.radial () we derived some relevant formulas needed in the calculation of the scalar one-loop effective action (in Euclidean spacetime), assuming SU(2) background gauge fields of the form

 (Case 1): Aμ(x)=2ημνaxνf(r)τa2, (1) (Case 2): Aμ(x)=2(ημνi^ui)xνg(r)τ32, (2)

where , , (or ) are the ’t Hooft symbols thooft (), and a unit 3-vector. Case 1 is inherently non-Abelian, while Case 2 has a fixed color direction and so is quasi-Abelian. These backgrounds are characterized by the radial profile functions and , respectively. In each case the spectral problem separates into partial waves due to the spherical symmetry.

Our method has been deliberately developed so that it can accommodate numerical input for and , since this situation often arises in quantum field theory applications. But, to illustrate the method more clearly, we choose here specific Ansätze for the radial profile functions. In Case 1, of ‘non-Abelian type’ (1), we choose the radial function of the form

 f(r)=1r2H(r),H(r)=(r/ρ)2α1+(r/ρ)2α, (3)

with free parameters and (under the regularity restriction ). [The BPST instanton solution belavin () corresponds to the choice ]. In Case 2, of the quasi-Abelian type (2), we choose the radial function of the form

 g(r)=B{1−tanh[β(√Br−ξ0)]}, (4)

with three free parameters , and (all taken to be positive). In the limit , (2) then approaches the case of uniform field strength . For finite , (2) represents a spherical bubble type potential with radius and wall thickness .

In this paper we calculate the renormalized scalar one-loop effective action (including its full finite part) in the gauge field background pertaining to the above two types. The effective action is computed for arbitrary choices of the parameters characterizing the shape of the background field, not relying on the background field being slowly or rapidly varying, or on the particle mass being large or small relative to the scales set by the background field. In performing this analysis, we have found that significantly greater calculational efficiency and precision can be attained by making a systematic use of higher-order quantum mechanical WKB-type approximations wkbpaper () for the large partial-wave contributions to the effective action. This permits the extension of the high partial wave radial-WKB approximation to lower and lower partial waves, and results in dramatic numerical improvements. Finally, we compare our results to the predictions based on the large mass expansion and the derivative expansion. To our knowledge, this kind of genuinely unambiguous comparison in four-dimensional gauge theory has not been made before.

This paper is organized as follows. In Sec. II we give a short outline of our numerically exact calculational scheme and also collect, for later use in the paper, relevant formulas from the large mass expansion and derivative expansion for the scalar one-loop effective action. Our detailed study on the one-loop effective action with non-Abelian-type backgrounds (1) is then presented in Sec. III. This is followed in Sec. IV by the corresponding study with quasi-Abelian backgrounds (2); in this case, the one-loop effective action is essentially that of scalar QED. In Sec. V we conclude with some related discussions and comments.

Ii Brief Summary of our Calculational Scheme and Other Approximation Methods

The calculational method developed in Refs. idet (); radial () can be summarized and streamlined as follows. For scalar fields in radially symmetric non-Abelian background gauge fields , it is possible to express the corresponding (Euclidean) one-loop effective action as a sum of individual partial-wave contributions, i.e., , with the -partial-wave term [including the appropriate degeneracy factor] given by radial functional determinants

 ΓJ(A;m)=ln⎛⎝det(−~D2J+m2)det(−~∂2J+m2)⎞⎠ (5)

or, equivalently, by the proper time representation

 ΓJ(A;m)=−∫∞0dsse−m2s∫∞0drtr{~ΔJ(r,r;s)−~ΔfreeJ(r,r;s)}. (6)

Here, is the scalar mass, () denotes the quadratic radial differential operator relevant to the -partial-wave in the given background (or with set to zero), and , represent the coincidence limits of the related proper-time Green functions specified by the radial ‘heat’ equations:

 [∂s−~D2J(r)]~ΔJ(r,r′;s)=0,(s>0) (7) s→0+:~ΔJ(r,r′;s)⟶δ(r−r′).

(The tildes over differential operators and Green’s functions imply that we are here considering reduced operators/functions after taking out various radial measure factors radial ()). While the quantities are finite individually, the partial wave series diverges and should be renormalized. We thus introduce the intermediate partial wave cutoff (a large, but finite value of ) and express the fully renormalized effective action by two separate terms

 Γren(A;m)=ΓJ≤JL(A;m)+ΓJ>JL(A;m), (8)

where

 ΓJ≤JL(A;m) ≡ JL∑J=0ΓJ(A;m)=JL∑J=0ln⎛⎝det(−~D2J+m2)det(−~∂2J+m2)⎞⎠, (9) ΓJ>JL(A;m) = limΛ→∞⎡⎣−∫∞0dss(e−m2s−e−Λ2s)∑J>JLFJ(s) (10) −1121(4π)2ln(Λ2μ2)∫d4xtr(FμνFμν)]

with

 FJ(s)=∫∞0drtr(~ΔJ(r,r;s)−~ΔfreeJ(r,r;s)). (11)

Notice that the renormalization counter-term is incorporated in the second term ; is the normalization mass, and (with ) denotes the background Yang-Mills field strengths. The first term, , may be evaluated numerically using the Gel’fand Yaglom method gy (); kleinert (); gvd () for one-dimensional functional determinants. As for the second term , analytically developed expressions valid for large enough were used in Refs. idet (); radial (); for this, we made essential use of the quantum mechanical radial WKB expansion and the Euler-Maclaurin summation formula. A crucial observation is that even though the partial-wave sum in (9) is quadratically divergent as we let , this divergence is exactly cancelled by a similar term originating from the second term in (10) in the large -limit. Thus, for the sum of the two terms, we secure a finite result in the large -limit – this is the essence of our numerically accurate scheme for the determination of the renormalized effective action. The “low” partial wave contribution is computed numerically, and the “high” partial wave contribution is computed analytically in the large limit using radial WKB. The ultraviolet cutoff dependence appears in the analytic high partial wave computation, and this allows standard renormalization techniques to be used, leading to the finite renormalized effective action. For details on our renormalization prescription (and on how to go from ours to other prescriptions), see Refs. idet (); radial (); kwon ().

Clearly, numerical efficiency of our calculational scheme hinges on how quickly the limit is attained; that is, on how much we can lower our partial wave cutoff to secure a reliable large- limit value for the above sum of the two terms. (Note that if we were able to calculate both parts, i.e., and exactly, their sum would be independent of the choice of the cutoff value . An interesting example of this situation is the effective action for massless quarks in a single BPST instanton background, where our technique reproduces exactly and analytically idet () ’t Hooft’s result thooft () .). In idet (), we chose the cutoff value to be rather large (of the order of 50), and used the Richardson extrapolation method bender () to reduce the numerical round-off error. There, it was sufficient to use the expression for with or smaller terms suppressed. But, in the present, more extensive, study, we have found that the computation is greatly improved, both in computing time and precision, by including [analytically calculated] higher-order WKB terms in the expression for , so that it becomes valid up to the accuracy. These higher-order WKB terms can be found straightforwardly from the -expansion radial (); lee2 () for the radial proper-time Green function. With this procedure, we were able to ensure that the renormalized effective action we calculate is independent of the (moderately large) -value with relative error of order , which is comparable to the final total error involved in our numerical computation of (9). In the end, we obtain comparable precision with much lower values of , of the order of 10 or 20. This improved precision is demonstrated in sections III and IV for the backgrounds in (1) and (2).

We also give the results of some popular approximation schemes for the one-loop effective action, which are supposed to capture accurate values in certain limits. First, we have the large mass expansion novikov (); kwon () of the scalar one-loop effective action which is obtained most easily with the help of the Schwinger-DeWitt proper-time expansion (or heat kernel expansion). Here, from the renormalized effective action, it is convenient to separate -dependent pieces (and use dimensional considerations) to write

 (Case 1): Γren(A;m)=16ln(μρ)∫d4x(4π)2trF2μν+~Γ(mρ) (12) (Case 2): Γren(A;m)=16ln(μ√B)∫d4x(4π)2trF2μν+~Γ(m√B) (13)

where dimensionless constants are not indicated in an explicit manner. The quantity or , which is independent of , can be expressed for large enough by an asymptotic series of the following form

 (14) ~Γ(m√B)=−16ln(m√B)∫d4x(4π)2trF2μν+∞∑n=3(n−3)!(m2)n−2∫d4x(4π)2tran(x,x). (15)

Here , denote appropriate coefficient functions in the Schwinger-DeWitt expansion: explicitly, for the traces of the - and -terms, we have novikov (); kwon ()

 tra3(x,x) = −16 tr[i215FκλFλμFμκ−120(DκFλμ)(DκFλμ)], (16) tra4(x,x) = 124 tr[−121FκλFλμFμνFνκ+11420FκλFμνFλκFνμ+235FκλFλκFμνFνμ (17) +435FκλFλμFκνFνμ+i635Fκλ(DμFλν)(DμFνκ)+i8105Fκλ(DλFμν)(DκFνμ) +170(DκDλFμν)(DλDκFνμ)],

where and , etc. Note that in these expressions for and , we have not assumed that satisfies the classical equations of motion. We also comment that while this large mass expansion is rather simple to use, it is in fact an asymptotic expansion, and so its regime of useful applicability is restricted to the mass being large relative to or , respectively. See Sec. III and Sec. IV for comparisons of the large mass expansions (14) and (15) with our exact numerical answers.

There is another well-known approximation method to the one-loop effective action, the derivative expansion. The leading order of the derivative expansion corresponds to using the Euler-Heisenberg constant field result heisenberg (); duff (); dunne-review (), but substituting the inhomogeneous fields for the homogeneous ones used to compute the Euler-Heisenberg effective action. This approximation is very simple to implement, and is expected to be a good approximation when the spacetime variation in the background gauge field strengths is sufficiently ‘slow’ so that we may regard their derivatives as small terms in the effective action. Subleading derivative expansion contributions can also be computed, but we will not consider them here. A systematic study of the validity range of this method is still lacking (although the Borel summability properties of the derivative expansion have been analyzed in a nontrivial soluble inhomogeneous QED example in cangemi ()). We remark here that if the background gauge fields are genuinely non-Abelian (as in our Case 1), the convergence character of the related expansion – the so-called covariant derivative expansion leutwyler (); yildiz (); gargett (); salcedo () – is less certain. In this paper we restrict ourselves to applying the derivative expansion with our quasi-Abelian backgrounds (2) only. In that case, the leading term in the derivative expansion is given by the Euler-Heisenberg formula heisenberg (); duff (); dunne-review ()

 ~ΓDE(m√B)= − ∫drr34∫∞0dss3e−m2s(E1ssinh(E1s)E2ssinh(E2s)−1+s26(E21+E22)) (18) − 16ln(m√B)∫d4x(4π)2trF2μν,

where and denote four eigenvalues of the matrix , with

 E1=12√F−√F2−G2,E2=12√F+√F2−G2, (19)

where

 F=12trFμνFμν,G=14trϵμνλκFμνFλκ. (20)

Consult Sec. IV to see how the predictions based on this method fare against the accurate numerical calculations.

Iii Non-Abelian backgrounds

iii.1 Properties of the Background Fields

We consider here a family of radial background fields described by (1) and (3), which resemble the single instanton configuration. The parameter is chosen to be in the range , so that is well-behaved at the origin . When takes a negative value it is convenient to cast the function in the form

 H(r)=11+(r/ρ)2|α|. (21)

Note that while our configuration with is simply the single instanton solution in the regular gauge idet (), by choosing the single anti-instanton solution in the singular gauge is also obtained. In Fig. 1, we have plotted the shape of the background function for several values of ’s.

For the gauge field (1), the corresponding field strength tensor is

 Fμν=τa2(4ημνaH(H−1)r2+2(xμηνλa−xνημλa)xλr4[2H(H−1)+rH′]), (22)

where . Alternatively, inserting the expression (3) for , we find

 Fμν=τa2(−ημνa−(xμηνλa−xνημλa)xλr2(1−α))4(rρ)2αr2(ρ2α+r2α)2. (23)

Note that the field strength tensor has a singularity at when , and the second part of (23), the part proportional to , vanishes if (i.e. for the single instanton solution). For these instanton-like radial background fields the corresponding classical action is readily evaluated:

 12∫d4xtrF2μν = 12π2∫∞0drr[r2(H′)2+4H2(H−1)2] (24) = 4π21+α2|α|

Note that this result does not depend on the scale parameter , and has a minimum value when , i.e., for the single instanton or single anti-instanton solution.

For the general vector potential considered here, the topological winding number is

 132π2ϵμνλτ∫d4xtrFμνFλτ=−∫∞0drddr(2H3−3H2)=±1. (25)

This means that all the configurations corresponding to positive values of belong to the class of winding number , and those corresponding to negative values of to the class of winding number . Further, notice that self-dual configurations occur when , which means , corresponding to the BPST instanton or anti-instanton for which .

iii.2 Large Mass Expansion

Before delving into the exact calculation of the one-loop effective action, we first present the result of the large mass expansion. In our background fields we find for the traces of leading coefficient functions and (see (16) and (17)), the following explicit results:

 ∫d4xtra3 = ∫∞0drρ2r6α−315(1+r2α)6[(10−13α2−33α4) (26) +r2α(5+20α+22α2+28α3+21α4)+(α→−α)], ∫d4xtra4 = (27) −8r2α(−91−273α−21α2+309α3+618α4+1164α5+694α6) +12(1337−5250α2−3399α4+11992α6)+(α→−α)].

Then, after performing the -integration, the large-mass expansion for the one-loop effective action is seen to take the form (we here give the result for the quantity , introduced in (12))

 ~ΓLM(mρ) = −(1+α2)12|α|ln(mρ)+π(5−10α2+11α4−6α8)1800α6sin(π/α)1(mρ)2+(α2−4)(α2−1) (28) ×π(−140+35α2−378α4−317α6+120α8)88200α8sin(2π/α)1(mρ)4+⋯.

Note that, for (or ), taking the limit (or ) in the right hand side of (28) should be understood. This large mass expansion result (28) will be compared with the numerically determined effective action later.

We make a comment on the length scale parameter here. The modified effective action does not depend on the renormalization mass scale , and so is a function only of the dimensionless combination . We may then set the size parameter during the calculation, without loss of generality, and readily restore it in the final result.

iii.3 Numerically Accurate Calculation of the Lower Angular Momentum Part

We now turn to our accurate effective action calculation based on (8). First consider the lower angular momentum part , given in (9). Note that, in the present backgrounds, the partial waves are specified by the quantum numbers , as described in detail in Ref. radial (). We are working with isospin , so . Using the notation of radial (), the radial Hamiltonian, representing in the given partial wave sector, assumes the form

 Hl,j≡−D2(l,j)=−∂2∂r2−3r∂∂r+Vl,j (29)

with

 Vl,j(r)=4l(l+1)r2+[j(j+1)−l(l+1)−34]4H(r)r2+3H2(r)r2, (30)

while in the absence of the background field

 Hfreel≡−∂2l=−∂2∂r2−3r∂∂r+4l(l+1)r2. (31)

The radial Hamiltonian is independent of the quantum numbers and ; this introduces the degeneracy factor in the partial wave sum below.

Having identified the relevant quantum numbers, the lower angular momentum part can be written as (here, serves as our partial wave cutoff)

 ΓJ≤JL(A;m)=L∑l=0(2l+1)l+1/2∑j=l−1/2(2j+1)ln(det(Hl,j+m2)det(Hfreel+m2)), (32)

The ratio of two determinants in (32) is determined, according to the Gel’fand-Yaglom method gy (); kleinert (); gvd (), by the ratio of the asymptotic values of two wave functions as

 det(Hl,j+m2)det(Hfreel+m2)=limR→∞(ψl,j(R)ψfreel(R)). (33)

Here and denote the solutions to the radial differential equations

 (Hl,j+m2)ψl,j(r) = 0, (34) (Hfreel+m2)ψfreel(r) = 0, (35)

which have the same small- behaviors, i.e.,

 r→0:ψl,j(r)∼rl,ψfreel(r)∼rl. (36)

Note that the solution to (35), which is the modified Bessel function

 ψfreel(r)=I2l+1(mr)r, (37)

grows exponentially fast at large , as do the numerical solutions to (34) for the operators . Thus, numerically, it is advantageous to consider the ratio, , which stays finite for all . In fact, since we compute the logarithm of the determinant, we can directly consider the logarithm of the ratio:

 S(l,j)(r)=lnψ(l,j)(r)ψfree(l)(r), (38)

which also has a finite value in the large limit. This function satisfies the differential equation

 d2S(l,j)dr2+(dS(l,j)dr)2+(1r+2mI′2l+1(mr)I2l+1(mr))dS(l,j)dr=U(l,j)(r), (39) U(l,j)(r)=Vl,j−4l(l+1)r2 U(l,j)(r)=4j(j+1)−4l(l+1)−3r2(1+r2α)+3r2(1+r2α)2, (40)

under the initial value boundary conditions

 S(l,j)(r=0)=0,S′(l,j)(r=0)=0. (41)

Noting that the eigenvalues of the total angular momentum equal for a given value of , it is convenient to combine the contributions and , which come with the same degeneracy factor . With this understanding, it is possible to express the amplitude (32) in the form

 ΓJ≤JL(A;m) = L∑l=0,12,1,…(2l+1)(2l+2)P(l), (42) P(l) ≡ S(l,l+12)(∞)+S(l+12,l)(∞). (43)

Here and denote the asymptotic (i.e., ) limits to the solutions of the differential equations in (39) with the potentials

 Vl,l+1/2(r)=4l(l+1)r2+(4l+3)H(r)r2+3H(r)(H(r)−1)r2, (44)

and

 Vl+1/2,l(r)=4l(l+1)r2+(4l+3)(1−H(r))r2+3(H(r)−1)H(r)r2, (45)

respectively.

Note that the potentials and have the same form except that in one expression gets replaced by in the other. For given in (3), we have , while ; i.e., the same expression as only with in the latter replaced by . Therefore, if we consider the effective action with the background parameter replaced by , the only change is that two potentials and are interchanged, and so the two quantities and in (43) are also interchanged. This shows that each partial wave contribution to the effective action with the background parameter is the same as the one with the parameter and thus two effective actions with and with have the same value. (This was true for the! classical action also). Similar behaviors, concerning the cases with , were observed already in Ref. idet (). Based on this observation, consideration of the effective action for positive values of is sufficient.

The ODE system specified by (39)-(41) can easily be solved numerically. (With and , analytic solution to this equation was found in idet ()). In Fig. 2 we plot the solutions for a few cases. It clearly shows that the solutions approach constant values in the limit.

In Fig. 3 we plot partial wave contributions with for and . Note that when the angular momentum becomes large. Since the degeneracy factor is quadratic, this implies that in (42) behaves as in the large limit. This divergent behavior will be canceled when we add the higher angular momentum contribution.

iii.4 WKB Calculation of the Higher Angular Momentum Part

We now calculate the higher angular momentum part , given in (10). This cannot be computed numerically (as we have done for ) because very large partial wave contributions lead naively to a divergent result and require careful renormalization to ensure a finite result. The large partial wave contribution depends on the regulating cutoff , whose effect must be identified and isolated for renormalization; and this cannot easily be done numerically. However, this quantity (incorporating renormalization) can be calculated analytically in a WKB-type asymptotic series, assuming that the partial wave cutoff is large enough. Here the higher angular momentum sum of the partial wave heat kernel, with given by (11), may be described more explicitly by the form

 ∑J=JL+12FJ(s) = ∫∞0dr∞∑l=L+12(2l+1)(2l+2){~Δ(l,l+12)(r,r;s)+~Δ(l+12,j=l)(r,r;s) (46) −~Δfree(l)(r,r;s)−~Δfree(l+12)(r,r;s)}.

Now, as explained in radial (), we may use the expansion for the modified radial proper-time Green function when is large. When is expanded in terms with increasing number of derivatives of the potential, the scaling is such that when the sum is approximated by the Euler-Maclaurin formula, this generates the large expansion. For a generic radial potential , this expansion has the following form:

 ~Δ(r,r;s) = 1√4πse−sV(r){1+(112s3(V′)2−16s2V′′) (47) + (1288(V′)4s6−11360(V′)2V′′s5+140(V′′)2s4+130V′V(3)s4−160V(4)s3) + ((V′)6s910368−17(V′)4V′′s88640+83(V′V′′)2s710080+1252(V′)3V(3)s7−6115120(V′′)3s6 −432520V′V′′V(3)s6−51008(V′)2V(4)s6+235040(V(3))2s5+192520V′′V(4)s5 +1280V′V(5)s5−1840V(6)s4)+O(1l8)}.

Note that the terms are collected according to the total number of derivatives on . [In addition to the terms already calculated in (3.16) of radial (), we have here included some higher order terms as well because they will be useful in finding the large expansion of ]. The large- series expression for can then be found by inserting (47) into (46) with the generic potential replaced by the potential or in (44) or (45). The summation over in (46) can be performed using the Euler-Maclaurin summation method. The result is tantamount to the systematic WKB series, as was shown in Ref. radial (). Since this procedure was described already in Appendix C of radial (), we will not repeat it here. The final result in the present potential can be presented as a series of the form

 ΓJ>JL = ∫∞0dr{Q2(r)L2+Q1(r)L+Qlog(r)lnL+Q0(r)+Q−1(r)1L+⋯}, (48)

where

 Q2(r)=8H(H−1)r√~r2+4, (49) Q1(r)=8(3~r2+8)r(~r2+4)3/2H(H−1), (50) Qlog(r)=−14r(4H2(H−1)2+r2H′2), (51) Q0(r)=16r(~r2+4)7/2[4(3~r6+49~r4+236~r2+352)H2(H−1)2 −6(22~r6+157~r4+352~r2+384)H(H−1)+16r(~r4+5~r2+4)(2H−1)H′ +r2(~r2+4)2{(3~r2+8)H′2−4(2H−1)H′′}]−Qlog(r)ln(μr√~r2+4+2), (52) Q−1(r)=1r(~r2+4)9/2[2(9~r6+36~r4−64~r2−256)H2(H−1)2 +(−6~r8+25~r6+368~r4+128~r2)H(H−1)+8r~r2(~r4+3~r2−4)(2H−1)H′ −2r2(~r2+4)2{2(~r2+2)H′2