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One-loop monodromy relations on single cuts. (English) Zbl 1383.81327
Summary: The discovery of colour-kinematic duality has led to significant progress in the computation of scattering amplitudes in quantum field theories. At tree level, the origin of the duality can be traced back to the monodromies of open-string amplitudes. This construction has recently been extended to all loop orders. In the present paper, we dissect some consequences of these new monodromy relations at one loop. We use single cuts in order to relate them to the tree-level relations. We show that there are new classes of kinematically independent single-cut amplitudes. Then we turn to the Feynman diagrammatics of the string-theory monodromy relations. We revisit the string-theoretic derivation and argue that some terms, that vanish upon integration in string and field theory, provide a characterisation of momentum-shifting ambiguities in these representations. We observe that colour-dual representations are compatible with this analysis.

MSC:
81U05 \(2\)-body potential quantum scattering theory
83E30 String and superstring theories in gravitational theory
Software:
CutTools; HyperInt
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[1] Bern, Z.; Carrasco, JJM; Johansson, H., New relations for gauge-theory amplitudes, Phys. Rev., D 78, (2008)
[2] Bern, Z.; Carrasco, JJM; Johansson, H., Perturbative quantum gravity as a double copy of gauge theory, Phys. Rev. Lett., 105, (2010)
[3] Kleiss, R.; Kuijf, H., Multi-gluon cross-sections and five jet production at hadron colliders, Nucl. Phys., B 312, 616, (1989)
[4] Bjerrum-Bohr, NEJ; Damgaard, PH; Vanhove, P., Minimal basis for gauge theory amplitudes, Phys. Rev. Lett., 103, 161602, (2009)
[5] Bjerrum-Bohr, NEJ; Damgaard, PH; Sondergaard, T.; Vanhove, P., Monodromy and Jacobi-like relations for color-ordered amplitudes, JHEP, 06, 003, (2010) · Zbl 1290.83015
[6] S. Stieberger, Open & closed vs. pure open string disk amplitudes, arXiv:0907.2211 [INSPIRE]. · Zbl 1284.81245
[7] Mafra, CR; Schlotterer, O.; Stieberger, S., Explicit BCJ numerators from pure spinors, JHEP, 07, 092, (2011) · Zbl 1298.81319
[8] Monteiro, R.; O’Connell, D., The kinematic algebra from the self-dual sector, JHEP, 07, 007, (2011) · Zbl 1298.81401
[9] Chiodaroli, M.; Jin, Q.; Roiban, R., Color/kinematics duality for general abelian orbifolds of N = 4 super Yang-Mills theory, JHEP, 01, 152, (2014) · Zbl 1333.81391
[10] Johansson, H.; Ochirov, A., Pure gravities via color-kinematics duality for fundamental matter, JHEP, 11, 046, (2015) · Zbl 1388.83017
[11] Chiodaroli, M.; Günaydin, M.; Johansson, H.; Roiban, R., Scattering amplitudes in \( \mathcal{N}=2 \) Maxwell-Einstein and Yang-Mills/Einstein supergravity, JHEP, 01, 081, (2015) · Zbl 1388.83772
[12] He, S.; Monteiro, R.; Schlotterer, O., String-inspired BCJ numerators for one-loop MHV amplitudes, JHEP, 01, 171, (2016) · Zbl 1388.81544
[13] Johansson, H.; Ochirov, A., Color-kinematics duality for QCD amplitudes, JHEP, 01, 170, (2016) · Zbl 1390.81697
[14] Mogull, G.; O’Connell, D., Overcoming obstacles to colour-kinematics duality at two loops, JHEP, 12, 135, (2015) · Zbl 1388.81868
[15] Johansson, H.; Kälin, G.; Mogull, G., Two-loop supersymmetric QCD and half-maximal supergravity amplitudes, JHEP, 09, 019, (2017) · Zbl 1382.83120
[16] Chiodaroli, M.; Günaydin, M.; Johansson, H.; Roiban, R., Explicit formulae for Yang-Mills-Einstein amplitudes from the double copy, JHEP, 07, 002, (2017)
[17] Bern, Z.; Davies, S.; Dennen, T., The ultraviolet structure of half-maximal supergravity with matter multiplets at two and three loops, Phys. Rev., D 88, (2013)
[18] Bern, Z.; Davies, S.; Dennen, T., Enhanced ultraviolet cancellations in \( \mathcal{N}=5 \) supergravity at four loops, Phys. Rev., D 90, 105011, (2014)
[19] Bern, Z.; Carrasco, JJ; Chen, W-M; Johansson, H.; Roiban, R., Gravity amplitudes as generalized double copies of gauge-theory amplitudes, Phys. Rev. Lett., 118, 181602, (2017)
[20] Tourkine, P.; Vanhove, P., Higher-loop amplitude monodromy relations in string and gauge theory, Phys. Rev. Lett., 117, 211601, (2016)
[21] Boels, RH; Isermann, RS, New relations for scattering amplitudes in Yang-Mills theory at loop level, Phys. Rev., D 85, (2012)
[22] Boels, RH; Isermann, RS, Yang-Mills amplitude relations at loop level from non-adjacent BCFW shifts, JHEP, 03, 051, (2012) · Zbl 1309.81317
[23] Feng, B.; Jia, Y.; Huang, R., Relations of loop partial amplitudes in gauge theory by unitarity cut method, Nucl. Phys., B 854, 243, (2012) · Zbl 1229.81304
[24] Brown, RW; Naculich, SG, Color-factor symmetry and BCJ relations for QCD amplitudes, JHEP, 11, 060, (2016)
[25] Brown, RW; Naculich, SG, BCJ relations from a new symmetry of gauge-theory amplitudes, JHEP, 10, 130, (2016) · Zbl 1390.81307
[26] He, S.; Schlotterer, O., New relations for gauge-theory and gravity amplitudes at loop level, Phys. Rev. Lett., 118, 161601, (2017)
[27] S. He, O. Schlotterer and Y. Zhang, New BCJ representations for one-loop amplitudes in gauge theories and gravity, arXiv:1706.00640 [INSPIRE].
[28] Cachazo, F.; He, S.; Yuan, EY, Scattering of massless particles in arbitrary dimensions, Phys. Rev. Lett., 113, 171601, (2014)
[29] Cachazo, F.; He, S.; Yuan, EY, Scattering of massless particles: scalars, gluons and gravitons, JHEP, 07, 033, (2014) · Zbl 1391.81198
[30] Bjerrum-Bohr, NEJ; Damgaard, PH; Sondergaard, T.; Vanhove, P., The momentum kernel of gauge and gravity theories, JHEP, 01, 001, (2011) · Zbl 1214.81145
[31] S. Hohenegger and S. Stieberger, Monodromy relations in higher-loop string amplitudes, arXiv:1702.04963 [INSPIRE]. · Zbl 1375.81202
[32] Feynman, RP, Quantum theory of gravitation, Acta Phys. Polon., 24, 697, (1963)
[33] R.P. Feynman, Closed loop and tree diagrams, in Selected papers of Richard Feynman, L.M. Brown ed., World Scientific, Singapore (1972).
[34] Brandhuber, A.; Spence, B.; Travaglini, G., From trees to loops and back, JHEP, 01, 142, (2006)
[35] Cachazo, F.; Svrček, P.; Witten, E., MHV vertices and tree amplitudes in gauge theory, JHEP, 09, 006, (2004)
[36] Catani, S.; Gleisberg, T.; Krauss, F.; Rodrigo, G.; Winter, J-C, From loops to trees by-passing feynman’s theorem, JHEP, 09, 065, (2008) · Zbl 1245.81117
[37] Bierenbaum, I.; Catani, S.; Draggiotis, P.; Rodrigo, G., A tree-loop duality relation at two loops and beyond, JHEP, 10, 073, (2010) · Zbl 1291.81381
[38] Bierenbaum, I.; Czakon, M.; Mitov, A., The singular behavior of one-loop massive QCD amplitudes with one external soft gluon, Nucl. Phys., B 856, 228, (2012) · Zbl 1246.81420
[39] Bierenbaum, I.; Buchta, S.; Draggiotis, P.; Malamos, I.; Rodrigo, G., Tree-loop duality relation beyond simple poles, JHEP, 03, 025, (2013) · Zbl 1371.81125
[40] Buchta, S.; Chachamis, G.; Draggiotis, P.; Malamos, I.; Rodrigo, G., On the singular behaviour of scattering amplitudes in quantum field theory, JHEP, 11, 014, (2014) · Zbl 1333.81149
[41] Nigel Glover, EW; Williams, C., One-loop gluonic amplitudes from single unitarity cuts, JHEP, 12, 067, (2008) · Zbl 1329.81283
[42] Britto, R.; Mirabella, E., Single cut integration, JHEP, 01, 135, (2011) · Zbl 1214.81295
[43] Britto, R.; Mirabella, E., External leg corrections in the unitarity method, JHEP, 01, 045, (2012) · Zbl 1306.81328
[44] G. Ossola, C.G. Papadopoulos and R. Pittau, Reducing full one-loop amplitudes to scalar integrals at the integrand level, Nucl. Phys.B 763 (2007) 147 [hep-ph/0609007] [INSPIRE]. · Zbl 1116.81067
[45] Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys.B 425 (1994) 217 [hep-ph/9403226] [INSPIRE]. · Zbl 1049.81644
[46] Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys.B 435 (1995) 59 [hep-ph/9409265] [INSPIRE]. · Zbl 1049.81644
[47] Britto, R.; Cachazo, F.; Feng, B., Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys., B 725, 275, (2005) · Zbl 1178.81202
[48] Forde, D., Direct extraction of one-loop integral coefficients, Phys. Rev., D 75, 125019, (2007)
[49] H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [INSPIRE]. · Zbl 1332.81010
[50] Caron-Huot, S., Loops and trees, JHEP, 05, 080, (2011) · Zbl 1296.81128
[51] Arkani-Hamed, N.; Bourjaily, JL; Cachazo, F.; Caron-Huot, S.; Trnka, J., The all-loop integrand for scattering amplitudes in planar N = 4 SYM, JHEP, 01, 041, (2011) · Zbl 1214.81141
[52] Ellis, RK; Giele, WT; Kunszt, Z.; Melnikov, K., Masses, fermions and generalized D-dimensional unitarity, Nucl. Phys., B 822, 270, (2009) · Zbl 1196.81234
[53] Ellis, RK; Kunszt, Z.; Melnikov, K.; Zanderighi, G., One-loop calculations in quantum field theory: from Feynman diagrams to unitarity cuts, Phys. Rept., 518, 141, (2012)
[54] Pius, R.; Sen, A., Cutkosky rules for superstring field theory, JHEP, 10, 024, (2016) · Zbl 1390.81463
[55] Sen, A., Unitarity of superstring field theory, JHEP, 12, 115, (2016) · Zbl 1390.81469
[56] Mason, L.; Skinner, D., Ambitwistor strings and the scattering equations, JHEP, 07, 048, (2014)
[57] Adamo, T.; Casali, E.; Skinner, D., Ambitwistor strings and the scattering equations at one loop, JHEP, 04, 104, (2014)
[58] Ohmori, K., Worldsheet geometries of ambitwistor string, JHEP, 06, 075, (2015)
[59] Geyer, Y.; Mason, L.; Monteiro, R.; Tourkine, P., Loop integrands for scattering amplitudes from the Riemann sphere, Phys. Rev. Lett., 115, 121603, (2015)
[60] Geyer, Y.; Mason, L.; Monteiro, R.; Tourkine, P., One-loop amplitudes on the Riemann sphere, JHEP, 03, 114, (2016) · Zbl 1388.81906
[61] Geyer, Y.; Mason, L.; Monteiro, R.; Tourkine, P., Two-loop scattering amplitudes from the Riemann sphere, Phys. Rev., D 94, 125029, (2016)
[62] D’Hoker, E.; Phong, DH, The geometry of string perturbation theory, Rev. Mod. Phys., 60, 917, (1988)
[63] Plahte, E., Symmetry properties of dual tree-graph n-point amplitudes, Nuovo Cim., A 66, 713, (1970)
[64] Bjerrum-Bohr, NEJ; Donoghue, JF; Vanhove, P., On-shell techniques and universal results in quantum gravity, JHEP, 02, 111, (2014) · Zbl 1333.83043
[65] Feng, B.; Huang, R.; Jia, Y., Gauge amplitude identities by on-shell recursion relation in S-matrix program, Phys. Lett., B 695, 350, (2011)
[66] Kawai, H.; Lewellen, DC; Tye, SHH, A relation between tree amplitudes of closed and open strings, Nucl. Phys., B 269, 1, (1986)
[67] Bern, Z.; Dixon, LJ; Perelstein, M.; Rozowsky, JS, Multileg one loop gravity amplitudes from gauge theory, Nucl. Phys., B 546, 423, (1999) · Zbl 0953.83006
[68] Bjerrum-Bohr, NEJ; Damgaard, PH; Feng, B.; Sondergaard, T., Gravity and Yang-Mills amplitude relations, Phys. Rev., D 82, 107702, (2010) · Zbl 1291.81230
[69] Broedel, J.; Schlotterer, O.; Stieberger, S., Polylogarithms, multiple zeta values and superstring amplitudes, Fortsch. Phys., 61, 812, (2013) · Zbl 1338.81316
[70] Mafra, CR, Berends-giele recursion for double-color-ordered amplitudes, JHEP, 07, 080, (2016) · Zbl 1390.81336
[71] Baadsgaard, C.; etal., New representations of the perturbative S-matrix, Phys. Rev. Lett., 116, (2016) · Zbl 1356.81208
[72] He, S.; Yuan, EY, One-loop scattering equations and amplitudes from forward limit, Phys. Rev., D 92, 105004, (2015)
[73] Cachazo, F.; He, S.; Yuan, EY, One-loop corrections from higher dimensional tree amplitudes, JHEP, 08, 008, (2016)
[74] Bern, Z.; Kosower, DA, Color decomposition of one loop amplitudes in gauge theories, Nucl. Phys., B 362, 389, (1991)
[75] V. Del Duca, L.J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys.B 571 (2000) 51 [hep-ph/9910563] [INSPIRE].
[76] Chester, D., Bern-carrasco-johansson relations for one-loop QCD integral coefficients, Phys. Rev., D 93, (2016)
[77] Primo, A.; Torres Bobadilla, WJ, BCJ identities and d-dimensional generalized unitarity, JHEP, 04, 125, (2016)
[78] Du, Y-J; Lüo, H., On general BCJ relation at one-loop level in Yang-Mills theory, JHEP, 01, 129, (2013) · Zbl 1342.81277
[79] F V. Tkachov, A theorem on analytical calculability of four loop renormalization group functions, Phys. Lett.B 100 (1981) 65.
[80] Chetyrkin, KG; Tkachov, FV, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys., B 192, 159, (1981)
[81] S. Laporta and E. Remiddi, The analytical value of the electron (g − 2) at order α\^{3}in QED, Phys. Lett.B 379 (1996) 283 [hep-ph/9602417] [INSPIRE].
[82] S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys.A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE]. · Zbl 0973.81082
[83] Bern, Z.; Kosower, DA, Efficient calculation of one loop QCD amplitudes, Phys. Rev. Lett., 66, 1669, (1991)
[84] Bern, Z.; Kosower, DA, The computation of loop amplitudes in gauge theories, Nucl. Phys., B 379, 451, (1992)
[85] 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].
[86] Schubert, C., Perturbative quantum field theory in the string inspired formalism, Phys. Rept., 355, 73, (2001) · Zbl 0988.81108
[87] Bjornsson, J., Multi-loop amplitudes in maximally supersymmetric pure spinor field theory, JHEP, 01, 002, (2011) · Zbl 1214.81134
[88] Badger, S.; Mogull, G.; Ochirov, A.; O’Connell, D., A complete two-loop, five-gluon helicity amplitude in Yang-Mills theory, JHEP, 10, 064, (2015) · Zbl 1388.81274
[89] Bern, Z.; Enciso, M.; Parra-Martinez, J.; Zeng, M., Manifesting enhanced cancellations in supergravity: integrands versus integrals, JHEP, 05, 137, (2017) · Zbl 1380.83275
[90] Mizera, S., Combinatorics and topology of Kawai-lewellen-tye relations, JHEP, 08, 097, (2017) · Zbl 1381.83126
[91] Casali, E.; Tourkine, P., Infrared behaviour of the one-loop scattering equations and supergravity integrands, JHEP, 04, 013, (2015)
[92] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory. Volume 2: loop amplitudes, anomalies and phenomenology, Cambridge University Press, Cambridge U.K. (1988).
[93] Panzer, E., Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun., 188, 148, (2015) · Zbl 1344.81024
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