More Like this

1. Two-loop superstring five-point amplitudes. Part III. Construction via the RNS formulation: even spin structures

   https://doi.org/10.1007/JHEP12(2021)063  

   D’Hoker, Eric ; Schlotterer, Oliver ( December 2021 , Journal of High Energy Physics)

   A bstract The contribution from even spin structures to the genus-two amplitude for five massless external NS states in Type II and Heterotic superstrings is evaluated from first principles in the RNS formulation. Using chiral splitting with the help of loop momenta this problem reduces to the evaluation of the corresponding chiral amplitude, which is carried out using the same techniques that were used for the genus-two amplitude with four external NS states. The results agree with the parity-even NS components of a construction using chiral splitting and pure spinors given in earlier companion papers [29] and [33]. 

   more » « less
2. Two-loop superstring five-point amplitudes. Part II. Low energy expansion and S-duality

   https://doi.org/10.1007/JHEP02(2021)139  

   D’Hoker, Eric ; Mafra, Carlos R. ; Pioline, Boris ; Schlotterer, Oliver ( February 2021 , Journal of High Energy Physics)

   null (Ed.)

   A bstract In an earlier paper, we constructed the genus-two amplitudes for five external massless states in Type II and Heterotic string theory, and showed that the α ′ expansion of the Type II amplitude reproduces the corresponding supergravity amplitude to leading order. In this paper, we analyze the effective interactions induced by Type IIB superstrings beyond supergravity, both for U(1) R -preserving amplitudes such as for five gravitons, and for U(1) R -violating amplitudes such as for one dilaton and four gravitons. At each order in α ′, the coefficients of the effective interactions are given by integrals over moduli space of genus-two modular graph functions, generalizing those already encountered for four external massless states. To leading and sub-leading orders, the coefficients of the effective interactions D 2 ℛ 5 and D 4 ℛ 5 are found to match those of D 4 ℛ 4 and D 6 ℛ 4 , respectively, as required by non-linear supersymmetry. To the next order, a D 6 ℛ 5 effective interaction arises, which is independent of the supersymmetric completion of D 8 ℛ 4 , and already arose at genus one. A novel identity on genus-two modular graph functions, which we prove, ensures that up to order D 6 ℛ 5 , the five-point amplitudes require only a single new modular graph function in addition to those needed for the four-point amplitude. We check that the supergravity limit of U(1) R -violating amplitudes is free of UV divergences to this order, consistently with the known structure of divergences in Type IIB supergravity. Our results give strong consistency tests on the full five-point amplitude, and pave the way for understanding S-duality beyond the BPS-protected sector. 

   more » « less
3. Integrating three-loop modular graph functions and transcendentality of string amplitudes

   https://doi.org/10.1007/JHEP02(2022)019  

   D’Hoker, Eric ; Geiser, Nicholas ( February 2022 , Journal of High Energy Physics)

   A bstract Modular graph functions (MGFs) are SL(2 , ℤ)-invariant functions on the Poincaré upper half-plane associated with Feynman graphs of a conformal scalar field on a torus. The low-energy expansion of genus-one superstring amplitudes involves suitably regularized integrals of MGFs over the fundamental domain for SL(2 , ℤ). In earlier work, these integrals were evaluated for all MGFs up to two loops and for higher loops up to weight six. These results led to the conjectured uniform transcendentality of the genus-one four-graviton amplitude in Type II superstring theory. In this paper, we explicitly evaluate the integrals of several infinite families of three-loop MGFs and investigate their transcendental structure. Up to weight seven, the structure of the integral of each individual MGF is consistent with the uniform transcendentality of string amplitudes. Starting at weight eight, the transcendental weights obtained for the integrals of individual MGFs are no longer consistent with the uniform transcendentality of string amplitudes. However, in all the cases we examine, the violations of uniform transcendentality take on a special form given by the integrals of triple products of non-holomorphic Eisenstein series. If Type II superstring amplitudes do exhibit uniform transcendentality, then the special combinations of MGFs which enter the amplitudes must be such that these integrals of triple products of Eisenstein series precisely cancel one another. Whether this indeed is the case poses a novel challenge to the conjectured uniform transcendentality of genus-one string amplitudes. 

   more » « less
4. Cyclic products of Szegö kernels and spin structure sums. Part I. Hyper-elliptic formulation

   https://doi.org/10.1007/JHEP05(2023)073  

   D’Hoker, Eric ; Hidding, Martijn ; Schlotterer, Oliver ( May 2023 , Journal of High Energy Physics)

   The summation over spin structures, which is required to implement the GSO projection in the RNS formulation of superstring theories, often presents a significant impediment to the explicit evaluation of superstring amplitudes. In this paper we discover that, for Riemann surfaces of genus two and even spin structures, a collection of novel identities leads to a dramatic simplification of the spin structure sum. Explicit formulas for an arbitrary number of vertex points are obtained in two steps. First, we show that the spin structure dependence of a cyclic product of Szegö kernels (i.e. Dirac propagators for worldsheet fermions) may be reduced to the spin structure dependence of the four-point function. Of particular importance are certaintrilinear relationsthat we shall define and prove. In a second step, the known expressions for the genus-two even spin structure measure are used to perform the remaining spin structure sums. The dependence of the spin summand on the vertex points is reduced to simple building blocks that can already be identified from the two-point function. The hyper-elliptic formulation of genus-two Riemann surfaces is used to derive these results, and its SL(2,ℂ) covariance is employed to organize the calculations and the structure of the final formulas. The translation of these results into the language of Riemannϑ-functions, and applications to the evaluation of higher-point string amplitudes, are relegated to subsequent companion papers.

    

   more » « less
5. Dimensional regularization and two-loop vacuum polarization operator: Master integrals, analytic results, and energy shifts

   https://doi.org/10.1103/PhysRevD.109.096020  

   Laporta, Stefano ; Jentschura, Ulrich D ( May 2024 , Physical Review D)

   We present a complete reevaluation of the irreducible two-loop vacuum-polarization correction to the photon propagator in quantum electrodynamics, i.e. with an electron-positron pair in the fermion propagators. The integration is carried out by reducing the integrations to a limited set of master integrals, which are calculated using integration-by-parts identities. Dimensional regularization is used in$D=4-2\epsilon$dimensions, and on-mass shell renormalization is employed. The one-loop effect is given to order$\epsilon$, to be combined with the$1/\epsilon$divergence of the two-loop amplitude. Master integrals are given. Final evaluations of two-loop energy shifts for$1S$,$2S$, and$2P$states are done analytically, and results are presented, with an emphasis on muonic hydrogen. For relativistic Dirac-Coulomb reference states, higher-order coefficients are obtained for the$Z\alpha$-expansion. We compare the results obtained to the existing literature.

   Published by the American Physical Society2024 

   more » « less