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Pramana – J. Phys. (2016) 87: 37 c Indian Academy of Sciences DOI 10.1007/s12043-016-1246-2

Why ? beyond the

ROMESHKKAUL

The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600 113, India E-mail: [email protected]

Published online 23 August 2016

Abstract. The Principle as a requirement that the heavy mass scales decouple from the physics of light mass scales is reviewed. In field theories containing elementary scalar fields, such as the Standard Model of electroweak interactions containing the Higgs particle, mass of the scalar field is not a natural parameter as it receives large radiative corrections. How supersymmetry solves this Naturalness Problem is outlined. There are also other contexts where the presence of elementary scalar fields generically spoils the high–low mass scales decoupling in the . As an example of this, the non-decoupling of possible Planck scale violation of Lorentz invariance due to effects from the physics at low scales in theories with elementary scalar fields such as the Higgs field is described. Here again supersymmetry provides a mechanism for ensuring that the decoupling of heavy–light mass scales is maintained.

Keywords. Naturalness problem; ; decoupling; supersymmetry; Standard Model; Higgs particle; violation of Lorentz invariance.

PACS Nos 11.30.Pb; 12.15.−y; 12.60.−i; 12.10.−g; 04.60.−m

1. Introduction this scale are not acceptable at all. Another name for this requirement is ‘Naturalness Principle’. Till recently, all elementary particles that were known A quantum field theory containing only -half to exist in Nature were only spin-half and fermions and gauge fields exhibits precisely this decou- spin-one gauge particles. With the discovery of Higgs pling. Such theories are called natural theories and the particle at the LHC in 2012, we now have the first ele- masses and the gauge couplings are natural parameters. mentary spin-zero particle. An elementary scalar field, Examples of such theories are: Quantum electrody- such as the Higgs field, introduces a completely new fea- namics (QED) and quantum chromodynamics (QCD). ture in quantum field theories containing such a field. The notion of Naturalness emerged in the late 1970s This new feature is a generic non-decoupling of the from the work of Wilson, Gildener and Weinberg and heavy mass scales from the physics of low mass scales. ’t Hooft [1,2]. A concise formulation is provided by ’t Hooft’s Doctrine of Naturalness [2]: A parameter A quantum field theoretic description for physical α(μ) at any energy scale μ in the description of physical processes with a characteristic smaller mass scale m L reality can be small, if and only if, there is an enhanced should not depend sensitively on the physics of larger in the limit α(μ) → 0. This implies a rule of mass scales m . This decoupling requirement is a rea- H thumb: quantum corrections to the parameter α (masses sonable expectation so that whatever low mass scale and couplings) should be proportional to a positive quantum theory we have, can describe the physics at n power of that parameter itself: (α)quantum ∼ α , n ≥ 1. that scale reliably. The only possible allowed depen- This would be ensured by the associated approxi- dence of the physics at low mass scale mL on the higher mate symmetry. Further, this property implies that the mass scale mH is in the form of its inverse powers enhanced symmetry holds even at the quantum level as and at the worst, a milder dependence through loga- α → 0. rithms of the high scale, but, as shall be discussed in Let us look at QED in some detail. The theory detail in the following, those with positive powers of describes the interaction of fermions λ of qf

1 37 Page 2 of 9 Pramana – J. Phys. (2016) 87: 37 such as an with electromagnetic radiation 2 In fact, such corrections to mL appear with quadratic through a Lagrangian density: divergences: L =−1 μν k2+m2 QED F Fμν m2 ∼−y2 d4k H 4 L (k 2−m2)2 +λ¯ iγ μ ∂ −ieq A − m λ. (1) H μ f μ e ∼− 2 2 2 2 y mH ln(mH/μ ), (5) Various parameters here, the electromagnetic where we have used dimensional regularization and e and electron mass me, can be naturally small. Limit m → minimal subtraction in the last step. Notice that this e 0 leads to an enhanced symmetry, the chiral 2 2 symmetry: separate conservation of the number of left- correction is proportional to mH and not to mL.There- and right-handed . It is for this reason that fore, there is no decoupling of the heavy mass scale one-loop correction to the electron mass in this theory from the light mass scale theory. Even in the limit → is given by logarithmically divergent diagrams and is mL 0, this correction does not go away. proportional to the electron mass itself. 2. Naturalness of electroweak theory 2 (me)1 loop ∼ e me ln . (2) Standard Model (SM) of has an ele- → In the limit me 0, this correction disappears reflect- mentary scalar particle, the Higgs particle. Discussion ing the fact that chiral symmetry is preserved by such above then implies that its mass is not protected by any perturbative quantum corrections in this limit. Also, symmetry against large radiative corrections. e → 0 results in enhanced symmetry: there is no Tree-level masses for the Higgs particle, gauge par- ± interaction and hence particle number of each type is ticles W ,√Z0 and the fermions in SM are given by: conserved. Again, quantum corrections to the coupling = = = mHiggs √λv, mW gv/2, mZ gv/(2cosθW ), e are logarithmically divergent and are, in the lowest m = Y v/ 2, where v is the expectation 3 f f order, proportional to e : value of the scalar field, λ is the quartic scalar coupling, 3 g is the gauge coupling, Yf is the Yukawa coupling of (e)1 loop ∼ e ln . (3) the fermions f to the scalar field and θW is the weak It is for this reason that described by the mixing angle. → interactions of electrons and is not disturbed Note that the limit v 0 does enhance classical by the fact there are other heavier charged fermions symmetry: (i) all particles being massless in this limit, such as the muon, tau-, top and others in we have scale invariance of the classical theory; (ii) the weak gauge are massless resulting in restored Nature: mμ ∼ 200me, mτ ∼ 3500me, ..., mtop ∼ 3.4 SU(2) gauge symmetry and additionally (iii) there is × 1011m . Effects of heavier fermions are decoupled e chiral symmetry due to zero masses of the fermions. from the physics of electrons and photons; showing up Yet, v is not a natural parameter. Consequently, masses at best through logarithmic dependence on them. of the Higgs particle, weak gauge bosons and fermions As against this, theories with elementary scalar are not natural. This is due to the Coleman–Weinberg fields have completely different behaviour. Elementary mechanism of radiative breaking of symmetry: quan- scalar fields spoil heavy–light decoupling: quantum tum fluctuations generate a non-zero quantum vacuum field theories with scalar fields are not natural. As an expectation value for the scalar field even when clas- example, consider a Yukawa theory of an elementary sically v is zero, breaking all these symmetries. There scalar field φ of low mass m coupled to a heavy L is no enhancement of symmetry at the quantum level λ of mass m : H in the limit where classical → 1 μ 1 2 2 ¯ μ v 0. L = ∂ φ∂μφ − m φ + λ iγ ∂μ − mH λ 2 2 L In the SM, one-loop radiative corrections to the ¯ + yλλφ; mH  mL. (4) Higgs mass come from the diagrams of the type where fermions f , gauge fields (W±,Z)and Higgs field H go In this theory, the light scalar mass mL is not a natu- around the loop (see figure 1). The diagrams contribute ral parameter. Smallness of mL cannot be protected by correction to the Higgs mass as any approximate symmetry against perturbative quan- 2 = 2 tum corrections involving heavy fermions in the loops. mHiggs α Pramana – J. Phys. (2016) 87: 37 Page 3 of 9 37

W+_, Z H f ++ H H + H H HH f

W+_, Z f H

H H H + + H H + H H H H Figure 1. Diagrams showing fermions f , Higgs field H and gauge fields (W±,Z) in the loop. with Naturalness breakdown scale of the SM: As we have 1 seen above, one-loop correction to Higgs mass α = (Aλ + Bg2 − CY2), 2 f due to quantum fluctuations of a size characterized 16π 2 = 2 by the scale  may be written as: mHiggs α . where A, B, C are numerical constants respectively Square of the vacuum expectation value of the scalar associated with the diagrams with scalar fields, gauge field, and also the masses of vector bosons W ± and fields and the fermions in the loops. Use dimensional Z0 and fermions would obtain similar corrections. For −1 regularization (with minimal subtraction) to write this coupling α ∼ (100) and mHiggs ∼ 100 GeV, if we correction as require that these radiative corrections to this mass do 2 ∼ 2 2 not exceed its value, mHiggs mHiggs,wehave 1 mHiggs m2 ∼ Aλm2 ln Higgs 2 Higgs 2 2 2 16π μ mHiggs (100 GeV) 2 = ∼ = (1000 GeV)2 m2 N α (100)−1 2 2 gauge +Bg m ln 2 gauge μ2 ≡ (1TeV) . (7) 2 This leads to an estimate of the naturalness breakdown mf − CY 2m2 ln . scale for the electroweak theory as:  ∼ 1TeV. f f μ2 N

Largest mass particle (top ) in the loops gives the 3. Large logarithmically divergent corrections dominant correction: in GUT m2 ∼−αm2 ln(m2 /μ2). (6) Higgs top top Not only are the quadratically divergent graphs with Thus, the radiative correction to scalar mass is gener- heavy mass fields going around the loops responsi- ically controlled by the highest mass in the loops. This ble for destabilizing the lower mass scale, but there is to be contrasted with QED where correction to the are also some log divergent graphs which contribute square of the electron mass is proportional not to the to this phenomenon [3]. These graphs appear generi- square of any other mass but only to square of the cally in any grand unified theory (GUT) of the QCD 2 ∼ 2 and electroweak SU(2) × U(1) model. electron mass itself: me αme. Consider a based on a gauge G Now from eq. (6), for top mass mtop = 175 GeV and the coupling factor α ∼ 1/100, the radiative contribu- which is spontaneously broken at two stages: tion m2 is still small for m = 125 GeV. But if F f Higgs Higgs G −→ G −→ G ; F  f. there were a much heavier particle in Nature, such as in 1 2 a Grand Unified Theory (containing the QCD and the This is achieved through vacuum expectation values of electroweak model), the radiative corrections to Higgs two scalar fields: vac = F ∼ M1 and φ vac = mass would be controlled by this heavy scale and hence f ∼ M2. It was in this context of GUTs that the Natu- very large. ralness Problem was noticed in its earliest versions by 37 Page 4 of 9 Pramana – J. Phys. (2016) 87: 37

H a new physical high scale of the order of 1011 GeV or 2 so linked to the mass of the right-handed neutrino. This F would imply that the radiative corrections would drag the Higgs mass to such high values. Even if we ignore F H H2 2 both these sources of possible high mass scales, there is = 19 yet another high physical mass scale, MPl 10 GeV in Nature, associated with quantum gravity. Radiative Figure 2. Large log divergent graph. corrections would draw the masses of electroweak the- ory to this high scale and hence their natural values Gildener and Weinberg [1] who realized that the rela- would be ∼1019 GeV and not the physical values char- tive stability of the smaller scale f as against the larger acterized by the low SM scale! All these suggest that scale F cannot be maintained under radiative quantum there has to be some new physics beyond 1 TeV such effects and this was given the name: Gauge Hierarchy that the SM with its characteristic scale of 100 GeV Problem. stays natural beyond this scale. Quadratically divergent graphs for the two-point cor- There are several proposed solutions to the Natural- relations of light scalar fields give large corrections ness Problem (for a review, see [4]). Of these, with ∼ 2 2 ∼ 2 ( F ) to its M2 ( f ). Besides these, there are also Higgs mass at 126 GeV, supersymmetry is the most large logarithmically divergent contributions from the promising solution. graphs involving large (∼F) three-point coupling [3]. An elementary property of quantum field theory These come from the mixed light–heavy field interac- which gives an extra minus sign for the radiative tion terms of the type: diagram with a fermion as against a boson field L ∼ κ 2φ2 = κ(H + F )2 (H + f )2 going around in the loop allows for the possibility int 1 2 that naturalness-violating effects due to bosonic and ∼···+ 2 + 2 +··· ; cH1 H2 dH1H2 fermionic quantum fluctuations can be arranged to can- c ∼ f, d ∼ F. (8) cel against each other. For this to happen, the various couplings and masses of bosons and fermions have to The relevant three-point vertex is from the interaction be related to each other in a highly restrictive manner. 2 term dH1H2 with coupling strength proportional to Further, for such a cancellation to hold at every order of the heavy mass scale, d ∼ F . As shown in figure perturbation, a symmetry between bosons and fermions 2, this leads to a logarithmically divergent two-point would be imperative. This is what supersymmetry does graph with light fields H2 on the external lines and one indeed provide. heavy (H1) and one light (H2) fields propagating on the internal lines in the loop. This diagram contributes A historical note: Supersymmetric solution of the 2 a correction to the light mass square, M2 ,givenby Naturalness Problem (or Non-decoupling Problem or Gauge Hierarchy Problem) was discovered in 1981 in M2 ∼ F 2 (F 2/μ2) 2 ln (9) Bangalore, requiring supersymmetry to become opera- which is proportional to the square of the larger mass tive at about 1 TeV for the masses of Standard Model scale due the presence of factor F 2 coming from the to be stable against radiative corrections: two interaction vertices. This results in a destabiliza- (i) In ref. [5], the quadratic divergences were shown tion of the light–heavy mass hierarchy. to be absent in a supersymmetric theory with spontaneously broken -free U(1) gauge 4. A window to physics beyond SM: symmetry. Supersymmetry (ii) In [3], absence of the naturalness-spoiling quad- ratic divergences as well as the large logarithmic In a Grand Unified Theory, perturbative quantum cor- divergences was shown in a supersymmetric rections tend to draw the smaller electroweak scale theory with two distinctly different scales, heavy (MEW ∼ 100 GeV) towards the GUT scale (MGUT ∼ F and light f, associated with sequential gauge 1016 GeV). Even without grand unification, now with symmetry breaking through vacuum expectation the established non-zero masses for neutrinos, though values of two sets of scalar fields. This de- very small, as indicated by the neutrino oscillations, monstrated that the hierarchy f 2/F 2 1is the see-saw mechanism for these masses also suggests radiatively maintained even when quantum Pramana – J. Phys. (2016) 87: 37 Page 5 of 9 37

corrections are included in a supersymmetric the quantum level (provided, in the presence of a U(1) framework. gauge symmetry, all the U(1) charges add up to zero). (iii) In ref. [6], it was demonstrated that (a) in a super- Coleman–Weinberg mechanism does not produce radi- symmetric theory with anomalous U(1) gauge ative breaking of the gauge symmetry in such theories. invariance (where the U(1) charges do not add Supersymmetry requires that bosons and fermions up to zero, QU(1) = 0), quadratic diver- come in families. For supersymmetric model build- gences are not absent; but in a theory which is ing, see ref. [10]. Simplest supersymmetric model is anomaly-free ( QU(1) = 0), these are absent the minimal supersymmetric Standard Model (MSSM) and (b) in a supersymmetric theory with anom- where every SM particle has a superpartner with oppo- aly-free SU(2) × U(1) gauge invariance, quad- site statistics: for the we have a Majorana ratic divergences due to the boson and fermion fermion, the photino, as its supersymmetric partner; fields in the loops cancel out completely with for the we have scalar sleptons; for the quarks no net quadratic divergences and hence such a scalar squarks; etc. theory is perfectly natural. Exact supersymmetry requires that all properties, (iv) In ref. [7], technicolour and supersymmetric except the spin, of particles in a supermultiplet are the solutions of the Naturalness Problem of SM are same: masses are equal and so are the couplings; elec- reviewed. troweak and colour quantum numbers are identical. (v) It was argued in [8] that, in a general GUT with But, supersymmetry cannot be an exact symmetry of two distinct mass scales, the decoupling of the Nature: otherwise we should already have seen a scalar high mass scale from the low mass scale is spoiled superpartner of an electron with the same mass and by the same features of elementary scalar fields charge. So supersymmetry has to be broken in a way as are responsible for the Coleman–Weinberg that the superpartners are much heavier than the SM radiative symmetry breaking. In a supersymmet- particles. Yet naturalness-violating effects should not ric theory, Coleman–Weinberg mechanism is not appear: in particular the quadratic divergences should operative and hence, the low–high mass scale not reappear in the radiative corrections. This indeed distinction holds even when quantum correc- happens if supersymmetry is spontaneously broken tions are incorporated. or explicitly broken by the so-called soft terms (i.e., (vi) In supersymmetric theories with spontaneously broken by masses only and not by dimensionless cou- broken U(1) gauge symmetry even when trace plings) in the action at a scale MSUSY ∼ 1TeV. of U(1) charges is zero, the D term can get MSSM has a whole variety of possible new interac- one-loop corrections, but that these are only tions: a large number of new free parameters (∼100) logarithmically divergent was proved in ref. [9]. with all possible soft supersymmetry breaking terms. This makes it difficult to make any robust and easily 5. Supersymmetric extension of the Standard verifiable predictions. An important requirement in Model supersymmetric theories is the suppression of un- wanted flavour changing (FCNC) which Supersymmetric theories with non-Abelian gauge in- are otherwise generically present in large sizes in such variances are always free of quadratic divergences. On theories. Sometimes, people make certain assumptions the other hand, those with U(1) gauge invariance have about the nature of the new interactions which reduces quadratically divergent radiative corrections propor- the number of the extra parameters. Different possible tional to the sum of U(1) charges of all the fields. If choices of these parameters lead to predictions with the U(1) charges sum to zero, quadratic divergences different possible masses and also different decay pat- are absent even in these theories. Supersymmetrized terns. One such model, the constrained minimal super- Standard Model is one such theory. symmetric model (cMSSM), has only a few extra free Also for theories with two widely separated scales parameters, five in all. such as a supersymmetric GUT, the large logarithmic So far, no evidence for supersymmetry has emerged divergences are also separately absent. from the 8 TeV data collected at LHC. This may change In supersymmetric theories with spontaneously bro- over time when more data become available. But, it ken gauge symmetries through non-zero vacuum expec- is perfectly possible that the simplest form of super- tation value (VEV) of elementary scalar fields, the limit symmetric model, i.e., cMSSM, is not the right picture. VEV → 0 does lead to an enhanced symmetry even at More involved supersymmetric models may have to be 37 Page 6 of 9 Pramana – J. Phys. (2016) 87: 37 explored. A MSSM with more parameters or a next-to- and gauge fields where only low–high mass scales minimal supersymmetric Standard Model (NMSSM) decoupling holds, this would indeed be realized. But, [6,11] or even a non-minimal model with more struc- as discussed by Collins et al [14], theories containing ture may be required. A recent example of a more elementary scalar fields would not exhibit such a prop- involved model is the gauge-mediated supersymmetry erty. We now present their argument for this behaviour breaking (GMSB) with an unconventional messenger in the following. content [12] as against the 5 and 5¯ of the Lorentz invariance implies a unique form of disper- grand unification gauge group SU(5) in the minimal sion relation for a particle: E2 − p2 − m2 = 0, in conventional GMSB model. units where velocity of light c = 1. A Lorentz non- It is important to realize that, except for the com- invariance effect would change the dispersion relation pelling naturalness argument which predicts supersym- to: E2 − p2 − m2 − (E, p) = 0, where  rep- metry as operative at about 1000 GeV, so far there are resents the result of all the self-energy graphs with a really not enough strong theoretical or experimental small Lorentz-violating contribution from the quantum constraints available to guide us to a reliable supersym- gravity effects. metric model. Besides, the properties of the Standard We may parametrize the Lorentz violation through a Model including the fact that Higgs mass is now known dimensionless parameter [14]: to be around 126 GeV, other important and strin- ∂ ∂ ∂ ∂ gent restrictions for supersymmetry model building ξ = lim + (p) . (10) → | | | | come from the requirement of sufficient suppression p 0 ∂p0 ∂p0 ∂ p ∂ p of flavour changing neutral currents (FCNCs) which For exact Lorentz invariance (p) would be a function otherwise can tend to be generically large in the 2 − 2 of the Lorentz-invariant combination p0 p implying supersymmetric theories. Hopefully, more experimen- that ξ is zero. Thus, ξ = 0 would provide a measure tal results from the LHC will provide enough discrim- of violation of Lorentz invariance. inating guidance that will finally result in the correct Now, as an example, consider a Yukawa theory of a supersymmetric model. fermion and a scalar field with their small masses given by the low scale mlow. Let us study the contribution to 6. Other non-decoupling problems: ξ for the scalar field from the correction to the scalar

Lorentz non-invariance two-point function (p0, p) due to a fermion loop in this theory: There can be many other physical situations where d4k (p) =−4iy2 [k · (k + p) + m2 ] presence of elementary scalar fields can cause the (2π)4 low same non-decoupling problems. These again would be ×[k2 −m2 +i]−1 cured by supersymmetry. We shall now discuss such an low − example which concerns possible Lorentz non-invari- ×[(p+k)2 −m2 +i] 1 low ance generated by quantum gravity effects at the Planck d4k 1 scale. =−2iy2 (2π)4 k2 − m2 + i Large quantum fluctuations in the gravitational field low would introduce granularity of space at extremely m2 −p2 −33 1 4 low short distances (∼ Planck = 10 cm). This would + 1 + , (k +p)2 + m2 + i k2−m2 + i imply a minimum spatial length beyond which no low low physical process can penetrate. This is in conflict where y is the Yukawa coupling. This integral has a with Lorentz invariance (LI) because LI implies that quadratic divergence which gets converted to a loga- we can make an arbitrarily large boost transformation rithmic divergence in ξ due to the momentum deriva- which would result in Lorentz contraction of lengths to arbitrarily small values. This violation of LI would tives in its definition above: reflect itself through a change of the dispersion relation 1 d4k k2 + k2 for a particle. These features are known to emerge in ξ =−16iy2 0 3 (2π)4 (k2 − m2 + i)3 theories of quantum gravity such as the loop quantum low gravity (LQG) as well as the [13]. 4m2 Do these Planck scale effects decouple from low- × 1 + low . k2 − m2 + i energy physics? In quantum field theory of fermions low Pramana – J. Phys. (2016) 87: 37 Page 7 of 9 37         ⎡ ⎤ This integral can be evaluated by Euclidean continua- |k|  |k| + |k| |k|  |k| f  f  2 f  f  tion of k0 to imaginary values ik4: ×⎣ ⎦ | | 2 + 2 4 2 1 2 2 k (k mlow) (d k)E k − k 4m ξ = 16y2 4 3 1− low . (11) (2π)4 (k2 + m2 )3 k2 + m2 y2 ( 4k) low low − 4 d E 3 (2π)4 We may use an ultraviolet cut-off  for the Euclidean         ⎡ = μ = 2 + 2 |k|  |k| + |k| |k|  |k| internal loop momentum k k kμ k4 k f f f f ×⎣   2   which is invariant under four-dimensional Euclidean | | 2 + 2 k (k mlow) rotations. It is straightforward to check that such a    ⎤ = | |  | | calculation yields the Lorentz symmetric answer ξ 0. 2|k |f k f k However, due to the possible Lorentz violations from −   ⎦ the Planck scale physics, the free fermion (k2 + m2 )2 low used in this calculation would get significantly modi- 4m2 fied at high scales. The momentum cut-off would have × 1− low k2 +m2 to be Lorentz violating, introducing different cut-offs low 2 ∞ for the k0 and |k| integrations. One way to intro- y =− x f(x)f(x) + xf (x)f (x) duce this order-one non-invariance is by introducing a 2 d 3π 0 Lorentz non-invariant cut-off by multiplying the free 2 m fermion propagator by a smooth function f(|k|/) +O low 2 which for the momenta much below the Planck-scale  ∼ = 2 ∞ 2 cut-off ( MPl) goes to 1, f(0) 1, so that the y  m = dxx f (x) 2 +O low , (12) low-energy propagator stays largely unaffected, and for 3π 2 2 high momenta this function goes to zero, f(∞) = 0, 0 to tame the ultraviolet behaviour in a Lorentz non- where prime denotes derivative with respect to the invariant manner. This would lead to a one-loop con- argument. Note that the leading effect is independent tribution to the two-point scalar function as ∼ ⎡     of the Planck scale cut-off ( MPl). That is, there is |k | |k + p| no m2 /M2 factor in the leading term and we have d4k f  f  low Pl (p) =−2iy2 ⎣ only a suppression. Hence, there ( π)4 2 − 2 + 2 k mlow i is no decoupling of the Planck scale effects from light     mass scale! This result implies a low-energy violation |k | |k + p| f  f  of Lorentz invariance of a size which is very large com- + pared to the measured limits on such non-invariance. (k + p)2 − m2 + i low It is important to emphasize that this non-decoupling 4m2 −p2 behaviour again emerges from those parts of the radia- × 1+ low . tive corrections in the two-point scalar correlation k2 − m2 + i low which, without the cut-off, are quadratically divergent. A simple example of such a cut-off function is The amount of non-invariance of Lorentz symmetry f(|k |/) = (1 + (k 2/2))−1. Such a regulator yields at low energies depends on the exact cut-off function = a large Lorentz violation at low energies as can be f(x). Clearly, if we replace f(x) 1 in the above seen by evaluating ξ from this (p) after Euclidean expression for ξ, we obtain, as expected, the Lorentz symmetric answer ξ = 0. continuation of k0 to ik4: ⎡    ⎤ We emphasize again that in theories with only gauge |k | |k | 4 f f 2 fields and fermions and no elementary scalar fields, 2 (d k)E ⎣   ⎦ 4mlow ξ = 16y 1 − where there are no quadratic divergences, this order, (2π)4 (k2 + m2 )3 k2 + m2 low low one quantum gravity-induced Lorentz violation at the Planck scale would radiatively percolate down to low × 2 − 1 2 k4 k energies in a highly suppressed form, with only order 3 m2 /M2 effects. y2 ( 4k) low Pl − 4 d E Obviously, in theories with elementary scalar fields, 3 (2π)4 supersymmetry again provides a protection mechanism 37 Page 8 of 9 Pramana – J. Phys. (2016) 87: 37 against the above discussed large radiative transmis- Hopefully, experimental discovery of supersymme- sion of Lorentz violation from Planck scale to low try, though very likely not in the simplest version as scales: there are no quadratic divergences in the scalar represented by the cMSSM, but as in a more general self-energy graphs in supersymmetric theories. In the MSSM framework, or even perhaps in a non-minimal supersymmetrized version of field theory example form, may happen in the near future. discussed above, exact supersymmetry will completely Besides the naturalness issues related to the masses cancel out the quadratically divergent contributions in the Standard Model, there are other places where in the two-point scalar correlator from graphs with similar problems arise. For example, generic non- boson fields going around the loops with those with decoupling of the (possible) Planck scale violation of fermion fields going around the loops. Consequently, Lorentz invariance due to quantum gravity effects in Planck scale violations of Lorentz invariance will leave theories with elementary scalar fields has the same ori- behind in ξ only a highly suppressed low-energy effect gin. Supersymmetry again can ensure decoupling of 2 2 [15] of a size O(mlow/MPl). But as supersymmetry is this Planck scale violation from the low-energy physics. softly broken at low energies below a scale MSUSY in This implies a suppressed low-energy violation of Nature, the Bose–Fermi cancellation will not be exact, Lorentz invariance as reflected by a variable velocity but will happen up to logarithmic effects: ξ ∼ ∼ ∼ 2 2 2 ∼ of light of a size ξ 4(c/c) y (MSUSY/MPl) y2(M2 /M2 )ln(M2 /M2 ). In the Standard −34 SUSY Pl Pl SUSY 10 for a supersymmetry breaking scale of MSUSY ∼ Model, radiative stability of the Higgs mass requires 103 GeV (which is required by the radiative stability of ∼ 3 MSUSY 10 GeV. Though approximate supersym- the 100 GeV scale of electroweak theory). metry provides a suppression, yet this discussion implies a profound result that quantum gravity effects predict a tiny violation of Lorentz invariance at low- Acknowledgements energy scales given by: ξ ∼ y2(M2 /M 2 ) ∼ SUSY Pl Comments by Gautam Bhattacharyya are gratefully ( )−1 ( 3/ 19)2 ∼ −34 ξ 100 10 10 10 . Non-zero value of acknowledged. This write-up is based on a talk at the modifies the Lorentz symmetry respecting dispersion conference on Supersymmetry and Dark , Centre  (p) ≡−p2 + m2c2 =−E2/c2 + p2 relation 0 ( ) for High Energy Physics, Indian Institute of Science, + m2c2 = c 0 by a change of velocity of light by an Bangalore, Indian, October 3–5, 2013. The author amount given by c/c = ξ/4+O(ξ2). The estimate of would like to thank the organizers of this meeting violation of Lorentz invariance here may be contrasted for excellent hospitality at Bangalore. The author also with the present day observational/experimental limits acknowledges the support of Department of Science on this violation as represented by the varying velocity and Technology, Government of India, through a J C of light as: c/c < 10−22. The violation of Lorentz Bose National Fellowship. invariance suggested above is significantly smaller, by some 12 orders magnitude, than this limit. References 7. Conclusion [1] E Gildener, Phys. Rev.D14, 1667 (1976); Phys. Lett. B 92, Quantum field theories with elementary scalar fields 111 (1980) do not exhibit low–high energy decoupling behaviour: E Gildener and S Weinberg, Phys. Rev.D13, 3333 (1976) such theories are not natural. The various low mass [2] G ’t Hooft, Naturalness, chiral symmetry, and spontaneous , in: Recent developments in field parameters are not stable under quantum radiative cor- theories edited by G ’t Hooft et al (Plenum Press, New York, rections which tend to drag them to the highest mass 1980) p. 135 scale. [3] Romesh K Kaul, Gauge hierarchy in a supersymmetric model, The Naturalness Problem of the SM has proved to CTS-TIFR preprint August 1981 (TIFR-TH-81-32),Phys. be an ideational fountain-head for a whole variety of Lett. B 109, 19 (1982) new beyond Standard Model (BSM) ideas over last [4] Romesh K Kaul, Naturalness and electro-weak symme- several decades. Supersymmetry is the most promising try breaking, Lectures at the Advanced School: From Strings to LHC-II (Bangalore, India, 11–18 December 2007), of these. Now is the time to confront these with experi- arXiv:0803.0381[hep-ph] ments at LHC. Surely, experimental search for super- [5] Romesh K Kaul and Parthasarathi Majumdar, Naturalness symmetry and related phenomenological developments in a globally supersymmetric gauge theory with elementary are the present day frontier of high-energy physics. scalar fields, CTS preprint May 1981 (Print-81-0373) Pramana – J. Phys. (2016) 87: 37 Page 9 of 9 37

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