Two-loop Correction to the Instanton Density for the Double Well Potential

Feynman diagrams in the instanton background are used for the calculation of the tunneling amplitude, up to the two-loops order. Some mistakes made in the previous works are corrected. The same method is applied to the next-order corrections to the ground state wave function. A motion in the double-well potential is a classic example, displaying tunneling phenomena in their simplest form. its discussion was revived with the development of semiclassical methods in quantum field theory, related to Euclidean classical solutions, the instantons [1, 2], (see e.g. good review [5]). Although many questions concerning ensemble of instantons in QCD remain unclear, it was recently demonstrated that the instanton-induced effects are more important than previously believed, dominating even such quantities as baryonic masses, splittings and wave functions [3]. Evaluation of the next-order corrections, going beyond the standard semiclassical theory, are of great interest. As it is technically difficult to do such calculations for the gauge theory, one may naturally start with the quantum mechanical example. Note also, that some of the higher-dimensional problems can be directly connected to the double-well considered here. This problem was first addressed in ref. [6]. Starting with the anharmonic oscillator, it was shown how to apply Feynman diagrams in quantum mechanics for a few simple problems∗, and then the two-loop diagrams in the instanton background were evaluated. The nontrivial point here is the new type of diagram, coming from Jacobian of the collective mode. Later an error in the Green’s function used was noticed by S̆.Olejnik [7], who corrected it and also made a new calculation of the instanton density, However, his results strongly disagree not only with those in [6], but also with numerical simulations of path integrals [8] and (which was discovered later) with the exact result† obtained by Zinn-Justin [9]. One may suspect either some fundamental problem of the method used; or a simple error. As we show in this letter, the method is fine and the error is found. Let us introduce our notations (same as in [5]). A particle of mass m=1 moves in the potential Vinst = λ(x 2 − η) (1) The instanton solution is Xinst(t) = ηtanh( 1 2ω(t − tc)) where tc is the instanton center and ω2 = 8λη2. Setting ω = 1, and shifting one minimum to the origin one gets the anharmonic oscillator potential V = 1 2x 2 − √ 2λx3 + λx4 with one small dimensionless parameter λ, to be used for the expansion. The classical action of the instanton solution S0 = 1 12λ is therefore large, and we are looking for the corrections to the wave functions of the first and second state (i=1,2), the ground state energy and the splitting in the following form Ψi(η) = Ψ 0 i (η)(1 +A/S0 + · · ·) (2) E0 = E 0 0(1 +B/S0 + · · ·) (3) δE = ∆E(1 + C/S0 + · · ·) (4) Here E0 0 = ω 2 and the (one-loop) semiclassical result for the splitting is ∆E = 2ω √ 6S0/πe −S0 . This part can be recommended as methodical examples in quantum field theory courses: here Feynman diagrams appear without singularities of divergences, and the results can be easily checked by standard means. Zinn-Justin work is based on solution of Schroedinger equation in a form of series expansion: thus it certainly cannot be generalized to the field theory context we are mainly interested in. Up to exponentially small corrections Ψ0(±η) and Ψ1(±η) are the symmetric and antisymmetric linear combinations of the ground state wavefunction of the perturbed harmonic oscillator, and therefore A1 = A2 = A. The transition amplitude for 1/∆E >> τ >> 1 can therefore be written as 〈−η | e | η〉 −→| Ψ0(η) | (1 + 2A/S0 + · · ·)× ×e0∆E(1 + C/S0 + · · ·)τ (5) Normalizing it to the transition amplitude 〈η | exp(−Hτ) | η〉 for the corresponding anharmonic oscillator, one can get rid of unwanted corrections A and B. In the Feynman path integral language, this ratio can be calculated by computing the diagrams in the instanton background ( with the instanton-based vertices and the instanton-based Green’s function ), and then by subtracting the diagrams for the anharmonic oscillator (see [6] for details). The instanton-based Green’s function should satisfy the standard differential equation and also be orthogonal to the zero mode. The result [7] is G(x, y) = G0(x, y)[2−xy+1/4 | x−y | (11−3xy)+(x−y)]+3/8(1−x)(1−y)[ln(G0(x, y))−11/3] (6) where x = tanh(t/2), y = tanh(t′/2) and G0(x, y) = 1− | x− y | −xy 1+ | x− y | −xy (7) The correction C we are interested in can be written as four diagrams for the instanton correction [6] C = a1 + b11 + b12 + c1 as shown in the Figure. Three first terms appear also for the anharmonic oscillator and should be subtracted, the last one comes from Jacobian of the zero mode and has no analogs in this case. Their analytic expressions are a1 = −3β ∫ ∞ −∞ dt(G(t, t)−G0(t, t)) = − 97 1680 (8) b11 = 3α 2 ∫ ∞ −∞ ∫ ∞ −∞ dtdt(tanh(t/2) tanh(t/2)G(t, t)−G0(t, t)) = − 53 1260 (9) b12 = 9 2 α ∫ ∞ −∞ ∫ ∞ −∞ dtdt(tanh(t/2) tanh(t/2)G(t, t)G(t, t)G(t, t) −G0(t, t)G0(t, t)G0(t, t)) = −39/560 (10) c1 = −9β ∫ ∞ −∞ ∫ ∞ −∞ dtdt tanh(t/2) cosh2(t/2) tanh(t/2)G(t, t)G(t, t) = −49/60 (11) from which follows C = −71/72. This is exactly the result of J. Zinn-Justin in [9] ‡ . For comparison, Alejnikov and Shuryak [6] and Shuryak [8] got C ≈ −1, and Olejnik [7] had C ≈ 0. The findings that (i) the correction has a coefficient C=O(1) and (ii) it is negative are important: they demonstrate that (i) the usual (one-loop) semiclassical formula does become inaccurate for not-so-large barriers (S ∼ 1), and (ii) that it overestimates the tunneling probability. The magnitude of the corresponding coefficient in gauge theories remains unknown, although this question is of great practical importance. Let us now switch to our second subject, the wave function correction (denoted A in (5)). Although it is much simpler than the problem discussed above (and has no dramatic history), it has a potential trap, which was not discussed for our knowledge and is worth mentioning. Now, one has the general anharmonic potential Vanhosc(x) = x 2/2+αx3 +βx4 and evaluates The error made in ref.[7] was found to be related with diagrams b11, b12, in which the time-dependent triple vertex those were erroneously included for the subtracted anharmonic oscillator part as well. the transition amplitude 〈x = 0 | e−Hanhoscτ | x = 0〉. For τ going to infinity it gives terms proportional to τ which are the energy corrections and constant terms which are the wavefunction corrections. Two different Green’s functions can be used, which satisfy both the same differential equation but vanish at different boundaries. The infinite boundary Green’s function is simply Ginfinite(t, t ′) = 1/2e−|t−t ′ | while the finite one is Gfinite(t, t ) = sinh(τ/2− t) sinh(t + τ/2)/ sinh(τ) (t ≥ t) = sinh(τ/2− t) sinh(t+ τ/2)/ sinh(τ) (t ≥ t) (12) Both give the same result for the energy shift, but different ones for the wave function correction. Up to the first order in β and α2 Ainfinite = −(29/96)α ; Afinite = (9/16)β − (167/96)α (13) Comparison of the two results to the corrections obtained by old-fashioned perturbation theory shows that Afinite is correct §. The lesson is as follows: unlike the energy correction (which corresponds to infinitely long paths), the wave function is a kind of a boundary effect, and therefore proper restrictions for the quantum paths (and Green’s functions) should not be ignored. In conclusion, we have calculated the tunneling amplitude (equal to the energy splitting or the instanton density) up to two-loops, using Feynman diagrams. Two important elements of the calculation, making it somewhat less standard, is the non-trivial background field and new effective vertices, which come the zero-mode Jacobian rather than from the action. Our results agree exactly with ref. [9], based on a different method, and with other works inside the accuracy. In ref.[7] the error is found. § Note that this observation does not invalidate the calculations above for the constant C: because the wave function corrections are the same for both the instanton and the anharmonic oscillator, up to exponentially small terms the constants A,B are cancelled whatever Green’s function is used. Extensions of this work can be (i) three and more loop corrections for quantum mechanical examples, and, more important (ii) possible two-loop calculations for gauge theories, in essentially the same setting. Such calculation would be of great interest, because the problem of large-size (or small action) instantons in QCD remains unsolved.