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Minhang, Rd., Dongchuan 800 University, Tong Jiao Shanghai .PROLOGUE I. and aua hlspyvru hlspyo Naturalness of Philosophy versus Philosophy Natural sn-a e nttt eateto hsc n Astrono and Physics of Department & Institute Lee Tsung-Dao Disclaimer Z oo eedsoee.This discovered. were boson ofrnet give to conference oa Senjanovi´c Goran HC nprdb h oko ioolse l[ pre- al high et Dimopoulos the of on work working unification, the 1981, and supersymmetry by sentiment In energy inspired low this of it. of analysis with cision whirlpool love the in in fell my- caught I was supersymmetry. energy self low of possibility beautiful small. very are how and they are, when they them research what protect the are of to numbers drive example the to why prime started asking the into that was issues This naturalness the protec- no behind. the is that symmetry there fact tive since The became problem, hierarchy small below. so infamous is not the ratio if mass - scale) to scale)/(Planck energies physics (weak TeV new the the at requires lie scale unification grand the or xlnto eidtesale fisms [ mass its of main smallnes the the as are behind emerged neutrinos mechanism explanation seesaw that the accepted and massive universally became finally thinking. be wishful can any models basically but achieve that, Models that achieve constructed masses? that particle of constructed SM view be in the can scale of same the spread the enormous should and the one world at the lie the all in of super-partners why mixings and Moreover, depending masses the models super-partners. about different assume of you myriads what on of to the collection because out a but turned - be pro- Model scales still Standard high it the TeV, Supersymmetric of above Minimal peril lies not the it from supersymmetry if us even to tects all, due after much started - so slowly seen not I being though faith, years my the lose Over to others it. many on as hooked I was prediction), supersymmetry energy low l a on tLPt be an- it to mixing when LEP weak but at - the found one was meanwhile bottom gle (and the correct to out close turned be that ’knew’ to everybody had - mass era top that in blasphemy a was That eurdtetpqakms oleaon 0 e [ GeV 200 around lie the smaller to of mass be quark to value top wrongly predicted the claimed required the then - - that angle suggested mixing weak I and ciano opig.S h nytu oiainfrlweeg su- energy low of su- for unification motivation no the true and only of nicely the theory, sake So fits the SO(10) couplings. for that the needed scale in is - same persymmetry mass with the neutrino nicely - with fits scale seesaw intermediate the an unification, grand of context h irrh rbe a rudt ecrdb the by cured be to argued was problem hierarchy The qal motn,a h einn ftecnuyit century the of beginning the at important, Equally hnmn fntr.Iagehr that here argue I nature. of phenomena w . ntes-aldheacypolm To problem. hierarchy so-called the on ea1,I311Tise tl and Italy Trieste, I-34151 11, iera iigpicpefrteftr four of future the for principle uiding etn n e ocnuisof centuries to led and Newton, d hsc rmisoiisa natural as origins its from physics tnadMdlpril physics, particle Model Standard hnhi204,China 200240, Shanghai y SKLPPC, my, θ W 0 = § . 3i codwt the with accord in 23 1 ,Bl Mar- Bill ], 3 .I the In ]. 2 ]. 2 persymmetry remained the hierarchy issue of the Higgs shows that the problem was never there. For a believer mass. in physics as natural philosophy, a solution to a problem On the other hand, with time the SM turned into a by definition requires well defined observable new phe- high precision theory, and thus clearly it had no problem nomena. One may argue that this is simply a question of of inconsistency of any kind - so why in the world would semantics, but semantics is important since it influences it have to be modified at the TeV scale? I started to the way we think and work. Dvali also points out what feel more and more that the explanation for the Higgs many of us felt for a long time - that the naturalness boson lightness could have to do with whatever could be program failed badly in the case of the cosmological con- the consistency requirement of some fundamental high stant, an additional argument against the obsession with energy physics, or some deep, presently unknown, princi- naturalness. ple. At the same time, low energy supersymmetry started In any case, I feel that time is ripe to discuss the to dominate the BSM model ad nauseam. To appreciate essence of what I believe is wrong with the hierarchy the degree of supersymmetry fever, think of this incredi- problem bandwagon. I start with a general, albeit brief, ble fact - any deviation from the SM is called exotics by discussion of naturalness and then move to the specific the LHC workgroups - even the attempts to understand issue of the Higgs mass. the origin of neutrino mass through minimal extensions of the SM. Not low energy supersymmetry though, in spite of the total lack of physical, phenomenological mo- II. NATURALNESS IN A NUTSHELL tivation. Meanwhile another important development took place The concept of naturalness has been nicely reviewed - an explicit realization of the possibility of the TeV cut- in [5]. Let me phrase it here once again for the sake of off of field theory in the context of large extra dimen- completeness. Take a physical parameter θ which we can sions, which implied that there may not even be high divide in the classical, tree-level, part and the quantum energy field theory [4]. Still, in spite of my original en- part of radiative corrections thusiasm for this striking possibility, I again felt that there was nothing urgent about the hierarchy problem, θ = θtree + θloop. (1) but rather that BSM physics should be based on physical, The parameter θ may be dimension-full (a particle mass, phenomenological arguments. The field stayed firmly on its grounds though, and I started to worry that this obses- say) or dimension-less (a coupling). If it happens that sion with naturalness of physics parameters was taking us θ ≫ θ ,. (2) in a wrong direction. After all, the great success of post- loop exp Galilean physics was precisely due to it having become a where θexp is an experimental value (or bound), we say natural philosophy, as called eloquently by Newton (and that the parameter θ is unnaturally small. It then re- emphasised lot by Weinberg in modern times) and prac- quires the fine-tuned cancellation between θtree and θloop, ticed by more than four centuries - a precise and predic- something that most of us would consider rather unap- tive study of natural phenomena driven by phenomeno- pealing. This has been named a naturalness, or fine- logical observations and philosophical implications, but tuning, problem. When θ is dimension-full it is called a never just the latter without the former. hierarchy problem, since it requires a hierarchy of scales. In short, high energy physics, instead of being Notice however that this is not a problem - it can al- natural philosophy, was being transformed into a ways be done at the needed precision and at a chosen philosophy of naturalness. order of perturbation theory - but it is certainly a puz- It has been some years that I wanted to write an essay zling question. As I will argue below, calling it a prob- along these lines and now I am additionally inspired by lem obscures the issue and creates confusion. There is a recent paper by Dvali [5], which I feel vindicates my definitely no problem of naturalness when a protective point of view. Let me be clear here: it is not that I feel symmetry keeps θloop small in perturbation theory, such that the hierarchy question is not fundamentally impor- as say chiral symmetry in the limit of vanishing fermion tant - I only claim that it is not urgent, in a sense of mass. requiring a change of low energy physics. Dvali’s work The case of enhanced symmetry in the limit θtree =0 provides an interesting and profound scenario of how this for a physical parameter θ has been called technical nat- could come about, by means of a consistency argument uralness, or naturalness a la ’t Hooft [7]. No need for fine (based on his early work with Vilenkin [6]) that could tuning in such a case, which provides a welcome relief. let us live with the low value of the weak scale, even I wish to emphazise here that there is a truly natural, if the new physics lies at the arbitrarily large scale. In intuitive definition of naturalness, something that hap- simple terms he argues that the vacuum in which live, pens in our daily experience. Small fermion masses are with the observed value of the Higgs field much smaller technically natural but equally important they should be than the Planck scale, may not be problem at all since it considered natural precisely since most of them are small. could be a product of inflationary evolution. I personally Most fermions are light compared to the electro-weak would not say that this provides the solution, but that it scale responsible for their masses, which is tantamount 3 to small Yukawa couplings. And yet, more often than SM Higgs doublet cannot live alone, since it must be not, a proponent of a new model will argue that the nat- embedded into a representation of a GUT gauge group. ural values for new Yukawa couplings ought to be of order A celebrated example is provided by the minimal SU(5) one. Even worse, in many instances simple models are theory where the SM Higgs lives in a 5-dimensional rep- made baroque upon learning that some couplings have to resentation 5H , consisting of the SM doublet D and a be very small. new heavy colour triple, weak singlet scalar T. The lat- There is much to be said about technical vs natural ter field must be heavy and live practically at the GUT naturalness, but suffice it to say that the naturalness con- scale MGUT since it mediates proton decay. The masses cept is often abused in demanding that a technically nat- of D and T gets split up by the vev of the adjoint Higgs ural small parameter θ vanish by ad-hoc use of a global 24H needed for the GUT scale symmetry breaking. In symmetry, instead of studying the experimental limits the process the D and T masses become (up to irrelevant on θ. In the case of protective symmetry, zero is not a Clebsch-Gordan coefficients) special point and thus any small value for θ is equally 2 2 2 2 2 2 acceptable. After all, zero is smaller than small. mD = m5 − MGUT , mT = m5 + MGUT (3) It would be tautology to say that the proton must be where m5 is the original symmetric mass term of the 5 absolutely stable due to the baryon number symmetry H field. Since m must be not much smaller than M which protects its longevity. It would be equally tau- T GUT (T mediates proton decay), this implies m5 ≃ M to tological to argue that proton-neutron mass difference GUT an astonishing precision. This is the infamous doublet- ought to vanish due to the isospin symmetry which pro- triplet (D-T) splitting problem, an explicit example of tects its smallness. This I am sure is obvious to every- mandatory fine-tuning. In the context of grand unifica- body, and yet such arguments are constantly used in lit- tion, where a new huge mass scale is a must, the issue of erature. The prime example is the celebrated Z2 symme- fine-tuning is clearly very painful. try [8] in the case of two Higgs doublets which guarantees What happens in the supersymmetric extension of the neutral current flavor conservation at the tree-level in all theory? The answer unfortunately is the same painful of the parameter space. This is a strange demand - af- fine-tuning and the reason is obvious. Once again one ter all the GIM mechanism works only for a small charm must split doublet and triplet multiplet masses, and once quark mass. The Z2 extreme limit was justifiably criti- again this is achieved by having the 24 acquire a huge cized in [9], since there is nothing special about the point H vev, of order M - hence precisely the same problem of exact global symmetry, but is pursued to this day and GUT all over. Supersymmetry per se offers no help - all it does used over and over. is to keep the fine-tuning stable as long as the scale of Let us however in what follows concentrate on a par- supersymmetry breaking is not far from the electro-weak ticular case of a dimension-full parameter, mass. As we one. said a fermion mass is protected by the chiral symme- This to me is a sufficient reason to be cautious about try. Similarly, a gauge boson mass is protected by a local supersymmetry. True, it predicted the gauge coupling gauge symmetry (which can actually be regained even for unification, but this can be also achieved naturally with- a massive gauge boson) so the issue is only relevant for out it, with a large seesaw scale. In the context of a scalar mass. the SO(10) grand unified theory, an intermediate see- saw scale provides sufficiently large neutrino mass and at the same time, predicts the unification of the SM gauge III. THE HIERARCHY PROBLEM couplings. The point is that low energy supersymmetry does not answer the mystery of the SM scale being so We are now led to supersymmetry, since in this case much below the GUT scale. Of course, one can employ there is no difference between scalars and fermions, and group theoretic tricks to achieve the D -T splitting with- the same chiral symmetry that protects fermion mass also out fine tuning, but it is typically done with a change at protects the scalar one. This is what makes supersymme- MGUT , a change that does not affect low energy physics. try so special from the hierarchy point of view and this is A particular approach stands out in that sense, Higgs as a why one would love the scale of supersymmetry breaking pseudo-Goldstone boson of an accidental symmetry [10]. in our world to be not too far from the Higgs mass scale, But even this appealing idea ends up having no impact on or the W mass. the low energy world that can be probed in any foresee- But how far is not too far? Notice how this becomes able future. True, it requires low energy supersymmetry immediately an aesthetic or emotional issue as to when to protect the Higgs from going to the GUT scale, but fine tuning is too much - a personal feeling more than the pseudo-Goldstone nature of the Higgs is hidden from the scientific criterium. In any case, the fine tuning - if the low energy observer. it is there - persists even with supersymmetry, no matter An important comment. Neither the desire to protect how low its breaking scale may end up being. the D − T splitting from loops effects, once fine-tuned, After all, in the SM there is no fine tuning whatsoever, nor the gauge coupling unification can give a quantita- not unless you bring up explicitly a new high energy scale, tive prediction for the scale of supersymmetry breaking such as in the case of grand unification. In that case, the (defined here as the difference in masses between parti- 4 cles and super-partners). Unlike what is often claimed, door? The answer lies in the physically attractive pos- this scale could easily be above the LHC reach without sibility of neutrino being Majorana particle. Since it is anything being lost. It is not even clear that the super- only its vanishing charge that separates neutrino from its partner’s masses ought to lie at one and the same scale, electro-weak sibling the electron, it is natural that it may even if one worries about the Higgs mass protection, be behind the smallness of its mass. And sure enough, it since lighter generations could have much heavier super- is precisely the Majorana nature of neutrino that leads partners without any visible effect. Just as it should to the seesaw mechanism. Now, one often argues that have never be claimed that supersymmetry ought to be the natural scale of seesaw is large, close to MGUT since observed at the LHC (or LEP as it was argued in the then the Dirac Yukawa coupling becomes of order one. eighties), it is senseless to say that it is being ruled out But is this true? by not being seen at the LHC. The low energy supersym- After all, large scales bring in the same hierarchy is- metry is as alive or as dead as it ever was. sue that one argues to be a real problem. It is surely In any case, I, for one, remain skeptical of the claim more technically natural to have a lower scale and small that new observable physics must stem from the solu- Yukawa couplings, in view of their self-protection. More- tion to this profound issue. We have no clue of a more over, the SM is all about small Yukawa couplings of fundamental theory of nature, and it could well be that charged fermions, so it is more natural for Dirac Yukawas the consistency mechanism behind it may unravel this to be small, in the natural sense of the word natural. mystery without an impact on the TeV, as argued by Notice however that once again we have fallen into the Dvali [5]. trap of being philosophical about physical scales, once Before I move to discuss the physical motivation for again we have let the naturalness criterium tell us where new physics at the energies accessible to the LHC or next new physics ought to lie. This is wrong. The scale is- hadron collider, a few words more about grand unifica- sue ought to be dictated by phenomenological considera- tion. Needless to say, it remains a beautiful, deeply moti- tions. So what about phenomenological arguments? The vated, extension of the SM with profound consequences: answer lies in the text-book consequence of the Majorana magnetic monopoles and proton decay. I myself have nature of neutrino: neutrinoless double beta decay. To spent years working on it, both ordinary and supersym- get a feel for what is expected, recall that the measure metric versions, trying hard (as many others) to come up of this process in the case of being mediated by neutrino with a predictive theory of proton decay branching ratios. Majorana mass is given by Except for the minimal, original SU(5) Georgi-Glashow m A ≃ G2 ν , (4) theory, [11] ruled out by experiment, no other model is ν F p2 truly predictive due to large threshold effects from many new states around the GUT scale, a discouraging fact for where p is a measure of neutrino virtuality and typically a believer in predictivity as an essential criterium in a on order of 100 MeV for nuclei of experimental interest. search for new physics. Since neutrinoless double beta is a d=9 six fermion pro- It may be worth mentioning that a minimal exten- cess, the coefficient above must be cut-off by the fifth sion [12] of the minimal SU(5) that can account for gauge power of mass - as it is. coupling unification and neutrino mass predicts new par- The present and near future experiments are probing −1 ticles at the TeV energies, but still, the model fails to neutrino Majorana mass to about mν ≃ 10 eV . In any make clear statements about proton decay channels. In case, from mν . 1eV , one has what follows I will not talk about grand unification any −17 −5 more, not because I do not find it fundamentally impor- Aν . 10 GeV . (5) tant but rather since I am here after the guidance for new Now imagine that tomorrow this process is observed. physics at the TeV energies - and grand unification does What would that imply? One possibility of course is a not make a convincing argument in its favor. non-vanishing neutrino Majorana mass, but it is far from being unique since the SM with Majorana neutrino needs an UV completion. But then the new physics may as well IV. IF NOT HIERARCHY, THEN WHAT? be the culprit behind this process, and moreover, it would have to be if the electrons that come out of the neutrino- So if physics is natural philosophy, as we should all be- less double beta decay were for example right-handed. Or lieve, and not a philosophy of naturalness - what could if it was established that the neutrino mass hierarchy is be the new physics at the TeV scale being probed by what we call normal, since in this case the neutrino con- the LHC? Instead of a hierarchy problem, what could tribution is expected to be too small to cause an effect be a motivation for the BSM physics? The answer is observable today (or tomorrow). The fact that neutrino- obvious. The only true failure of the SM is the predic- less double beta decay is not really a probe of neutrino tion of the vanishing neutrino mass - clearly the origin mass has been argued already sixty years ago by Feinberg of neutrino mass ought to be the door to new physics. and Goldhaber and echoed by Pontecorvo [13], and yet The question then is why should we hope that the LHC to this day often ignored. In our seesaw paper [3] Moha- or the next generation hadron collider could open that patra and I have even provided an explicit realization in 5 the form of the RH neutrino and the RH gauge boson as then that high energy physics is alive and well, we have mediators of this process. a plethora of experiments sensitive to new physics scale In any case, whatever new physics may lie behind neu- that may be accessible if not to the LHC itself, at least trinoless double beta, it would have to be quite low in to the future hadron collider. And it has nothing to do order to cause an observable effect. To see this, write with the naturalness or hierarchy issues, issues that are its contribution in the effective form, for simplicity and more emotional or philosophical rather than phenomeno- illustration imagined with a single new mass scale logical. Simply, once again, at the risk of boring the −5 reader to death: neutrinoless double beta, the process ANP ≃ Λ (6) that touches into the essence of up to now sacred law of lepton number conservation, if observed could well point From (6), one has in turn Λ ≥ T eV . In other words, out to new physics. And this new physics would have all the cut-off for this process is on the order of few TeV, the chance in the world to lie at the few TeV energies as tailor-made for the LHC. And new physics is a direct I explained in detail above. lepton number violation at hadron colliders in a form of A particularly illustrative example of such new physics same sign lepton pairs accompanied by jets as argued by is the Left-Right Symmetric Model [18] that led originally Keung and myself [15] more than thirty years ago. It to the prediction of non-vanishing neutrino mass. This seemed almost science fiction at that time, but the LHC theory started as an attempt to attribute maximal parity has made it an exciting reality, and today both CMS and violation to its spontaneous symmetry breaking and over ATLAS are actively pursuing it [16]. Let me not dwell the years turned into the self-contained predictive theory on it, the interested reader should consult [17]. of neutrino mass. In particular it can untangle the see- In summary, if new physics were to cause neutrino- saw mechanism, allowing for the verifiable Higgs origin of less double decay, it could lie tantalisingly close to the neutrino mass, as originally shown with Miha Nemevˇsek LHC energies. Hard to imagine a better motivation for and Vladimir Tello [19], analogous to the Higgs origin of observable new physics, suggested by pure phenomeno- charged fermion masses in the SM. This remarkable re- logical considerations - and yet it is rarely mentioned. sult came as a surprise even to the authors (I can vouch Ironically, neutrinoless double beta decay could actually for at least one of them) and in recent years Tello and be a probe of the new physics behind neutrino Majorana I carried this program to the bitter end, see e.g. [20] for mass and not the probe of the mass itself. This possi- a recent discussion and references therein. It should be bility would be even more exciting for it would open the stressed - and it cannot be overstressed - that left-right door to lepton number violating processes at today’s or symmetry is not used here in tautological sense by pre- near future hadron colliders. dicting that the parameters that violate it must vanish. Rather, it simply says that the dominant breaking is dy- V. EPILOGUE namical with resulting physical consequences. Also, left- right symmetry is not something chosen ad-hoc in order to control the unknown part of a parameter space, as is By now the reader has hopefully grasped the essence of often done in model building. It is the first symmetry my trouble with the present day atmosphere in our field, that the child sees and it shapes our understanding of but let me make sure once again that I get my message the world, culminating with its breaking in weak interac- across as clearly as possible. I have recently come back tions. Its fundamental role is similar to the one played by from a major conference and as usual I heard speaker af- Lorentz symmetry or even more basically by the invari- ter speaker talk about the shortcomings and the problems 1 ance under the laws of mechanics discovered by Galileo. of the SM: the hierarchy problem, the strong CP prob- Imposing such symmetries is surely justifiable as long lem, the fermion mass problem and so on and so forth, all as we do not claim that that they must be exact or eter- used as a guidance for the search of new physics. And yet nal. none of these problems based on naturalness are problems In short, we can have phenomenological and experi- at all, on the contrary they impose no constraint what- mental facts guide us to beyond the SM physics, as they soever on the parameter space of the SM. The only prob- did over and over in the past. All that is needed is a lem of the SM is a lack of problems. Even the strong CP sense of discipline and rigour and avoiding at all cost violation imposes no constraint on its parameter space, pessimistic proclamations of the end of phenomenologi- strangely enough. cally based particle physics. These proclamations are the True, there is dark matter but we have no idea at all result of the incredibly optimistic feelings that all that is what it is and if it ends up being black holes of one type left to be done is to explain why numbers are what they or another, it would not require new physics whatsoever. are (and not what they are not). The history of physics is Fortunately the SM did fail, and it failed loudly and clearly by predicting a vanishing neutrino mass. Ar- guably then, the most obvious and the only true phe- nomenological road to observable BSM physics ought to 1 I am grateful to G. Dvali for emphasising this point regarding be the origin of neutrino mass. I find it mind boggling the importance of left-right symmetry, the very reason for my that this is not universally shared. To me everything says original entusiasm. 6 full of such social movements that wanted to end physics hierarchy and naturalness issues discussed here. He is and they have always been rather damaging, to put it grateful to Charanjit Aulakh, Alessio Maiezza, Alejandra mildly. Think of the bootstrap, Regge poles and such Melfo, Miha Nemevˇsek, Fabrizio Nesti, Vladimir Tello that were to represent the end of fundamental physics in and Francesco Vissani for many fruitful discussions over the sixties and think of all the incredible progress that the years on a number of issues discussed above. He was being made precisely while the doom was preached: also uses the opportunity to apologise to them for com- quarks with color and its gauging and the asymptotic plaining too often against the bandwagon of naturalness freedom that led to QCD as a theory of strong interac- (and other wagons) instead of writing this a long time tions; the electro-weak gauge theory, the ideas of spon- ago. Thanks are due to Gia Dvali, Alejandra Melfo and taneous symmetry breaking with Nambu-Goldstone and Vladimir Tello for their encouragement, and help in im- Higgs mechanisms, some of the greatest achievements in proving the quality and clarity of my presentation. I am the history of physics. And incredibly enough, almost also grateful to Alessio Maiezza and Fabrizio Nesti for no attention was being paid to these developments when careful reading of the manuscript. they were taking place, maybe precisely due to the pre- vailing atmosphere that there was nothing basic to be uncovered any more. How can you recognize a new de- velopment when you sustain that such a thing cannot exist? I initiated this essay during the visit to the theory divi- sion of Fermilab last summer and did most of the writing Acknowledgments while visiting the particle physics division of Tsung-Dao Lee institute in Shanghai. I am grateful to both institu- The author acknowledges a quarter of century of di- tions for their warm hospitality, in particular to Wai-Yee alectic and often fierce debates with Gia Dvali on the Keung and Stephen Parke for making me feel at home.

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