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Evanescent Effects Can Alter Ultraviolet Divergences in Quantum Gravity Without Physical Consequences

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Evanescent Effects Can Alter Ultraviolet Divergences in Quantum Gravity Without Physical Consequences week ending PRL 115, 211301 (2015) PHYSICAL REVIEW LETTERS 20 NOVEMBER 2015 Evanescent Effects can Alter Ultraviolet Divergences in Quantum Gravity without Physical Consequences Zvi Bern,1,2,3 Clifford Cheung,2 Huan-Hang Chi,4 Scott Davies,1 Lance Dixon,2,4 and Josh Nohle1 1Department of Physics and Astronomy, University of California at Los Angeles, Los Angeles, California 90095, USA 2Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA 3Department of Physics, CERN Theory Division, CH-1211 Geneva 23, Switzerland 4SLAC National Accelerator Laboratory, Stanford University, Stanford, California 94309, USA (Received 3 August 2015; published 17 November 2015) Evanescent operators such as the Gauss-Bonnet term have vanishing perturbative matrix elements in exactly D ¼ 4 dimensions. Similarly, evanescent fields do not propagate in D ¼ 4; a three-form field is in this class, since it is dual to a cosmological-constant contribution. In this Letter, we show that evanescent operators and fields modify the leading ultraviolet divergence in pure gravity. To analyze the divergence, we compute the two-loop identical-helicity four-graviton amplitude and determine the coefficient of the associated (nonevanescent) R3 counterterm studied long ago by Goroff and Sagnotti. We compare two pairs of theories that are dual in D ¼ 4: gravity coupled to nothing or to three-form matter, and gravity coupled to zero-form or to two-form matter. Duff and van Nieuwenhuizen showed that, curiously, the one-loop trace anomaly—the coefficient of the Gauss-Bonnet operator—changes under p-form duality transformations. We concur and also find that the leading R3 divergence changes under duality transformations. Nevertheless, in both cases, the physical renormalized two-loop identical-helicity four-graviton amplitude can be chosen to respect duality. In particular, its renormalization-scale dependence is unaltered. DOI: 10.1103/PhysRevLett.115.211301 PACS numbers: 04.65.+e, 11.15.Bt, 11.25.Db, 12.60.Jv Although theories of quantum gravity have been studied pure gravity diverges, as demonstrated explicitly by Goroff for many decades, basic questions about their ultraviolet and Sagnotti [7] and confirmed by van de Ven [8]. (UV) structure persist. One subtlety is the trace anomaly [1] In this Letter, we investigate the UV properties of the which, at one loop, provides the coefficient of the Gauss- two-loop amplitude for the scattering of four identical- Bonnet (GB) term. The physical significance of this relation- helicity gravitons, including the effect of p-form duality ship has not been settled, however. In particular, Duff and van transformations. We use dimensional regularization, which Nieuwenhuizen showed that the trace anomaly changes forces us to consider the effects of evanescent operators like under duality transformations of p-form fields, suggesting the GB term. By definition, an evanescent operator is that theories related through such transformations are independent of other operators in D dimensions, but either quantum-mechanically inequivalent [2]. In response, vanishes, or is a total derivative, or becomes a linear Siegel argued that this effect is a gauge artifact and, therefore, combination of other operators in four dimensions. We not physical [3]; Fradkin, Tseytlin, Grisaru et al. have also show that the GB counterterm is required to cancel argued that duality should hold at the quantum level [4]. subdivergences and reproduce the two-loop counterterm Furthermore, for D ¼ 4 external states, one-loop divergen- coefficient found previously [7,8]. ces in gravity theories coupled to two-form antisymmetric Evanescent operators are well studied in gauge theory tensors are unchanged under a duality transformation relating (see, e.g., Ref. [9]), where they can modify subleading two-forms to zero-form scalars [5]. However, as we shall see, corrections. In contrast, we find that evanescent effects can intuition based on one-loop analyses can be deceptive. alter the leading UV divergence in gravity. (Effects of the As established in the seminal work of ’t Hooft and Veltman GB term have also been studied in renormalizable, but 2 [6], puregravity is finite at one loop because the only available nonunitary, R gravity [10].) Despite this change in the UV on-shell counterterm is the GB term, which integrates to zero divergence, the physical dependence of the renormalized in a topologically trivial background. While amplitudes with amplitude on the renormalization scale remains unchanged. external matter fields diverge at one loop, amplitudes with This break in the link between the UV divergence and the only external gravitons remain finite. At two loops, however, renormalization-scale dependence is unlike familiar one- loop examples. We arrive at a similar conclusion when comparing the divergences and renormalization-scale dependences in gravity coupled to scalars versus antisym- Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- metric-tensor fields. bution of this work must maintain attribution to the author(s) and Pure gravity is defined by the Einstein-Hilbert (EH) the published article’s title, journal citation, and DOI. Lagrangian 0031-9007=15=115(21)=211301(6) 211301-1 Published by the American Physical Society week ending PRL 115, 211301 (2015) PHYSICAL REVIEW LETTERS 20 NOVEMBER 2015 2 pffiffiffiffiffiffi one loop [6]. In a topologically nontrivial space, the integral L ¼ − −gR; ð1Þ EH κ2 over the GB term gives the Euler characteristic. When matter is added to the theory, the four-graviton amplitude is κ2 ¼ 32π ¼ 32π 2 where GN =MP and the metric signature is still UV finite at one loop, although divergences appear in ðþ−−−Þ L . We also augment EH by matter Lagrangians for amplitudes with external matter states. one of the following: n0 scalars, n2 two-form fields Using the unitarity method, we verified Eq. (5) by (antisymmetric tensors), or n3 three-form fields, considering the one-loop four-graviton amplitude with external states in arbitrary dimensions and internal ones 1pffiffiffiffiffiffiXn0 1pffiffiffiffiffiffiXn2 ¼ 4 − 2ϵ L ¼ − ∂ ϕ ∂μϕ L ¼ − μνρ in D dimensions. On-shell scattering amplitudes 0 2 g μ j j; 2 6 g HjμνρHj ; 2 j¼1 j¼1 are sensitive only to the coefficient of the Rμνρσ operator, 2 2 because the R and Rμν operators can be eliminated by field 1pffiffiffiffiffiffiXn3 L ¼ − − μνρσ ð Þ redefinitions at leading order in the derivative expansion. 3 8 g HjμνρσHj : 2 j¼1 The GB combination is especially simple to work with in dimensional regularization since there are no propagator Here, ϕj is a scalar field and Hjμνρ and Hjμνρσ are the field corrections in any dimension [15]. strengths of the two- and three-form antisymmetric-tensor For the case of antisymmetric tensors coupled to gravity, fields Ajμν and Ajμνρ. The index j labels distinct fields. another relevant one-loop four-point divergence is that of Standard gauge fixing for the two- and three-form actions, two gravitons and two antisymmetric tensors, which is as well as for L , leads to a nontrivial ghost structure. We generated by the operator EH avoid such complications by using the generalized unitarity 2 Xn2 κ 1 1 pffiffiffiffiffiffi μν αρσ method [11–13], which directly imposes appropriate L ¼ −g R ρσH μναH : ð Þ RHH 2 ð4πÞ2 ϵ j j 6 D-dimensional physical-state projectors on the on-shell j¼1 states crossing unitarity cuts. Like the GB term, this operator is evanescent. In particular, Under a duality transformation, in four dimensions, the in D ¼ 4, we can dualize the antisymmetric tensors to two-form field is equivalent to a scalar, scalars, which collapses the Riemann tensor into the Ricci i α scalar and tensor. Under field redefinitions, they can be H μνρ ↔ pffiffiffi εμνρα∂ ϕ ; ð3Þ j 2 j eliminated in favor of the dualized scalars, removing the one-loop divergence in two-graviton two-antisymmetric- and the three-form field is equivalent to a cosmological- tensor amplitudes with D ¼ 4 external states. The four- constant contribution via pffiffiffiffiffi scalar amplitude does diverge. The change in Eq. (5) under duality transformations is 2 Λj H μνρσ ↔ pffiffiffi εμνρσ : ð4Þ j 3 κ central to the claim by Duff and van Nieuwenhuizen of quantum inequivalence under such transformations [2]. As usual, we expand the graviton field around a flat-space Here, we analyze their effects on the two-loop amplitude. ¼ η þ κ background: gμν μν hμν. Similarly, we expand the First, let us note that our unitarity-based evaluation of Eq. (5) scalar, two-form field, and three-form field around trivial sews together physical, gauge-invariant tree amplitudes. This background values. It is interesting to note that the three- explicitly demonstrates that the numerical coefficient of 2 form field has been proposed as a means for neutralizing the Rμνρσ term in Eq. (5) is gauge invariant, in contrast to the cosmological constant [14]. implications of Ref. [3]. This gauge invariance suggests that For a theory with n0 scalars, n2 two-forms and n3 three- by two loops, Eq. (5) could lead to duality-violating con- forms coupled to gravity, the one-loop UV divergence takes tributions to nonevanescent operators. To see if this happens, the form of the GB term [1,2,7] we must account for subdivergences and renormalization. 1 1 53 91 At two loops, pure gravity diverges in D ¼ 4.The L ¼ þ n0 þ n2 − n3 GB ð4πÞ2 ϵ 90 360 360 2 coefficient of this divergence was determined by Goroff pffiffiffiffiffiffi and Sagnotti [7] from a three-point computation in the − ð 2 − 4 2 þ 2 Þ ð Þ × g R Rμν Rμνρσ ; 5 standard MS regularization scheme and later confirmed by which is proportional to the trace anomaly. The calculations vandeVen[8] of the trace anomaly and of the UV divergence are 2 209 κ 1 1 pffiffiffiffiffiffi αβ γδ ρσ L 3 ¼ − − ð Þ essentially the same, except that we replace a graviton R 1440 2 ð4πÞ4 ϵ gR γδR ρσR αβ; 7 polarization tensor with a trace over indices.
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