Motivated by the absence of signals of new physics at the LHC, which seems to imply the presence of large mass hierarchies, we investigate the theoretical possibility that these could arise dynamically in new strongly-coupled gauge theories extending the standard model of particle physics. To this purpose, we study lattice data on non-Abelian gauge theories in the (near-)conformal regime---specifically, SU(2) with Nf=1 and 2 dynamical fermion flavours in the adjoint representation. We focus our attention on the ratio R between the masses of the lightest spin-2 and spin-0 resonances, and draw comparisons with a simple toy model in the context of gauge/gravity dualities. For models in which large anomalous dimensions arise dynamically, we show indications that this mass ratio can be large, with R>5. Moreover, our results suggest that R might be related to universal properties of the IR fixed point. Our findings provide an interesting step towards understanding large mass ratios in the non-perturbative regime of quantum field theories with (near) IR conformal behaviour

Ed Bennett

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 Cronfa -  Swansea University Open Access Repository   _____________________________________________________________   This is an author produced version of a paper published in :Proceedings of the 34th annual International Symposium on Lattice Field Theory (LATTICE2016)                                                                                       Cronfa URL for this paper:http://cronfa.swan.ac.uk/Record/cronfa32043_____________________________________________________________ Conference contribution :Athenodorou, A., Bergner, G., Elander, D., Lin, C., Lucini, B., Piai, M. & Bennett, E. (2017).  Large mass hierarchiesfrom strongly-coupled dynamics. Proceedings of the 34th annual International Symposium on Lattice Field Theory(LATTICE2016),         _____________________________________________________________  This article is brought to you by Swansea University. Any person downloading material is agreeing to abide by theterms of the repository licence. Authors are personally responsible for adhering to publisher restrictions or conditions.When uploading content they are required to comply with their publisher agreement and the SHERPA RoMEOdatabase to judge whether or not it is copyright safe to add this version of the paper to this repository. http://www.swansea.ac.uk/iss/researchsupport/cronfa-support/  PoS(LATTICE2016)232Large mass hierarchies from strongly-coupleddynamicsAndreas Athenodorouab, Ed Bennettycd , Georg Bergnere, Daniel Elander f , C.-J.David Lingh, Biagio Lucinic, and Maurizio PiaicaDepartment of Physics, University of Cyprus, POB 20537, 1678 Nicosia, CyprusbComputation-based Science and Technology Research Center, The Cyprus Institute, 20 KavafiStr., Nicosia 2121, CypruscCollege of Science, Swansea University, Singleton Park, Swansea SA2 8PP, UKdKobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI), NagoyaUniversity, Nagoya 464-8602, JapaneUniversität Bern, Institut für Theoretische Physik, Sidlerstr. 5, CH-3012 Bern, SwitzerlandfNational Institute for Theoretical Physics, School of Physics, and Mandelstam Institute forTheoretical Physics, University of the Witwatersrand, Johannesburg, Wits 2050, South AfricagInstitute of Physics, National Chiao-Tung University, Hsinchu 30010, TaiwanhCNRS, Aix Marseille Universit, Universit de Toulon, Centre de Physique Thorique, UMR 7332,F-13288 Marseille, FranceMotivated by the absence of signals of new physics at the LHC, which seems to imply the presenceof large mass hierarchies, we investigate the theoretical possibility that these could arise dynam-ically in new strongly-coupled gauge theories extending the standard model of particle physics.To this purpose, we study lattice data on non-Abelian gauge theories in the (near-)conformalregime—specifically, SU(2) with Nf = 1 and 2 dynamical fermion flavours in the adjoint repre-sentation. We focus our attention on the ratio R between the masses of the lightest spin-2 andspin-0 resonances, and draw comparisons with a simple toy model in the context of gauge/gravitydualities. For models in which large anomalous dimensions arise dynamically, we show indica-tions that this mass ratio can be large, with R > 5. Moreover, our results suggest that R mightbe related to universal properties of the IR fixed point. Our findings provide an interesting steptowards understanding large mass ratios in the non-perturbative regime of quantum field theorieswith (near) IR conformal behaviour.34th annual International Symposium on Lattice Field Theory24-30 July 2016University of Southampton, UKSpeaker.yE-mail: e.j.bennett@swansea.ac.ukc Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/PoS(LATTICE2016)232Large mass hierarchies from strongly-coupled dynamics Ed Bennett1. IntroductionThe Higgs particle [1, 2] is the first example in nature of a boson the mass of which is notprotected by symmetry arguments. Its low-energy Effective Field Theory (EFT) description interms of a weakly-coupled scalar field is fine-tuned, as additive renormalisation makes it sensitiveto unknown physics up to high scales. This is the hierarchy problem, one of the main motivationsto investigate theoretical extensions of the Standard Model (SM).Searches at the LHC currently show no new states not predicted by QCD [3, 4], suggesting thatshould such states exist (and have non-vanishing couplings to the SM), then they must have masseslarge enough to have escape detection so far. If a new strongly-coupled theory is responsible forelectroweak symmetry breaking and all the physical phenomena connected with it (reviewed in [5]),it would provide an elegant and conclusive solution to the hierarchy problem(s) of the electroweaktheory; however, such a theory would need to exhibit a large mass hierarchy between the would-beHiggs and other states of the theory.We want to identify models that dynamically produce a large mass hierarchy between com-posite states, that cannot be explained in simple terms by symmetry arguments in a low energyEFT context (and without fine-tuning). We are aiming at something more than what in QCD iscaptured by the chiral Lagrangian, or Heavy Meson Chiral Perturbation Theory (cPT), that ex-plain the masses and properties of pions and of heavy-light mesons, respectively. We want to findappropriate physical observables that allow such an identification to be assessed in a clean, un-ambiguous way distinctive from simple arguments formulated at weak coupling on the basis ofinternal symmetries.In these proceedings, we provide one interesting step in this direction, and suggest possibleways to further develop this challenging research program in the future. Our starting point isidentifying an observable that we can use to characterise a hierarchy in a strongly-coupled theory.Such a quantity would need to have a value near 1 for QCD, which does not exhibit such a hierarchy,and a significantly larger value for theories of phenomenological interest.To this end we observe that, irrespectively of the microscopic details, all Lorentz-invariantfour-dimensional field theories admit a stress-energy tensor Tmn . Correlation functions involvingTmn can be analysed in terms of their scalar (trace) part and tensor (transverse and traceless) part.A well-defined observable is the ratioR=MTM0where MT is the mass of the lightest spin-2 composite state, while M0 is the mass of the lightestspin-0 state. This quantity is defined universally, it can be computed explicitly in a wide varietyof models, it is scheme-independent, and it is not directly controlled by internal global symmetriesof the theory. It is legitimate to compare R computed in theories with completely different internalsymmetries and symmetry-breaking patterns. This is a particularly welcome feature in the con-text of gauge theories with fermionic field content, where the physics of chiral symmetry and itsbreaking introduces non-trivial model-dependent features.2. Mass deformation and infrared behaviourTo better understand the analysis we will perform, we must have an idea of how R will vary1PoS(LATTICE2016)232Large mass hierarchies from strongly-coupled dynamics Ed BennettFigure 1: Schematic representation of as = g2=(4p) as a function of E for three representative choices offermion mass m. In yellow the massless m= 0 case, in which the only scale is L. In purple a case in whichthe mass m is very large. In green a case in which m L.for a conformal theory as it is deformed by some mass m. In the absence of the deforming mass,the theory has a single scale L; the mass distorts the IR behaviour. The consequences of this aresketched in Fig. 1. For m&L, the large mass limit (plotted in purple), the scale L is inaccessible,so the theory is indistinguishable from one that confines. In the absence of a deforming mass, thetheory is IR conformal, as shown in yellow. In the intermediary range meanwhile, the two scalesare both visible—a crossover region is seen where the conformal behaviour is visible, before theconfining behaviour takes over at scales near the deforming mass.This allows us to probe how the spectrum will depend on m. In the region of small, finitedeforming mass, then spectral masses will scale asM µ m1=D ;where D is the scaling dimension, equivalent to 1+g, where g is the anomalous dimension. Sincethe finite lattice volume introduces an absolute IR cutoff, then an appropriate scaling variable toconsider isx= Lm1=D :and so the ith state mass scales asLMi = fi(x) :To first-order for a state of massM0, this givesLM0 µ x :which we can then substitute back in to giveLMi = fi(LM0) :2PoS(LATTICE2016)232Large mass hierarchies from strongly-coupled dynamics Ed BennettThus if we take mass ratios, then the dependence on the lattice extent drops out, allowing us tocompare data at different volumes:MiM j=fi(LM0)f j(LM0)We therefore expect four regimes for the ratio R. At large m, then the physics of interest isquenched out, so R should become consistent with the value seen in pure Yang–Mills theory. ForSU(2), this value has been found in lattice studies to be R= 1:44(4) [6]. Meanwhile at small m, theregime will depend on the lattice extent L. At sufficiently large volumes, then the observed physicsincludes the small E region that shows confining properties. At very small volumes, then the scaleL is no longer observable, and so the theory observed is that of the “femto-universe”. Previousstudies indicate that R  1 in this region [7, 8, 9]. However, there is an intermediary region of Lwhere it is sufficiently large to observe the conformal behaviour, without being so large as to bedistorted by the confining effects. This is the region of interest, which can be extrapolated to achiral limit value, which allows R to be determined for the conformal case.3. Predictions from gauge–gravity dualitySince we have constructed R to allow comparison between disparate theories, we choose toexplicitly draw this comparison. As a reference theory, we choose a toy model constructed makinguse of the principles of gauge–gravity duality. While full details of the model are given in [10], theimportant fact to note here is that it is constructed within the bottom-up approach to holography, insuch a way that the only operator deforming the theory and driving it away from conformality hasscaling dimension D. Thus for a theory investigated on the lattice found to have a particular scalingdimension, we may compare to the toy model with the corresponding value of D.The methodology used allows for the full spectrum of composite spin-0 and spin-2 states tobe computed numerically as a function of D. By taking the ratio of the lowest-lying state for eachcase, we may then find the value of R predicted from the gauge–gravity theory. The main result ofthe gravity analysis is that the value of R is a monotonically increasing function of D, as shown bythe spectrum of scalar and tensor states in Fig. 2.4. Lattice resultsWe consider lattice results from two theories: SU(2) gauge theory with one and two flavours ofadjoint Dirac fermions. Both have been studied on the lattice using the Wilson plaquette action andthe Wilson fermion action, and found to show signals of being in or near the conformal window[11, 12]. The two-flavour theory has been found (at b = 2:25) to have anomalous dimensiong = 0:371(20) [13], corresponding to D = 1:371. The one-flavour theory has been found (atb = 2:05) to have anomalous dimension g = 0:925(25), corresponding to D = 1:925(25) [11].The relevant region of the gauge–gravity plot for each range of D is shown in Fig. 2 as the left andright red highlight respectively, giving values RNf=2  1:95(4) and RNf=1  6:53+1:50 0:91.In each case we make use of configurations from the existing lattice studies mentioned above,which were not optimised for this study. As such a range of both deforming mass m and latticeextent L were considered. The results for R found in both cases are plotted in Figs. 3 and 4. In3PoS(LATTICE2016)232Large mass hierarchies from strongly-coupled dynamics Ed BennettFigure 2: The mass M of the composite spin-0 (black) and spin-2 (blue) states, as a function of the scalingdimension D. Masses are normalised by the mass of the lightest scalar M0, thus the lowest-lying blue linerepresents the ratio R. The relevant regions for the two theories considered are shown in red.Figure 3: The ratio R = MT=M0 computed for SU(2) with two adjoint Dirac flavours (black points). Thepink and blue bands, and the red line, are described in the text.each case, the prediction from the gauge–gravity model at corresponding D is shown as a pinkband, while the numerical result for pure-gauge SU(2) is shown as a blue band. The red line is theresult from the GPPZ model [14], on which the gauge–gravity model is based, which has D = 1and R=p2.In the two-flavour case, at large values of LM0, R closely matches the pure gauge value—asanticipated at large M0. At small values of LM0, R is again near the pure-gauge value but showssigns of descending below it as predicted by the femto-universe; this is expected for small valuesof L. In the intermediary range, R approaches (and takes values compatible with) the predictions4PoS(LATTICE2016)232Large mass hierarchies from strongly-coupled dynamics Ed Bennettof the gauge–gravity model for a theory with the same scaling dimension.Meanwhile in the one-flavour case, at large values of LM0, the behaviour is similar to the two-flavour theory—R is compatible with the pure gauge result. Once again, as LM0 gets smaller, theratio R gets larger, and once again enters a range compatible with the predictions of the gauge–gravity model at the same scaling dimension. However in this case we do not see the other side ofthis peak, heading towards the femto-universe.Figure 4: The ratio R=MT=M0 computed for SU(2) with one adjoint Dirac flavour (black points). The pinkand blue bands, and the red line, are described in the text.5. ConclusionsWe have calculated the measure R=MT=M0 for a variety of theories, both lattice gauge theo-ries with adjoint matter and toy gauge–gravity models. We see an unexpected degree of agreementbetween the two when the scaling dimension is held constant; if this were a hint of universality,then it would suggest that theories of phenomenological interest that have a large anomalous di-mension 1 would also by necessity exhibit the large hierarchy of scales necessary to explain theirabsence in LHC searches.A dedicated study of each theory is necessary, performing scans separately of L and m, toensure that the observed behaviour is not an artefact. Further work is also needed to probe othertheories of phenomenological interest and investigate whether the observed hints of possible uni-versality are also observed there.AcknowledgementsNumerical computations were executed in part on the HPC Wales systems, supported by theERDF through the WEFO (part of the Welsh Government), on the Blue Gene Q system at theHartree Centre (supported by STFC) and on the DiRAC Blue Gene/Q Shared Petaflop system at5PoS(LATTICE2016)232Large mass hierarchies from strongly-coupled dynamics Ed Bennettthe University of Edinburgh. The latter is operated by the Edinburgh Parallel Computing Centreon behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded byBIS National E-infrastructure capital grant ST/K000411/1, STFC capital grant ST/H008845/1, andSTFC DiRAC Operations grants ST/K005804/1 and ST/K005790/1. DiRAC is part of the NationalE-Infrastructure.References[1] ATLAS collaboration, G. 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Large mass hierarchies from strongly-coupled dynamics

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