Contents

# Contents

## Idea

In quantum field theory (and string theory) the scattering amplitudes (string scattering amplitudes) “where nothing external scatters,” hence for no incoming and no outgoing states, are called vacuum amplitudes.

## Properties

### As generating functionals for all other amplitudes

As functions of source fields, the vacuum amplitudes, or rather the vacuum energy, serve as the generating functionals for all correlators/n-point functions (e.g. Scrucca, 1.6).

### One loop contribution and zeta functions

The 1-loop vacuum amplitudes are regularized traces over Feynman propagators/Dirac propagators. These are the incarnations of special values of zeta functions, L-functions and eta functions in physics:

context/function field analogytheta function $\theta$zeta function $\zeta$ (= Mellin transform of $\theta(0,-)$)L-function $L_{\mathbf{z}}$ (= Mellin transform of $\theta(\mathbf{z},-)$)eta function $\eta$special values of L-functions
physics/2d CFTpartition function $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of complex structure $\mathbf{\tau}$ of worldsheet $\Sigma$ (hence polarization of phase space) and background gauge field/source $\mathbf{z}$analytically continued trace of Feynman propagator $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$analytically continued trace of Feynman propagator in background gauge field $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$analytically continued trace of Dirac propagator in background gauge field $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s$regularized 1-loop vacuum amplitude $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / regularized fermionic 1-loop vacuum amplitude $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / vacuum energy $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$
Riemannian geometry (analysis)zeta function of an elliptic differential operatorzeta function of an elliptic differential operatoreta function of a self-adjoint operatorfunctional determinant, analytic torsion
complex analytic geometrysection $\theta(\mathbf{z},\mathbf{\tau})$ of line bundle over Jacobian variety $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$zeta function of a Riemann surfaceSelberg zeta functionDedekind eta function
arithmetic geometry for a function fieldGoss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number fieldHecke theta function, automorphic formDedekind zeta function (being the Artin L-function $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the trivial Galois representation)Artin L-function $L_{\mathbf{z}}$ of a Galois representation $\mathbf{z}$, expressible “in coordinates” (by Artin reciprocity) as a finite-order Hecke L-function (for 1-dimensional representations) and generally (via Langlands correspondence) by an automorphic L-function (for higher dimensional reps)class number $\cdot$ regulator
arithmetic geometry for $\mathbb{Q}$Jacobi theta function ($\mathbf{z} = 0$)/ Dirichlet theta function ($\mathbf{z} = \chi$ a Dirichlet character)Riemann zeta function (being the Dirichlet L-function $L_{\mathbf{z}}$ for Dirichlet character $\mathbf{z} = 0$)Artin L-function of a Galois representation $\mathbf{z}$ , expressible “in coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function

### Vanishing of the vacuum amplitude and supersymmetry

In the presence of supersymmetry 1-loop vacuum amplitudes are typically supposed to vanish.

For the type II superstring, see e.g. (Palti). For the heterotic superstring see e.g. Han 89.

In view of the above relation of 1-loop vacuum amplitudes to special values of L-functions such vanishing reminds one of the Riemann hypothesis. See (ACER 11).

### For particles

Discussions for particles includes

• Claudio Scrucca, Advanced quantum field theory pdf

• Ori Yudilevich, Calculating Massive One-Loop Amplitudes in QCD, Utrecht 2009 (pdf)

• Robbert Rietkerk, One-loop amplitudes in perturbative quantum field theory, Utrecht 2012 (pdf)

### For strings

Lecture notes for 1-loop vacuum amplitudes for strings (vacuum string scattering amplitudes) include

• José Edelstein, Lecture 8: 1-loop closed string vacuum amplitude, 2013 (pdf)

• Eran Palti, The IIA/B superstring one-loop vacuum amplitude (pdf)

• Seung Kee Han, Vanishing vacuum amplitude of four-dimensional heterotic string theory compactified on N=2 superconformal field theory, Phys. Rev. D 39, 2322 – Published 15 April 1989 (web)

And with relation to the Riemann hypothesis:

• Carlo Angelantonj, Matteo Cardella, Shmuel Elitzur, Eliezer Rabinovici, Vacuum stability, string density of states and the Riemann zeta function,JHEP 1102:024,2011 (arXiv:1012.5091)