Abstract
Starting from the superstring amplitude describing interactions among D-branes with a constant world-volume field strength, we present a detailed analysis of how the open string degeneration limits reproduce the corresponding field theory Feynman diagrams. A key ingredient in the string construction is represented by the twisted (Prym) super differentials, as their periods encode the information about the background field. We provide an efficient method to calculate perturbatively the determinant of the twisted period matrix in terms of sets of super-moduli appropriate to the degeneration limits. Using this result we show that there is a precise one-to-one correspondence between the degeneration of different factors in the superstring amplitudes and one-particle irreducible Feynman diagrams capturing the gauge theory effective action at the two-loop level.
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References
E. Witten, The super period matrix with Ramond punctures, J. Geom. Phys. 92 (2015) 210 [arXiv:1501.02499] [INSPIRE].
E. D’Hoker and D.H. Phong, The super period matrix with Ramond punctures in the supergravity formulation, arXiv:1501.02675 [INSPIRE].
R. Pius, A. Rudra and A. Sen, Mass renormalization in string theory: general states, JHEP 07 (2014) 062 [arXiv:1401.7014] [INSPIRE].
R. Pius, A. Rudra and A. Sen, String perturbation theory around dynamically shifted vacuum, JHEP 10 (2014) 070 [arXiv:1404.6254] [INSPIRE].
A. Sen, Off-shell amplitudes in superstring theory, Fortschr. Phys. 63 (2015) 149 [arXiv:1408.0571] [INSPIRE].
E. D’Hoker and M.B. Green, Zhang-Kawazumi invariants and superstring amplitudes, arXiv:1308.4597 [INSPIRE].
E. D’Hoker, M.B. Green, B. Pioline and R. Russo, Matching the D 6 R 4 interaction at two-loops, JHEP 01 (2015) 031 [arXiv:1405.6226] [INSPIRE].
P. Tourkine, Tropical amplitudes, arXiv:1309.3551 [INSPIRE].
E. Witten, Superstring perturbation theory revisited, arXiv:1209.5461 [INSPIRE].
E. D’Hoker, Topics in two-loop superstring perturbation theory, arXiv:1403.5494 [INSPIRE].
L. Magnea, S. Playle, R. Russo and S. Sciuto, Multi-loop open string amplitudes and their field theory limit, JHEP 09 (2013) 081 [arXiv:1305.6631] [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Nonlinear electrodynamics from quantized strings, Phys. Lett. B 163 (1985) 123 [INSPIRE].
A. Abouelsaood, C.G. Callan Jr., C.R. Nappi and S.A. Yost, Open strings in background gauge fields, Nucl. Phys. B 280 (1987) 599 [INSPIRE].
C. Bachas and M. Porrati, Pair creation of open strings in an electric field, Phys. Lett. B 296 (1992) 77 [hep-th/9209032] [INSPIRE].
L. Magnea, R. Russo and S. Sciuto, Two-loop Euler-Heisenberg effective actions from charged open strings, Int. J. Mod. Phys. A 21 (2006) 533 [hep-th/0412087] [INSPIRE].
J.-L. Gervais and A. Neveu, Feynman rules for massive gauge fields with dual diagram topology, Nucl. Phys. B 46 (1972) 381 [INSPIRE].
L. Crane and J.M. Rabin, Super Riemann surfaces: uniformization and Teichmüller theory, Commun. Math. Phys. 113 (1988) 601 [INSPIRE].
E.J. Martinec, Conformal field theory on a (super-)Riemann surface, Nucl. Phys. B 281 (1987) 157 [INSPIRE].
S.B. Giddings and P.C. Nelson, The geometry of super Riemann surfaces, Commun. Math. Phys. 116 (1988) 607 [INSPIRE].
E. D’Hoker and D.H. Phong, The geometry of string perturbation theory, Rev. Mod. Phys. 60 (1988) 917 [INSPIRE].
E. Witten, Notes on super Riemann surfaces and their moduli, arXiv:1209.2459 [INSPIRE].
C. Lovelace, M-loop generalized Veneziano formula, Phys. Lett. B 32 (1970) 703 [INSPIRE].
M. Kaku and L. Yu, The general multi-loop Veneziano amplitude, Phys. Lett. B 33 (1970) 166 [INSPIRE].
V. Alessandrini, A general approach to dual multiloop diagrams, Nuovo Cim. A 2 (1971) 321 [INSPIRE].
D.I. Olive, Operator vertices and propagators in dual theories, Nuovo Cim. A 3 (1971) 399 [INSPIRE].
V. Alessandrini and D. Amati, Properties of dual multiloop amplitudes, Nuovo Cim. A 4 (1971) 793 [INSPIRE].
C. Montonen, Multiloop amplitudes in additive dual-resonance models, Nuovo Cim. A 19 (1974) 69 [INSPIRE].
P. Di Vecchia, R. Nakayama, J.L. Petersen, J. Sidenius and S. Sciuto, BRST invariant N -reggeon vertex, Phys. Lett. B 182 (1986) 164 [INSPIRE].
P. Di Vecchia, M. Frau, A. Lerda and S. Sciuto, A simple expression for the multiloop amplitude in the bosonic string, Phys. Lett. B 199 (1987) 49 [INSPIRE].
P. Di Vecchia, K. Hornfeck, M. Frau, A. Lerda and S. Sciuto, N -string, g-loop vertex for the fermionic string, Phys. Lett. B 211 (1988) 301 [INSPIRE].
P. Di Vecchia et al., N -point g-loop vertex for a free bosonic theory with vacuum charge Q, Nucl. Phys. B 322 (1989) 317 [INSPIRE].
P. Di Vecchia et al., N -point g-loop vertex for a free fermionic theory with arbitrary spin, Nucl. Phys. B 333 (1990) 635 [INSPIRE].
R. Russo and S. Sciuto, Twisted determinants on higher genus Riemann surfaces, Nucl. Phys. B 669 (2003) 207 [hep-th/0306129] [INSPIRE].
K. Aoki, E. D’Hoker and D.H. Phong, Two loop superstrings on orbifold compactifications, Nucl. Phys. B 688 (2004) 3 [hep-th/0312181] [INSPIRE].
R. Russo and S. Sciuto, Twisted determinants and bosonic open strings in an electromagnetic field, Fortschr. Phys. 52 (2004) 678 [hep-th/0312205] [INSPIRE].
I. Antoniadis, K.S. Narain and T.R. Taylor, Open string topological amplitudes and gaugino masses, Nucl. Phys. B 729 (2005) 235 [hep-th/0507244] [INSPIRE].
J. Scherk, Zero-slope limit of the dual resonance model, Nucl. Phys. B 31 (1971) 222 [INSPIRE].
M.B. Green, J.H. Schwarz and L. Brink, N = 4 Yang-Mills and N = 8 supergravity as limits of string theories, Nucl. Phys. B 198 (1982) 474 [INSPIRE].
M.L. Mangano, S.J. Parke and Z. Xu, Duality and multi-gluon scattering, Nucl. Phys. B 298 (1988) 653 [INSPIRE].
M.L. Mangano and S.J. Parke, Multiparton amplitudes in gauge theories, Phys. Rept. 200 (1991) 301 [hep-th/0509223] [INSPIRE].
Z. Bern and D.A. Kosower, A new approach to one loop calculations in gauge theories, Phys. Rev. D 38 (1988) 1888 [INSPIRE].
Z. Bern and D.A. Kosower, Efficient calculation of one loop QCD amplitudes, Phys. Rev. Lett. 66 (1991) 1669 [INSPIRE].
Z. Bern and D.A. Kosower, Color decomposition of one loop amplitudes in gauge theories, Nucl. Phys. B 362 (1991) 389 [INSPIRE].
Z. Bern and D.A. Kosower, The computation of loop amplitudes in gauge theories, Nucl. Phys. B 379 (1992) 451 [INSPIRE].
Z. Bern and D.C. Dunbar, A mapping between Feynman and string motivated one loop rules in gauge theories, Nucl. Phys. B 379 (1992) 562 [INSPIRE].
Z. Bern, A compact representation of the one loop N-gluon amplitude, Phys. Lett. B 296 (1992) 85 [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, One loop corrections to five gluon amplitudes, Phys. Rev. Lett. 70 (1993) 2677 [hep-ph/9302280] [INSPIRE].
M.J. Strassler, Field theory without Feynman diagrams: one loop effective actions, Nucl. Phys. B 385 (1992) 145 [hep-ph/9205205] [INSPIRE].
M.G. Schmidt and C. Schubert, On the calculation of effective actions by string methods, Phys. Lett. B 318 (1993) 438 [hep-th/9309055] [INSPIRE].
M.G. Schmidt and C. Schubert, The worldline path integral approach to Feynman graphs, hep-ph/9412358 [INSPIRE].
C. Schubert, Perturbative quantum field theory in the string inspired formalism, Phys. Rept. 355 (2001) 73 [hep-th/0101036] [INSPIRE].
P. Dai and W. Siegel, Worldline Green functions for arbitrary Feynman diagrams, Nucl. Phys. B 770 (2007) 107 [hep-th/0608062] [INSPIRE].
F. Bastianelli, O. Corradini and E. Latini, Higher spin fields from a worldline perspective, JHEP 02 (2007) 072 [hep-th/0701055] [INSPIRE].
P. Di Vecchia, L. Magnea, A. Lerda, R. Russo and R. Marotta, Renormalization constants from string theory, hep-th/9602055 [INSPIRE].
A. Frizzo, L. Magnea and R. Russo, Systematics of one loop Yang-Mills diagrams from bosonic string amplitudes, Nucl. Phys. B 604 (2001) 92 [hep-ph/0012129] [INSPIRE].
P. Di Vecchia, L. Magnea, A. Lerda, R. Marotta and R. Russo, Two loop scalar diagrams from string theory, Phys. Lett. B 388 (1996) 65 [hep-th/9607141] [INSPIRE].
A. Frizzo, L. Magnea and R. Russo, Scalar field theory limits of bosonic string amplitudes, Nucl. Phys. B 579 (2000) 379 [hep-th/9912183] [INSPIRE].
R. Marotta and F. Pezzella, Two loop ϕ 4 diagrams from string theory, Phys. Rev. D 61 (2000) 106006 [hep-th/9912158] [INSPIRE].
L. Magnea and R. Russo, String derivation of two loop Feynman diagrams, AIP Conf. Proc. 415 (1997) 347 [hep-ph/9708471] [INSPIRE].
L. Magnea and R. Russo, Two loop gluon diagrams from string theory, AIP Conf. Proc. 407 (1997) 913 [hep-ph/9706396] [INSPIRE].
B. Körs and M.G. Schmidt, Two loop Feynman diagrams in Yang-Mills theory from bosonic string amplitudes, hep-th/0003171 [INSPIRE].
J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string, Cambridge University Press, Cambridge U.K. (1998) [INSPIRE].
R.R. Metsaev and A.A. Tseytlin, On loop corrections to string theory effective actions, Nucl. Phys. B 298 (1988) 109 [INSPIRE].
K. Hornfeck, Three-reggeon light-cone vertex of the Neveu-Schwarz string, Nucl. Phys. B 293 (1987) 189 [INSPIRE].
L. Álvarez-Gaumé, G.W. Moore and C. Vafa, Theta functions, modular invariance and strings, Commun. Math. Phys. 106 (1986) 1 [INSPIRE].
M. Frau, I. Pesando, S. Sciuto, A. Lerda and R. Russo, Scattering of closed strings from many D-branes, Phys. Lett. B 400 (1997) 52 [hep-th/9702037] [INSPIRE].
R. Russo and S. Sciuto, The twisted open string partition function and Yukawa couplings, JHEP 04 (2007) 030 [hep-th/0701292] [INSPIRE].
P. Di Vecchia, L. Magnea, A. Lerda, R. Russo and R. Marotta, String techniques for the calculation of renormalization constants in field theory, Nucl. Phys. B 469 (1996) 235 [hep-th/9601143] [INSPIRE].
P. Vanhove, The physics and the mixed Hodge structure of Feynman integrals, Proc. Symp. Pure Math. 88 (2014) 161 [arXiv:1401.6438] [INSPIRE].
P. Di Vecchia and A. Liccardo, D branes in string theory, II, hep-th/9912275 [INSPIRE].
E. Witten, Notes on supermanifolds and integration, arXiv:1209.2199 [INSPIRE].
P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Quantum dynamics of a massless relativistic string, Nucl. Phys. B 56 (1973) 109 [INSPIRE].
C.B. Thorn, A world sheet description of planar Yang-Mills theory, Nucl. Phys. B 637 (2002) 272 [hep-th/0203167] [INSPIRE].
M. Headrick, grassmann.m: a package that teaches Mathematica how to manipulate Grassmann variables (2015), http://web.archive.org/web/20150317172836/ http://people.brandeis.edu/∼headrick/Mathematica/grassmann.m.
D. Friedan, Notes on string theory and two dimensional conformal field theory, in M.B. Green et al. eds., Unified string theories, World Scientific (1986), pp. 162-213.
L.J. Dixon, D. Friedan, E.J. Martinec and S.H. Shenker, The conformal field theory of orbifolds, Nucl. Phys. B 282 (1987) 13 [INSPIRE].
A.A. Tseytlin, Open superstring partition function in constant gauge field background at finite temperature, Nucl. Phys. B 524 (1998) 41 [hep-th/9802133] [INSPIRE].
C. Bachas, D-brane dynamics, Phys. Lett. B 374 (1996) 37 [hep-th/9511043] [INSPIRE].
M. Berkooz, M.R. Douglas and R.G. Leigh, Branes intersecting at angles, Nucl. Phys. B 480 (1996) 265 [hep-th/9606139] [INSPIRE].
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Magnea, L., Playle, S., Russo, R. et al. Two-loop Yang-Mills diagrams from superstring amplitudes. J. High Energ. Phys. 2015, 146 (2015). https://doi.org/10.1007/JHEP06(2015)146
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DOI: https://doi.org/10.1007/JHEP06(2015)146