In grand unified theories with large numbers of fields, renormalization
effects significantly modify the scale at which quantum gravity becomes strong.
This in turn can modify the boundary conditions for coupling constant
unification, if higher dimensional operators induced by gravity are taken into
consideration. We show that the generic size of, and the uncertainty in, these
effects from gravity can be larger than the two-loop corrections typically
considered in renormalization group analyses of unification. In some cases,
gravitational effects of modest size can render unification impossible.Comment: 3 pages, to appear in the proceedings of 16th International
  Conference on Supersymmetry and Unification of Fundamental Interactions
  (SUSY08), Seoul, Korea, June 16-21 200

David Reeb

Deog Ki Hong

Pyungwon Ko

Stephen D. H. Hsu

Xavier Calmet

English

arXiv

In grand unified theories with large numbers of fields, renormalization
effects significantly modify the scale at which quantum gravity becomes strong.
This in turn can modify the boundary conditions for coupling constant
unification, if higher dimensional operators induced by gravity are taken into
consideration. We show that the generic size of, and the uncertainty in, these
effects from gravity can be larger than the two-loop corrections typically
considered in renormalization group analyses of unification. In some cases,
gravitational effects of modest size can render unification impossible.
 Comment: 3 pages, to appear in the proceedings of 16th International
  Conference on Supersymmetry and Unification of Fundamental Interactions
  (SUSY08), Seoul, Korea, June 16-21 200

Calmet, Xavier

Hsu, Stephen D.H.

Reeb, David

16th International Conference on Supersymmetry and the Unification of Fundamental Interactions (SUSY08)

DIAL UCLouvain

arXiv:0809.3953v1  [hep-ph]  23 Sep 2008Quantum Gravitational Effects and Grand UnificationXavier Calmet∗, Stephen D. H. Hsu† and David Reeb†∗Catholic University of Louvain, Center for Particle Physics and Phenomenology, 2, Chemin du Cyclotron,B-1348 Louvain-la-Neuve, Belgium†Institute of Theoretical Science, University of Oregon, Eugene, OR 97403, USAAbstract. In grand unified theories with large numbers of fields, renormalization effects significantly modify the scale atwhich quantum gravity becomes strong. This in turn can modify the boundary conditions for coupling constant unification,if higher dimensional operators induced by gravity are taken into consideration. We show that the generic size of, and theuncertainty in, these effects from gravity can be larger than the two-loop corrections typically considered in renormalizationgroup analyses of unification. In some cases, gravitational effects of modest size can render unification impossible.Keywords: grand unification, supersymmetry, quantum gravityPACS: 12.10.Kt 04.60.-m 12.10.DmLEP data hint towards a unification of the couplingconstants of the standard model, or possibly of its su-persymmetric version, at a large energy scale of order1016 GeV [1]. However, this scale is uncomfortably closeto the Planck scale – the energy at which quantum gravi-tational effects become strong. Such effects can alter theboundary conditions on coupling constant unification atthe grand unified scale [2], and, since their precise size isonly determined by Planck scale physics, introduce un-certainties in predictions of grand unification.We identify an additional uncertainty, arising from therenormalization of the quantum gravity scale itself. Wefind that the Planck scale is reduced significantly in mod-els with large numbers of particles (e.g., of order 103species, common in many grand unified models, and of-ten mostly invisible at low energies). This in turn leads toadditional uncertainties in the low energy coupling val-ues associated with unification (see Fig. 1); these uncer-tainties are generically as large as the two-loop correc-tions to the renormalization group equations that havebecome part of the standard analysis of grand unification.Our results suggest that low energy results alone cannot,with any high degree of confidence, either suggest or ruleout grand unification.The strength of the gravitational interaction is mod-ified, i.e. renormalized, by matter field fluctuations[3, 4]. One finds that the effective Planck mass de-pends on the energy scale µ as M(µ)2 = M(0)2 −µ212pi(N0 + N1/2−4N1), where N0, N1/2 and N1 are thenumbers of real spin zero scalars, Weyl spinors and spinone gauge bosons coupled to gravity. M(0) = MPl isthe Planck mass at low energies – i.e., it is directly re-lated to Newton’s constant G = M(0)−2 in natural unitsh¯ = c = 1. Related calculations performed in string the-ory, which presumably take into account quantum grav-ity effects, lead to the same behavior for the running ofthe Planck mass [5].If the strength of gravitational interactions is scaledependent, the true scale µ∗ at which quantum grav-ity effects are large is one at which M(µ∗) ∼ µ∗. Thiscondition means that fluctuations in spacetime geome-try at length scales µ−1∗ will be unsuppressed. It hasbeen shown in [4] that the presence of a large numberof fields can dramatically impact the value µ∗. For ex-ample, it takes 1032 scalar fields to render µ∗ ∼ TeV,thereby removing the hierarchy between weak and grav-itational scales. In many grand unified models, which westudy here, the large number of fields can cause the truescale µ∗ of quantum gravity to be significantly lower thanthe naive value MPl ∼ 1019 GeV. In fact, from the aboveequations,µ∗ = MPl/η , (1)where, for a theory with N ≡ N0 + N1/2−4N1,η =√1 +N12pi. (2)We will exhibit examples of grand unified theorieswith N ∼O(103), so that the scale of quantum gravity isup to an order of magnitude below the naive Planck scale.In such models, corrections to the unification conditionsfrom quantum gravity are much larger than previouslyconsidered [2]. In this paper, we show that the genericsize of these effects can be larger than the two-loop cor-rections usually considered in RG analyses of unifica-tion, and that in some cases even modestly sized grav-itational effects can render unification impossible. Suchlarge uncertainties might impact whether one considersapparent unification of couplings to be strong evidencefor grand unification or supersymmetry.The breaking of a grand unified gauge group down tothe standard model group SU(3)×SU(2)×U(1) via Higgsmechanism typically involves several scalar multiplets,which can be in large representations. Furthermore, thetotal number of these scalar degrees of freedom in theform of Higgs bosons is typically much larger than thenumber of gauge bosons, so N = N0 +N1/2−4N1 can belarge. In this paper, we mainly consider supersymmet-ric grand unified theories, since they naively lead to bet-ter unification results compared to non-supersymmetricmodels [1].For example, SUSY-SU(5) with three families alreadyhas N = 165, i.e. η = 2.3. In SUSY-SO(10) models, thenumbers are larger: the minimal supersymmetric SO(10)model uses 126, 126, 210 and 10 Higgs multiplets, yield-ing N = 1425 or η = 6.2. Some models use even moremultiplets, others use fewer and smaller ones, althoughthe model with the smallest representations 10, 16, 16and 45 – yielding N = 270 and η = 2.9 – leads toR-parity violation and other problems. We thus haveη ∼ 5 for most reasonable SUSY-SO(10) models. Otherunification groups considered in the literature includeE8× E8, which is motivated by string theory and re-quires both 248 and 3875 Higgs multiplets, clearly yield-ing even bigger renormalization effects on MPl.Quantum gravity effects have been shown to affect theunification of gauge couplings [2]. The lowest order ef-fective operators induced by a quantum theory of gravityare of dimension five, such as [2]cµˆ∗Tr(GµνGµνH), (3)where Gµν is the grand unified theory field strength andH is a scalar multiplet. This operator is expected to beinduced by strong non-perturbative effects at the scale ofquantum gravity, so has coefficient c∼ O(1) and is sup-pressed by the reduced true Planck scale µˆ∗ = µ∗/√8pi =ˆMPl/η with ˆMPl = 2.43×1018 GeV. Note, there is someambiguity as to whether the Planck scale µ∗ or the re-duced Planck scale µˆ∗ should be used [2]. Our main pointhere is the gravitational enhancement η of this opera-tor due to renormalization of the quantum gravity scale,which has not been taken into consideration previously.To be as concrete and unambiguous as possible, wewill first examine these gravitational effects in the exam-ple of SUSY-SU(5). Operators similar to (3) are presentin all grand unified theory models and an equivalent anal-ysis applies. The following analysis can be carried oververbatim to SO(10) models [6].In SU(5) the multiplet H in the adjoint represen-ation acquires, upon symmetry breaking at the unifi-cation scale MX , a vacuum expectation value 〈H〉 =MX (2,2,2,−3,−3)/√50piαG, where αG is the valueof the SU(5) gauge coupling at MX . Inserted into theoperator (3), this modifies the gauge kinetic terms ofSU(3)×SU(2)×U(1) below the scale MX to− 14(1 + ε1)FµνFµνU(1)−12(1 + ε2)Tr(FµνF µνSU(2))(4)−12(1 + ε3)Tr(FµνF µνSU(3))with ε1 = ε2/3 =−ε3/2 =√25√picη√αGMXˆMPl.After a finite field redefinition Aiµ → (1 + εi)1/2 Aiµthe kinetic terms have familiar form, and it is thenthe corresponding redefined coupling constants gi →(1 + εi)−1/2 gi that are observed at low energies and thatobey the usual RG equations below MX , whereas it is theoriginal coupling constants that need to meet at MX inorder for unification to happen. In terms of the observ-able rescaled couplings, the unification condition there-fore reads:αG = (1 + ε1)α1(MX) = (1 + ε2)α2(MX ) (5)= (1 + ε3)α3(MX) .Numerical unification results using this boundary con-dition are shown in Fig. 1. Leaving the parametersα3(MZ) and MSUSY open in some range in order to com-pare the size of the corrections from the new boundarycondition to experimental uncertainties, we evolved thegauge couplings under two-loop RG equations of theSM/MSSM with SUSY breaking scale MSUSY, takingas fixed α1(MZ) = 0.016887, α2(MZ) = 0.03322. Then,testing each pair (α3(MZ), MSUSY) in the wide range ofparameters of Fig. 1 for unification according to (5), itturns out that for every pair perfect unification happensfor exactly one value of the coefficient c of (3).Our results show that, e.g., in a theory with η ∼ 5,unification depends quite sensitively on the size of thegravitational operator: reasonable values of the coeffi-cient c ∼ O(1) can give unification for quite a largerange of low-energy couplings αi(MZ) and parametersMSUSY, so unification does not seem to be a very spe-cial feature. On the other hand, even a slight change tothe value of c requires quite large adjustments in initialconditions αi(MZ) for unification to still happen. Thisis very unsatisfying since the value of c is determinedonly by some deeper theory of quantum gravity abovethe scale MX , i.e., grand unification cannot be predictedor claimed based on low-energy observations alone, andtherefore loses most of its beauty. More severely yet, theeffects of the gravitational operator can be so large that,if quantum gravity determines the sign of this operator tobe positive with c > 4/η (which is quite natural for the-ories with large particle content), then unification cannothappen for any experimentally allowed parameters of theSM/MSSM model, see Fig. 1.Improving the precision of theoretical predictions andexperimental values seems unnecessary and meaning-less: e.g., for the parameter values α3(MZ) = 0.108,FIGURE 1. For η fixed by the particle content of the theory,solid lines are contours of constant c such that, under the pres-ence of the gravitationally induced and enhanced operator (3),SUSY-SU(5) perfectly unifies at two loops for given values ofthe initial strong coupling constant α3(MZ) and SUSY break-ing scale MSUSY. Over the whole range, unification happensfor some value of the coefficient c, with unification scale andunified coupling between MX = 9.3× 1014 GeV, αG = 0.033(lower right corner) and MX = 5.5× 1016 GeV, αG = 0.045(upper left).MSUSY = 103 GeV, MX = 1016 GeV and αG = 0.0389 fa-vored by Amaldi et al. [1] to yield good unification, tableI compares the shifts α2i (MX)−α1i (MX ) in theoreticalpredictions due to inclusion of two-loop running to thesplittings αG−αG/(1+εi) required by the the boundarycondition (5). These splittings are shown for η ∼ 5 andc = −1, but would be larger or smaller proportional tocη . The table shows that the generic size of, and uncer-tainty in, the effects from gravity is larger than the two-loop corrections. Thus, two-loop computations do actu-ally not improve evidence for unification.TABLE 1. The upper half of the table shows shifts inthe predictions for the values of the coupling constantsat MX = 1016 GeV due to inclusion of two-loop running.These shifts are comparable in size or even smaller thanthe necessary splittings between the αGi due to (5) in thecase η = 5, c =−1 (lower half).i 1 2 3α1i (MX ) 0.03815 0.03767 0.03814α2i (MX ) 0.03897 0.03899 0.03868δαi = α2i −α1i 8.2×10−4 13.2×10−4 5.4×10−4δαi/α1i +2.1% +3.5% +1.4%εi(cη =−5) −0.0167 −0.0503 +0.0335αG(MX ) 0.0389 0.0389 0.0389αGi = αG/(1+ εi) 0.0396 0.0410 0.0376δi = αG−αGi −6.6×10−4 −20.6×10−4 12.6×10−4δi/αG −1.7% −5.3% +3.2%Similarly, the uncertainty in the value of the coeffi-cient c is far greater than experimental uncertainties inmeasurements of SM/MSSM parameters. For example,the parameter range α3(MZ) = 0.108±0.005, MSUSY =103±1 GeV quoted in [1] is covered by varying the coef-ficient c in the small range −2/η < c < 2/η , see Fig. 1.In particular, previous attempts to pin down α3(MZ) orsin2 θW by demanding gauge coupling unification seeminvalid without further knowledge about c. Also, claim-ing that SUSY unification is favored by, e.g., LEP dataseems far-fetched. Without actually observing proton de-cay it is hard to claim convincing evidence for unifica-tion of the gauge interactions of the standard model atsome higher scale. Finally, as can be seen from Fig. 1,the unification scale that would be compatible with cur-rent experimental values of α3(MZ) is of the order ofMX ∼ 1016 GeV, which, depending on the specific modelunder consideration, might be uncomfortably low withrespect to proton decay. Phrased another way, given thecurrent measurements of αi(MZ), the operator (3) cannotbe used to shift the unification scale MX to values muchabove 1016 GeV (this possibility was discussed in pastanalyses [2]).Many predictions of grand unified theories are subjectto uncertainties due to quantum gravitational corrections.We have shown that these uncertainties are significantlyenhanced in models with large particle content (e.g., oforder 103 matter fields), including common variants ofSU(5), SO(10) and E8×E8 unification. Since the quan-tum gravitational corrections and, potentially, most of thelarge number of matter fields appear only at very highenergies, it seems that low energy physics alone cannot,with a high degree of confidence, either suggest or ruleout grand unification. Model builders should perhaps fa-vor smaller matter sectors in order to minimize these cor-rections and obtain calculable, predictive results.ACKNOWLEDGMENTSThis work is supported in part by the Department ofEnergy under DE-FG02-96ER40969. X.C. is a "Chargéde recherches du F.R.S.-FNRS."REFERENCES1. U. Amaldi, W. de Boer and H. Furstenau, Phys. Lett. B260, 447 (1991).2. C. T. Hill, Phys. Lett. B 135, 47 (1984); Q. Shafi andC. Wetterich, Phys. Rev. Lett. 52, 875 (1984); L. J. Halland U. Sarid, Phys. Rev. Lett. 70, 2673 (1993).3. F. Larsen and F. Wilczek, Nucl. Phys. B 458, 249 (1996).4. X. Calmet, S. D. H. Hsu and D. Reeb, Phys. Rev. D 77,125015 (2008).5. E. Kiritsis and C. Kounnas, Nucl. Phys. B 442, 472 (1995).6. X. Calmet, S. D. H. Hsu and D. Reeb, arXiv:0805.0145[hep-ph].

Quantum Gravitational Effects and Grand Unification

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In grand unified theories with large numbers of fields, renormalization effects significantly modify the scale at which quantum gravity becomes strong. This in turn can modify the boundary conditions for coupling constant unification, if higher dimensional operators induced by gravity are taken into consideration. We show that the generic size of, and the uncertainty in, these effects from gravity can be larger than the two-loop corrections typically considered in renormalization group analyses of unification. In some cases, gravitational effects of modest size can render unification impossible. © 2008 American Institute of Physics.SCOPUS: cp.pinfo:eu-repo/semantics/publishe

Hsu, Stephen D H

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Quantum gravitational effects and grand unification

In grand unified theories with large numbers of fields, renormalization effects significantly modify the scale at which quantum gravity becomes strong. This in turn can modify the boundary conditions for coupling constant unification, if higher dimensional operators induced by gravity are taken into consideration. We show that the generic size of, and the uncertainty in, these effects from gravity can be larger than the two-loop corrections typically considered in renormalization group analyses of unification. In some cases, gravitational effects of modest size can render unification impossible

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