NNaturalness without new particles
Anson Hook Maryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742, USA
Abstract
We demonstrate that the physics which resolves naturalness problems need not take the form of new particles and cansometimes manifest itself as higher dimensional operators. As a proof of principle, we present a simple model where the scaleof new particles is parametrically separated from that estimated via naturalness arguments applied to self-quartic couplings.In this example, new particles appear far above the scale m/ √ λ , where m is the mass of the particle and λ is its self-quarticcoupling. The shift symmetry responsible for resolving the naturalness problem involves higher dimensional operators ratherthan new particles. The Electroweak Hierarchy problem is one of the driving forces in particle physics (see e.g. Ref. [1] for a reviewof the current state of affairs). When the naturalness paradigm is applied to this problem, it is typically stated thatnew particles must appear by the TeV scale modulo accidental fine tunings. In this article, we provide an interestingproof of principle that the physics which resolves naturalness problems need not involve new particles.Consider a complex scalar Φ which is charged under a Z N symmetry so thatΦ → e πiN Φ . (1)The most general Lagrangian allowed by symmetries is V = − f | Φ | + | Φ | + Φ N Λ N − + · · · , (2)where we take all coefficients to be O (1) numbers. In the context of axions, this exact potential has been consideredmany times when discussing Planck suppressed corrections to the axion potential, see e.g. Ref.[2]. At the renormal-izable level, there is an accidental U (1) symmetry. The leading-order operator that breaks this symmetry is shownabove and is a higher-dimensional operator whose origin is unimportant. Later, we will give a simple example wherefermions with a mass (cid:38) f can be integrated out to generate it. The radial mode is integrated out at an energy scale f . At low energies there is a pseudo-Goldstone boson with the potential V = − (cid:15) cos N φf , (cid:15) ∼ f N Λ N − . (3)Expanding the potential, we see that the mass and quartic couplings of φ are m φ ∼ (cid:15) N f , λ ∼ (cid:15) N f . (4)We will be concerned with an imaginary experimentalist who has discovered the pseudo-Goldstone boson and measuredits mass and quartic coupling. Said person would apply the standard naturalness arguments and expect the scale ofnew particles to be Λ ∼ m √ λ ∼ fN . (5)However, note that there are new particles, namely the radial mode, at f rather than f /N . Thus by taking N large, wehave a proof of principle that the mass scale of new particles can be parametrically separated from where naturalnessarguments typically expect them to be. If higher harmonics were generated instead, e.g. V ∼ cos m ( N φ/f ), the a r X i v : . [ h e p - ph ] F e b expectation would be that new particles are at the scale f / ( N √ m ) further separating the scale where new physics isexpected and the scale at which it actually appears.The reason why the IR experimentalist incorrectly estimated the scale of new physics is because measuring therenormalizable Lagrangian was not enough information in order to determine the symmetries of the theory. If theIR experimentalist proceeded to look for higher dimensional operators, he/she would eventually discover the higherdimensional operator (cid:15) N φ f ∼ λ φ m (6)It is important to note that this operator becomes strongly coupled at a scale (cid:29) f so that it would be difficult to findexperimentally. Upon measuring this operator, the experimentalist would note that this dimension six operator isalgebraically related to the quartic coupling and mass terms and would suspect that there is a shift symmetry in thetheory. The useful operators of the theory are not φ , φ and φ but instead only a single operator in the theory hasbeen discovered, cos Nφf . As the coefficient of this operator, (cid:15) , goes to zero there is an enhanced full shift symmetryfor φ . It is thus technically natural to set (cid:15) small and there is no hierarchy/naturalness problem. The dimensionfulscale associated with the symmetry, f /N , is independent of the scale of new particles and thus cannot be used topredict the appearance of said particles.At this point, we give a quick demonstration of how the higher-dimensional operator in Eq. 2 can be generated.There are many ways to generate this operator, e.g. from theory space [3] or from discrete symmetries [4]. We willfollow the discrete symmetry approach of adding to the theory a set of N vector-like fermions Ψ j and Ψ cj , whichtransform under a Z N symmetry as Ψ ( c ) j → Ψ ( c ) j +1 , (7)with j = 0 and j = N identified. Writing down the most general renormalizable Lagrangian, we have L ⊃ f | Φ | − | Φ | + N (cid:88) j =1 (cid:16) m f + ye πij/N Φ (cid:17) Ψ j Ψ cj . (8)Taking m f > yf , we can integrate out the fermions Ψ, which generates the higher-dimensional operator of interest( y N Φ N /m N − f ) as well as some U (1) symmetry preserving operators.We now turn to the implications of this example for the naturalness argument. We have shown that the physicswhich resolves naturalness problems could take the form of higher dimensional operators rather than new particles.The natural question is to wonder if the solution to the Electroweak Hierarchy problem can also take the form ofhigher dimensional operators. Currently we do not know whether or not this is possible. However it is clear thatthe moral of the story is that if such a solution were to exist, that one would search for higher dimensional operatorsrather than new particles. Without an explicit example, it is not clear which higher dimensional operators to look foror what value to expect them to take (e.g. one might expect an operator of the form ∼ λ (cid:0) HH † (cid:1) /m H ). The scalethat suppresses these higher dimensional operators might be large, but one hopes that they are in reach of currentexperiments.To conclude, we have demonstrated through an explicit counter-example that the solution to a naturalness problemneed not involve new particles. While this example has nothing to say about gauge and Yukawa quadratic divergencesor about the cosmological constant, it would be interesting if similar models could be constructed for these othercouplings. This example suggests an interesting new approach to the naturalness paradigm. Perhaps new physics willappear in the form of higher dimensional operators rather than new particles. Acknowledgments
We are very grateful to Zackaria Chacko, Junwu Huang, Matt Strassler, Raman Sundrum, and Gustavo MarquesTavares for useful discussions. This research was supported in part by the NSF under Grant No. PHY-1620074 andby the Maryland Center for Fundamental Physics (MCFP). [1] M. Dine, Ann. Rev. Nucl. Part. Sci. , 43 (2015), 1501.01035. [2] M. Kamionkowski and J. March-Russell, Phys. Lett. B282 , 137 (1992), hep-th/9202003.[3] N. Arkani-Hamed, A. G. Cohen, and H. Georgi, Phys. Rev. Lett. , 4757 (2001), hep-th/0104005.[4] A. Hook, Phys. Rev. Lett.120