Functional Determinants in Quantum Field Theory 2

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where is some differential operator. Clearly, to be able to vary the determinant D efficiently, one needs a clever way to evaluate the determinant. Lattice gauge theory: integrating out quarks leads to fermion determinant factors • 4 ¯ ψ ψ¯ e− d x ψ[D/−m]ψ = det (D/ m) (3) D D − Z R which are needed for dynamical fermion computations in QCD. There are still fundamental challenges in evaluating such determinants within the Monte Carlo approach to lattice gauge theory, especially at finite density and temperature. Faddeev-Popov determinants: the determinant of the Faddeev-Popov operator • plays an important role in gauge fixing and confinement. Mathematical physics: the determinant of a partial differential operator, such as a • Dirac or Klein-Gordon operator, encodes interesting spectral information.

2. What we know

Exact results for functional determinants exist only for certain special cases. For example, the gauge theory effective action [the logarithm of the determinant of a Dirac or Klein-Gordon operator] for background fields with (covariantly) constant field strength can be expressed as simple proper-time integrals, building on the seminal work of Heisenberg and Euler [1, 2, 3]. Mathematically, these expressions are closely related to the multiple zeta and gamma functions [4]. This constant curvature case is soluble because we can choose a Fock-Schwinger gauge with A = 1 F x , in which case µ − 2 µν ν the spectral problem decomposes into a set of harmonic oscillator problems, for which the spectral information is very simple. Similarly, in gravitational problems, for special spaces such as harmonic spaces or symmetric spaces, the spectral problem is sufficiently separable and simple that the determinant can be evaluated [5]. For example, the determinant of Dirac or Klein-Gordon operators on spheres, cones and tori are known in quite explicit form [5, 6]. Remarkably, this type of analysis can also be extended to general Riemann surfaces [7, 8], with important applications to worldsheet problems in string theory and with beautiful applications in number theory and differential geometry. Things become more interesting when including general non-constant curvatures [either gauge or gravitational]. Then there are a few especially symmetric cases that are still soluble, but they are all one-dimensional. For example, in gauge theory, if the field strength points in a particular fixed direction, and has a magnitude that is a function of 2 only one coordinate [say xµ], and is of the form F sech (k xµ), then the corresponding [Dirac or Klein-Gordon] spectral problem is hypergeometric, and the determinant can be computed explicitly. Also, for massless quarks in an instanton background, the fermion determinant is known. Such a computation was pioneered by ’t Hooft for the single su(2) instanton [9], but has since been generalized to instantons of full ADHM form [10], to finite temperature [11], and to calorons with nontrivial holonomy [12]. These computations all rely on the fact that the Green’s functions for massless particles in Functional Determinants in Quantum Field Theory 3

an instanton background are known explicitly. The generalization to nonzero masses is nontrivial, and is discussed below in Section 8. Some general bounds are known, based on lattice regularizations and generalizations of Kato’s inequality for Dirac and Klein-Gordon operators. For example [13, 14],

2 2 2 2 det [m D/ ] det m Dµ − 1 ; − 1 . (4) 2 2 h i det [m ∂/ ] ≤ det m2 ∂2 ≥ − µ − h i While very general, these bounds are still quite weak, and do not contain information about the phase of the determinant, which can be physically significant. Tighter bounds have been found recently in QED [15], but general results are still relatively limited. For Schrödinger operators with general potentials, or Dirac/Klein-Gordon operators with arbitrary gauge and/or gravitational backgrounds, we rely heavily on approximate methods. One such approximation is the heavy mass expansion. Define ∞ ds 2 ln det m2 + = tr ln m2 + = e−m s tr e−s D (5) D D − 0 s h i h i Z n o Then an inverse mass expansion follows (after renormalization) from the small s asymptotic expansion of the heat kernel operator [16] 1 ∞ tr e−s D sk a [ ] . (6) ∼ (4πs)d/2 k D n o kX=0 Here the a [ ] are known functionals of the potentials appearing in . Similarly, a k D D derivative expansion can be derived by expanding the heat kernel trace tr e−s D about the soluble constant background case, in powers of derivatives of the background.n o This corresponds to resumming all non-derivative terms in the inverse mass expansion. While very general, these expansions are asymptotic, with higher terms becoming rapidly unwieldy, and so have a somewhat limited range of application. Another approach is to define the functional determinant via a zeta function 1 ζ(s) , (7) ≡ λs Xλ where the sum is over the spectrum of the relevant differential operator. Then, by the ′ formal manipulations ζ (0) = λ ln λ = ln ( λ λ), one defines the determinant as ′ − − det = e−ζ (0) . P Q (8) The problem becomes one of analytically continuing ζ(s) from the region in which it converges [typically (s) > d/2] to the neighbourhood of s = 0. For example, in a ℜ radially separable problem, one might estimate the spectrum using WKB phase shifts 1 dδ [and the corresponding spectral function ρ(k) = π dk ], thereby obtaining information about the zeta function. The zeta function approach is complementary to the so-called replica method, where one defines the logarithm of an operator by considering positive integer powers n of the operator, and then analytically continues to n = 0. There are also non-trivial interesting one dimensional quantum mechanical problems for which the determinant can be computed from the zeta function [17]. Functional Determinants in Quantum Field Theory 4

There are several special features of one-dimensional problems that deserve mention. First, the issue of renormalization does not really enter in 1d. Second, the theory of inverse scattering allows us to characterize a potential in terms of the associated scattering data. This is especially useful for solving variational and gap equations, as the variational problem can then be reformulated in terms of the scattering data, which is more direct. Beautiful applications of this idea occur in interacting 1+1 dimensional QFT’s such as the Gross-Neveu model, Sine-Gordon model and their generalizations [18, 19]. [Inverse scattering has been generalized in higher dimensions for radial problems [20]]. A third special feature of 1d problems is the Gel’fand Yaglom theorem [21, 22], which provides a way to compute the determinant of an ordinary diﬀerential operator without computing its eigenvalues. The basic idea is that in 1d one can deﬁne the Green’s function without using an eigenfunction expansion, simply from the product of two independent solutions.

3. Gel’fand-Yaglom theorem: ordinary diﬀerential operators

The simplest form of this result can be stated as follows [21, 22, 23, 24, 25, 26]. Consider a Schrödinger operator on the interval x [0, L], with Dirichlet boundary conditions: ∈ d2 + V (x) ψ(x)= λ ψ(x) ; ψ(0) = ψ(L)=0 (9) "−dx2 # Then to compute the determinant it is in fact not necessary to compute the infinite discrete set of eigenvalues λ ,λ ,... . Instead, solve the related initial value problem { 1 2 } d2 + V (x) φ(x)=0 ; φ(0)=0 ; φ′(0)=1 (10) "−dx2 # Then d2 det + V (x) = φ(L) . (11) "−dx2 # It is worth pausing to appreciate the simplicity of this result, as well as its practical utility. It is straightforward to implement numerically. The determinant follows, without having to compute any eigenvalues, let alone multiply them all together! Actually, (11) is not quite right, as we can only define the ratio of two determinants [27], and (11) should be understood in this sense. Often in physical applications one is computing the determinant relative to the corresponding determinant for the free operator, for which the initial value problem is usually known in closed form [in this case it is numerically better to solve directly the initial value problem for the ratio]. As an illustration, consider the following simple example of the massive Helmholtz operator 2 d2 + m2 . The Dirichlet spectrum is λ = m2 + nπ , so from the eigenvalues: − dx2 n L h i 2 2 2 nπ d 2 ∞ m + ∞ 2 det 2 + m L mL sinh(m L) − dx 2 = 2 = 1+ = (12) h d i nπ n π m L det 2 n=1 n=1 " # − dx Y L Y h i Functional Determinants in Quantum Field Theory 5

On the other hand, if we use the Gel’fand-Yaglom result, we solve the initial value problems: d2 + m2 φ = 0, to find φ(x)= sinh(mx) ; and d2 φ = 0, to find φ (x)= x − dx2 m − dx2 0 0 for the masslessh operator.i Then (11) implies d2 2 det 2 + m φ(L) sinh(m L) − dx = = (13) h d2 i det φ0(L) m L − dx2 These clearly agree,h buti the point is that if m2 were replaced by a nontrivial potential V (x), the first approach, from the eigenvalues, would be extremely difficult, while the Gel’fand-Yaglom approach is still easy. As another example, consider the Pöschl- 2 Teller system with H = [ d + m2 j(j + 1) sech2 x], with integer j. This is a − dx2 − reflectionless potential with j bound states at E = m2 l2, (l = 1,...j), and phase j − shift δ(k)=2 l=1 arctan(l/k). The determinant follows directly from the spectrum as d2 2 2 j ∞ det[ P2 + m j(j + 1)sech x] dk dδ ln − dx − = ln(m2 l2)+ ln(m2 + k2)  d2 2  det[ 2 + m ] − 0 π dk − dx Xl=1 Z   Γ(m)Γ(m + 1) = ln (14) Γ(m j)Γ(m + j + 1) − On the other hand, the solutions to the Gel’fand-Yaglom initial value problems (10) are j−m+2 j−m+1 1 Γ Γ φ(x) = 2 2 P m( th L) Qm(th x) Qm( th L)P m(th x) 4m j+m+2 j+m+1 j − j − j − j Γ 2 Γ 2 1 φ (x)= (cosh(m L) sinh(m x) + sinh(m L) cosh(m x)) (15) (j=0) m on the interval x [ L, L]. The P m and Qm are Legendre functions. Then ∈ − j j d2 2 2 det[ 2 + m j(j + 1)sech x] φ(L) Γ(m)Γ(m + 1) − dx − = lim = (16) d2 2 →∞ det[ + m ] L φ(j=0)(L) Γ(m j)Γ(m + j + 1) − dx2 − Similarly, one can define a general reflectionless potential with N bound states at 2 2 2 d2 E = m κi , (i =1,...N), by V (x)= m 2 dx2 ln det A(x) where A(x) is the N N − c c − − × matrix A = δ + i j e (κi+κj )x. The c are (constant) moduli which affect the shape ij ij κi+κj i N of the potential, but not the spectrum. From the phase shift δ(k)=2 i=1 arctan(κi/k), one finds the determinant P d2 2 N det[ 2 + m + V (x)] m − dx = exp 2 arctanh (17) d2 2 det[ + m ] "− κi # − dx2 Xi=1 It is instructive to verify this numerically for various potentials using (10) and (11). The basic result (11) generalizes in several ways. First, it generalizes to other boundary conditions [24, 26, 28], characterized by two 2 2 matrices M and N as × ψ(0) ψ(L) M ′ + N ′ =0 (18) ψ (0) ! ψ (L) ! 2 Construct two independent solutions u of d + V (x) φ(x) = 0 such that (1),(2) − dx2 ′ h i u(1)(0)=1 ; u(1)(0) = 0 ′ u(2)(0)=0 ; u(2)(0)=1 (19) Functional Determinants in Quantum Field Theory 6

Then the inﬁnite dimensional determinant reduces to a simple 2 2 determinant × 2 d u(1)(L) u(2)(L) × det 2 + V (x) = det2 2 M + N ′ ′ (20) "−dx # " u(1)(L) u(2)(L) !# For example, 1 0 0 0 Dirichlet : M = , N = det = u(2)(L) 0 0 ! 1 0 ! ⇒

0 0 0 1 ′ Neumann : M = , N = det = u(1)(L) 0 1 ! 0 0 ! ⇒ Periodic (P) : M = 1, N = 1 det = 2 u (L)+ u′ (L) − ⇒ − (1) (2) ′ Antiperiodic (AP) : M = 1, N = 1 det = 2 + u (L)+ u (L) ⇒ (1) (2) ′ In the P/AP cases we recognize (u(1)(L)+ u(2)(L)) as the Floquet-Bloch discriminant, which takes values 2 for P/AP eigenfunctions. Second, these results generalize ± straightforwardly to coupled systems of ODE’s [24, 28], and also to linear ODE’s of any order [29]. Third, they generalize to Sturm-Liouville problems [28]. For example, consider the radial operator in d dim., with Dirichlet b.c.’s − − d2 l + d 3 l + d 1 + 2 2 + V (r) . (21) M(l) ≡ −dr2 r2

Define the function φ(l)(r) as the solution to the radial initial value problem + m2 φ (r)=0 ; φ (r) rl+(d−1)/2 , as r 0 . (22) M(l) (l) (l) ∼ → Then (this shouldh also bei understood as applying to ratios of determinants) det + V (r) = φ (R) . (23) M(l) (l) As an explicit example,h consideri the 2d radial Helmholtz problem, with eigenvalues in 2 2 j(l),n terms of the zeros j(l),n of the Jl Bessel function: λ(l),n = m + R . Then j 2 2 ∞ m2 + (l),n ∞ 2 det (l) + m R m R M = = 1+ (24) h i j 2   det (l) n=1 (l),n n=1 j(l),n ! M Y R Y h i   On the other hand, if we use the Gel’fand-Yaglom result (23), we readily solve the relevant initial value problems to find φ(l) in terms of the modified Bessel function Il:

φ(l)(R) l! Il(m R) free = l . (25) φ(l) (R) m R 2 The agreement between (24) and (25) expresses the product formula for Il(x).

4. Gel’fand-Yaglom theorem: partial diﬀerential operators?

In generalizing from ordinary to partial diﬀerential operators, it is natural to consider ﬁrst the case of separable operators. But the na¨ıve extension fails. Formally, we simply take a product over the radial determinants [each of which is evaluated as above] for all Functional Determinants in Quantum Field Theory 7

possible angular momenta, with the appropriate degeneracies. However, this product diverges [24]. For example, for the 2d radial Helmholtz system (25)

2 det (l) + m l!2l I (m R) 1 ln M = ln l O , l (26)  deth i (m R)l ! ∼ l →∞ M(l)   The degeneracy factor ish 1 foril = 0, and 2 for all l 1, so the angular momentum ≥ sum over l diverges. This should not be so surprising from a physical point of view, as we expect to require renormalization in dimension higher than 1. To formulate the problem more precisely, consider the radially separable operators = ∆+ V (r), and M − free = ∆, where ∆ is the Laplace operator in IRd, and V (r) is a radial potential M − vanishing at infinity as r−2−ǫ for d = 2 and d = 3, and as r−4−ǫ for d = 4. Since V = V (r), we can separate variables and consider the Schrödinger-like radial operator in (21). For dimension d 2, the radial eigenfunctions ψ have degeneracy M(l) ≥ (l) (2l + d 2)(l + d 3)! deg(l; d) − − . (27) ≡ l!(d 2)! − Formally, we write ∞ 2 det [ + m2] det (l) + m ln M = deg(l; d) ln M . (28) det [ free + m2]! det h free + m2i  M Xl=0 M(l)  h i Each term in the sum is computed straightforwardly using (23), but the l sum is divergent. However, this divergence can be understood physically, leading to a finite and renormalized determinant ratio [30]: ∞ det[ + m2] φ ( ) ∞ φ ( ) drrV (r) ln M = ln (0) ∞ + 2 ln (l) ∞ 0 det[ free + m2]! φfree( )  φfree( ) − 2l  d=2 (0) l=1  (l) R  M ∞ ∞ X ∞   µr   + drrV ln + γ  Z0 2 2 ∞ ∞ det[ + m ] φ(l)( ) 0 drrV (r) ln M = (2l + 1) ln ∞ 1 det[ free + m2]!  φfree( ) − 2(l + )  d=3 l=0  (l) R 2  M X ∞ det[ + m2]   ln M =   det[ free + m2]! d=4 M ∞ ∞ ∞ 3 2 2 φ(l)( ) 0 drrV (r) 0 drr V (V +2m ) (l + 1) ln free ∞ + 3  φ ( ) − R 2(l + 1) R 8(l + 1)  Xl=0  (l) ∞  1 ∞   µr  drr3 V (V +2m2) ln + γ +1 . (29) − 8 Z0 2 Here γ is Euler’s constant, and µ is a renormalization scale, which is essential for physical applications. A conventional renormalization choice is to take µ = m in (29). In each of (29), the sum over l is convergent once the indicated subtractions are made. The

function φ(l)(r) is deﬁned in (22). Notice that the results (29) state once again that the determinant is determined by the boundary values of solutions of [ + m2] φ = 0, with the only additional information M Functional Determinants in Quantum Field Theory 8

being a ﬁnite number of integrals involving the potential V (r). We also stress the computational simplicity of (29); no phase shifts or eigenvalues need be computed to evaluate the renormalized determinant. These results have been derived [30] using the zeta function formalism. Scattering theory [31] and standard contour manipulations [28, 32] yield an integral representation

[valid for (s) > d/2] of the zeta function in terms of the Jost function fl(ik): ℜ ∞ sin πs ∞ ∂ ζ(s)= deg(l; d) dk [k2 m2]−s ln f (ik) . (30) π − ∂k l Xl=0 mZ The technical problem is the analytic continuation of (30) to a neighborhood about s = 0. This analytic continuation relies on the uniform asymptotic behavior of the Jost

function fl(ik), which follows from standard results in scattering theory [31]: ∞ ∞ r ln f (ik)= drrV (r)K (kr)I (kr) drrV (r)K2(kr) dr′ r′ V (r′)I2(kr′)+ (V 3) l ν ν − ν ν O Z0 Z0 Z0 where ν l + d 1. Then the calculation of the asymptotics of the Jost function ≡ 2 − reduces to the known uniform asymptotics of the modiﬁed Bessel functions Kν and Iν. asym 2 Using these asymptotics, we can deﬁne ln fl (ik) as the O(V ) and O(V ) parts of this uniform asymptotic expansion, and then by construction, ∞ ζ′ (0) deg(l; d) [ln f (im) ln f asym(im)] . (31) f ≡ − l − l Xl=0 is now convergent. Adding back the subtracted terms in a dimensional regularization scheme, leads [30] to the results in (29).

5. Comparison With Feynman Diagram Approach

These determinants can also be derived in a Feynman diagrammatic approach, where the perturbative expansion in powers of the potential V is det [ + m2] ∞ ( 1)k+1 ln M − A(k) det [ free + m2]! ≡ k M kX=1 1 1 = + + ... , (32) − 2 3 Here the solid dots denote insertions of the potential V . If the potential is radial, then with dimensional regularization one can express the renormalized 4d determinant as [33] 2 ∞ ∞ det [ + m ] 2 ln M = (l + 1) ln fl(im) drrV (r)Kl+1(mr) Il+1(mr) " det [ free + m2]!# − 0 M d=4 Xl=0 Z ∞ r 2 ′ ′ ′ 2 ′ (1) 1 (2) + drrV (r)Kl+1(mr) dr r V (r )Il+1(mr ) + Afin Afin (33) Z0 Z0 − 2 where the finite contributions from the first and second order Feynman diagrams in the MS scheme are [33] (note the small typo in equation (4.32) of [33]): 2 ∞ (1) m 3 Afin = drr V (r) − 8 Z0 Functional Determinants in Quantum Field Theory 9

∞ 2 √ 2 2 √ 2 2 (2) 1 3 ˜ 4m + q 4m + q + q Aﬁn = dq q V (q) 2 ln (34) 128π4 0 " − q √4m2 + q2 q !# Z − Here V˜ (q) is the 4d Fourier transform of the radial potential V (r). These expressions (33) and (34) look quite diﬀerent from (29), but in fact they can be shown to be completely equivalent, using some Bessel identities [30]. However, it is clear that (29) is much easier to evaluate, especially if (as often happens) V (r) is known only numerically.

6. Removing Zero Modes

In certain quantum field theory applications the determinant may have zero modes, and correspondingly one is actually interested in computing the determinant with these zero modes removed, together with a collective coordinate factor [23]. For example, consider the case of a self-interacting scalar field theory in d dim Euclidean d space with Euclidean action S[Φ] = d x [∂µΦ∂µΦ+ U(Φ)], where there is a [radially symmetric] classical solution Φ (r) solving ∆Φ = dU , with Φ′(0) = 0, and Φ 0 as cl R dΦ → r . The fluctuation operator about this classical solution is a radial operator, with →∞ d2U 2 V (r) = dΦ2 m . In the l = 1 sector the corresponding radial operator Φ=Φcl(r) − hash a di-fold degenerate zero mode associated with translational invariance: M(l=1) Φ′ (r) ψ (r)= cl . (35) zero Φ′′(0) Including the collective coordinate contribution [23] due to translational invariance, the required determinant factor is [the prime denotes exclusion of zero mode(s)] −1/2 d/2 ′ 2 S[Φ ] det (l=1) + m cl M . (36) 2π !  det h free + m2 i M(l=1)   Since the determinant is expressedh in terms ofi a solution to an ODE, we can use simple ODE theory to show that (generalizing the d = 1 [34] and d = 4 [35] analyses) ′ det + m2 ∞ d−1 2 d (l=1) 0 drr ψzero(r) M = − ′ , (37) h free 2 i d 1 free det + m " limR→∞ R( 2R ψzero(R) ψzero(R))# M(l=1) − free h d/2 d i d/2−1 Here, ψzero(r) = 2 Γ( 2 + 1) Id/2(r)/r , and the classical solution behaves as Kd/2−1(R) ∞ ∞ Φcl(R) Φ Rd/2−1 , as R , for some constant Φ . Thus the R limit exists ∼ →∞ →∞ d ∞ d−1 2 Γ( 2 +1)S[Φcl] and is simple. Furthermore, from the virial theorem, drr ψ (r)= 2 , 0 zero d/2 ′′ π (Φcl(0)) so we arrive at the simple expression for the net contributioR n of this l = 1 mode: −1/2 d/2 ′ 2 S[Φcl] det (l=1) + m d/2−1 ′′ d/2 M = (2π) Φ∞ Φcl(0) (38). 2π !  det h free + m2 i | | M(l=1) h i   This expresses the zero mode factorh solely in termsi of the asymptotic properties of the

classical solution Φcl, which are already known numerically as part of the determination

of Φcl and the ﬂuctuation potential V (r). No further computation is needed. Functional Determinants in Quantum Field Theory 10

7. Application 1: ﬂuctuation determinant for false vacuum decay

The phenomenon of nucleation drives first order phase transitions in many applications in physics. The semiclassical analysis of the rate of such a nucleation process was pioneered by Langer [36], who identified a semiclassical saddle point solution that gives the dominant exponential contribution to the rate, with a prefactor to the exponential given by the quantum fluctuations about this classical solution. The nucleation rate is given by the quantum mechanical rate of decay of a metastable ”false” vacuum, Φ−, into the ”true” vacuum, Φ+. Decay proceeds by the nucleation of expanding bubbles of true vacuum within the metastable false vacuum [36, 37, 38, 39, 40, 41]. The semiclassical prefactor requires computing the determinant of the differential operator associated with quantum fluctuations about the classical solution. This is a technically difficult problem, but it is ideally suited to the above results for determinants of radially separable operators, because the fluctuation operator is radial. The expression (42) below can be applied to any metastable field potential U(Φ) [35], without relying on the ”thin-wall” limit (the limit of degenerate vacua) . For definiteness, consider the specific (rescaled) quartic potential: U(Φ) = 1 Φ2 1 Φ3 + α Φ4, where α characterizes the shape of the 2 − 2 8 potential [the thin-wall limit is α 1]. The decay rate per unit volume and unit time → is − S [Φ ] 2 det′ ( 2 + U ′′(Φ )) 1/2 cl cl cl −Scl[Φcl]−δctS[Φcl] γ = − ′′ e , (39) 2π ! det ( 2 + U (Φ−))

− where the prime on the determinant means that the zero modes (corresponding to

translational invariance) are removed. Here Φcl is a classical solution known as the “bounce” solution [39], deﬁned below, and the prefactor terms in (39) correspond to quantum ﬂuctuations about this bounce solution. The second term in the exponent,

δctS[Φcl], denotes the counterterms needed for renormalizing the classical action Scl. The ﬁrst step in computing the false vacuum decay rate γ is to ﬁnd the classical bounce

solution, Φcl(r), which is a radially symmetric stationary point of the classical Euclidean

action, interpolating between the false and true vacuum. The bounce Φcl(r) solves the nonlinear ordinary differential equation 3 3 α Φ′′ Φ′ + Φ Φ2 + Φ3 =0 (40) − cl − r cl cl − 2 cl 2 cl ′ with boundary conditions Φ (0) = 0, Φ (r) Φ− 0, as r . Φ (r) must be cl cl → ≡ → ∞ cl computed numerically, for example by the method of shooting, adjusting the initial value Φ (0) until the boundary conditions at r = are satisfied. Given the bounce cl ∞ solution, Φcl(r), the corresponding radial fluctuation potential is 3α U ′′(Φ (r))=1 3Φ (r)+ Φ2 (r) . (41) cl − cl 2 cl There are three different types of eigenvalue of the fluctuation operator, each having a different role physically and mathematically. (i) Negative Mode: The l = 0 sector has a negative eigenvalue mode of the fluctuation operator, and is responsible for the instability leading to decay. This mode contributes a factor to the decay rate γ related Functional Determinants in Quantum Field Theory 11

to the absolute value of the determinant of the l = 0 fluctuation operator [36, 39]. (ii) Zero Modes: In the l = 1 sector, there is a four-fold degenerate zero eigenvalue of the fluctuation operator. Integrating over the corresponding collective coordinates Scl produces the factors of 2π in (39). In computing the rate γ, we need the determinant of the fluctuation operator with the zero mode removed [36, 39, 40]. This can be evaluated simply using the formula (38), in terms of the asymptotic behavior of the classical bounce solution Φ (r). (iii) Positive Modes: For l 2, the fluctuation operator has positive cl ≥ eigenvalues, each of degeneracy (l +1)2. For each l, the associated radial determinant is computed numerically using (23). But the sum over l is divergent and the answer must

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Α 0.9 0.92 0.94 0.96 0.98

0.18

Figure 1. Comparison as α 1 of the exact numerical result [dots and solid blue → line] from (42) with the analytic thin-wall limit result 9 1 2π 0.16788.. of [42]. 32 − 9√3 ≈ h i be renormalized. Using the MS scheme, the renormalized eﬀective action is [35]:

1 π 3 2 α 3 Γ = ln T ( ) 2 ln Φ∞ Φ Φ + Φ MS 2 (0) ∞ − 2 0 − 2 0 2 0 ∞ ∞ ∞ 1 1 3 1 2 2 0 drrV (r) 8 0 drr V (V + 2) + (l + 1) ln T(l)( ) + 2 ( ∞ − R (l + 1) R (l + 1)3 ) l=2 X∞ ∞ 3 1 3 1 r drrV (r)+ drr V (V + 2) γE ln (42) − 4 Z0 16 Z0 2 − − 2 The first term is from the negative mode, the second from the zero mode(s) [using (38)], the third term from the numerical computation of the higher modes, and the last term is the renormalization counterterm contribution. The expression (42) can be applied to any field potential U(Φ), without relying on the thin-wall limit. A comparison with the analytic thin-wall computation of Konoplich and Rubin [42] is plotted in Fig. 1, and shows excellent agreement. This approach can also be extended to false vacuum decay in curved space, where the structure of the bounce solutions [43], and the corresponding fluctuation equations [44], is much richer. Functional Determinants in Quantum Field Theory 12

8. Application 2: determinant for massive quarks in instanton background

The fermion determinant in an instanton background is known in the massless (chiral) limit [9], and in the heavy quark limit [45]. Here we compute it for any m [46]. The ˜S relevant part of the renormalized effective action, Γren(m), behaves as 1 α(1/2)+ (ln m + γ ln2) m2 + ... , m 0 ˜S 2 − → Γren(m)=  (43)  ln m 1 17 232 7916  + + , m − 6 − 75m2 − 735m4 2835m6 − 148225m8 ··· →∞  where α(1/2) = 0.145873. The small mass expansion [9] is based on the known ≃ massless propagators in an instanton background, while the large mass expansion [45] follows most easily from the Schwinger-DeWitt or heat kernel expansion. Since the instanton is self-dual, we can simplify things slightly and work with the scalar Klein-Gordon operator. For a single su(2) instanton, the spectral problem 1 separates into partial waves [9], with radial Schrödinger-like operators (take isospin 2 ) ∂2 3 ∂ 4l(l + 1) 4(j l)(j + l + 1) 3 (l,j) + + − , (44) H ≡ "−∂r2 − r ∂r r2 r2 +1 − (r2 + 1)2 # Here l =0, 1 , 1, 3 , , and j = l 1 , and there is a degeneracy factor of (2l+1)(2j +1). 2 2 ··· | ± 2 | Since the ”potential” involves l and j, we cannot directly use the result (29). In order to compare with ’t Hooft’s massless quark analysis [9], we use a Pauli-Villars regulator Λ, and define the renomalized effective action by charge renormalization. To extract the renormalized effective action we need to consider the Λ limit in conjunction → ∞ with the infinite sum over l. Split the partial wave sum into two parts: L ∞ S S S ΓΛ(m)= ΓΛ,(l)(m)+ ΓΛ,(l)(m) (45) l=0, 1 ,... l=L+ 1 X2 X 2 where L is a large but finite integer. The first sum involves low partial wave modes, and the second sum involves the high partial wave modes. For the low partial wave modes, we can remove the regulator and evaluate the (finite) sum using the (numerical) Gel’fand-Yaglom result (23). But this sum diverges quadratically with L. In the second sum in (45) we cannot take the large L and large Λ limits blindly, as each leads to a divergence. Radial WKB is a good approximation for the high l modes. With WKB we can compute analytically the large Λ and large L divergences of the second sum in (45), using the WKB approximation for the corresponding determinants [46]: ∞ 1 1 m2 ΓS (A; m) lnΛ+2L2 +4L + ln L Λ,(l) ∼ 6 − 6 2 l=L+ 1 ! X 2 127 1 m2 m2 1 + ln2+ m2 ln2+ ln m + O (46) " 72 − 3 2 − 2 # L It is important to identify the physical role of the various terms in (46). The first term is the expected logarithmic charge renormalization counterterm. The next three terms Functional Determinants in Quantum Field Theory 13

˜S Γren(m) 0.15 0.1 0.05 m 0.5 1 1.5 2 2.5 3 -0.05 -0.1 -0.15 -0.2

S Figure 2. Plot of our numerical results for Γ˜ren(m) from (47), compared with the analytic extreme small and large mass limits [dashed curves] from (43). The dots denote numerical data points from (47), and the solid line is a ﬁt through these points.

give quadratic, linear and logarithmic divergences in L. These divergences cancel exactly against corresponding divergences in the ﬁrst sum in (45), which are found numerically. Combining the numerical results for low partial wave modes with the radial WKB ˜S results for the high partial wave modes the renormalized eﬀective action Γren(m) is

L 2 ˜S 2 1 m Γren(m) = lim  (2l + 1)(2l + 2)P (l)+2L +4L + ln L L→∞ − 6 2 l=0, 1 ,... !  X2 2 2 127 1 m m +  ln2+ m2 ln2+ ln m . (47) " 72 − 3 2 − 2 #) ˜S This is ﬁnite for any mass m. Figure 2 shows these results for Γren(m), compared with the analytic small and large mass expansions in (43). The agreement is remarkable [46]. It is a highly nontrivial check on the WKB computation that the correct mass dependence rises to interpolate between the large mass and small mass regimes. As an interesting analytic check, the formula (47) also provides a simple computation [46] of ’t Hooft’s leading small mass result α(1/2) = 17 1 ln2+ 1 2ζ′( 1)=0.145873.... − 72 − 6 6 − −

9. Concluding remarks

In this talk I have reviewed some of the reasons for considering determinants of partial differential operators in quantum field theory. Mathematically, much is known for ordinary differential operators, especially through the work of Gel’fand-Yaglom [21, 22] and its extensions [24, 26, 28], but considerably less is known for partial differential operators with non-trivial potentials. The new mathematical results [30] reported here are the formulas in (29), which provide simple expressions for the determinant of a radially separable partial differential operator of the form ∆ + m2 + V (r). This − Functional Determinants in Quantum Field Theory 14 generalizes the (Dirichlet) Gel’fand-Yaglom result (11) to higher dimensions. These results lead to many direct applications in quantum field theory, where they extend the class of solvable fluctuation determinant problems away from the restrictive class of constant background fields, or one dimensional background fields, to the more general class of separable higher dimensional background fields. This includes, for example, applications in quantum field theory to the study of quantum fluctuations in the presence of vortices, monopoles, sphalerons, instantons, domain walls (branes), etc ... A number of further generalizations could be made. First, it is practically important to include directly the matrix structure that arises from Dirac-like differential operators and from non-abelian gauge degrees of freedom. This has been done in various ways for certain applications [47, 48, 49, 50, 51], but a simpler unified formalism along the lines of the explicit formulas (29) is probably possible, as in one dimension [52]. Second, a practical extension to finite temperature and density would be useful in various physical applications. Third, the biggest technical challenge is to ask if the restriction of separability can be relaxed. A promising current approach is the numerical worldline loop method [53]. Without symmetry, computing functional determinants analytically remains a difficult problem, and any progress will be interesting.

Acknowledgments

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