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Emerging Internal Symmetries from Effective Spacetimes ∗ Manfred Lindner, Sebastian Ohmer

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Emerging Internal Symmetries from Effective Spacetimes ∗ Manfred Lindner, Sebastian Ohmer Physics Letters B 773 (2017) 231–235 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Emerging internal symmetries from effective spacetimes ∗ Manfred Lindner, Sebastian Ohmer Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany a r t i c l e i n f o a b s t r a c t Article history: Can global internal and spacetime symmetries be connected without supersymmetry? To answer this Received 9 August 2017 question, we investigate Minkowski spacetimes with d space-like extra dimensions and point out under Accepted 15 August 2017 which general conditions external symmetries induce internal symmetries in the effective 4-dimensional Available online 24 August 2017 theories. We further discuss in this context how internal degrees of freedom and spacetime symmetries Editor: M. Cveticˇ can mix without supersymmetry in agreement with the Coleman–Mandula theorem. We present some specific examples which rely on a direct product structure of spacetime such that orthogonal extra di- mensions can have symmetries which mix with global internal symmetries. This mechanism opens up new opportunities to understand global symmetries in particle physics. © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction mix spacetime and internal symmetries in a relativistic theory is a strong theoretical argument for supersymmetry and supersym- The nature of spacetime is still a great mystery in fundamen- metric extensions of the Standard Model of particle physics are tal physics and it might be a truly fundamental quantity or it therefore widely studied. However, there is no experimental evi- could be an emergent concept. An appealing and most minimal- dence for supersymmetry, see e.g. [5–7], and it is a fine question istic approach would be if spacetime and propagating degrees of to ask: Are there alternative ways to circumvent the Coleman– freedom would have a common origin on equal footing. In such a Mandula theorem? scenario, spacetime is thus an emergent quantity and there seems The answer to this question is: Yes. We therefore relax the to be no reason for it to be restricted to a 4-dimensional Poincaré assumption that spacetime is described by the 4-dimensional symmetry apart from low energy phenomenology. The only excep- Poincaré symmetry. We then investigate new alternative scenarios tion are additional time-like dimensions which typically lead to to mix global spacetime and internal symmetries. Next, we review inconsistencies when requiring causality [1,2], while there is no the Coleman–Mandula theorem to understand how to circumvent consistency problem with additional space-like dimensions. Addi- the theorem with extra space dimensions. In section 3, we discuss tional space-like dimensions have therefore been widely studied. translational invariant extra dimensions and show how momen- If spacetime and particles consist of the same building blocks, tum conservation can be interpreted as new internal symmetry. then a fundamental connection of these low energy quantities We then go further in section 4 and consider extra dimensions de- should exist at high energies. Early attempts in this direction have scribed by rotational invariant spacetimes which lead to “hidden” led to the Coleman–Mandula no-go theorem [3]. The no-go the- spins. Finally, we investigate how rotational and internal symme- orem shows under general assumptions that a symmetry group tries can mix if the rotational symmetry group is compact in sec- accounting for 4-dimensional Minkowski spacetime and internal tion 5. Such scenarios can for example lead to an explanation of symmetries has to factor into the direct product of spacetime and the three Standard Model families. We conclude and give an out- internal symmetries. This implies that spacetime and particle sym- look for further investigations in section 6. metries cannot mix in relativistic interacting theories. One way to circumvent the no-go theorem is to study graded 2. Coleman–Mandula no-go theorem symmetry algebras which introduce fermionic symmetry gener- ators and are known as supersymmetries [4]. The possibility to The Coleman–Mandula theorem [3,8,9] states, if G is a con- nected symmetry group of the S-matrix and * Corresponding author. (i) G has a subgroup which is locally isomorphic to the Poincaré E-mail address: [email protected] (S. Ohmer). group, http://dx.doi.org/10.1016/j.physletb.2017.08.026 0370-2693/© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 232 M. Lindner, S. Ohmer / Physics Letters B 773 (2017) 231–235 (ii) all physical particles have positive definite mass and there ex- Utilizing this property, we will introduce additional space-like ists only a finite number of particles below an energy thresh- dimensions which transform rotationally according to a compact old Emax, symmetry group in section 5. We then illustrate how global space- (iii) the S-matrix is an analytic function of the Mandelstam vari- time and internal symmetries can mix. This can then give rise to ables s and t, new symmetries which may be the origin of the family and flavor (iv) the S-matrix is non-trivial at almost all energies, structure of the Standard Model. (v) generators of G are representable as integrals in momentum space, 3. Translational symmetries then G is locally isomorphic to the Poincaré group times an inter- First, we consider the simple and well known example of a nal symmetry group. D-dimensional theory, D = 4 + d, with d extra dimensions where It is important to develop a physical intuition for the Coleman– spacetime is described by Mandula theorem [10,11]. A physical scattering amplitude has to respect all symmetries of the theory and thus the number of in- M4 × d , (2) dependent variables describing the scattering process is reduced. with M4 the 4-dimensional Minkowski spacetime and d the Requiring that the theory respects the laws of special relativity im- additional d-dimensional space. The spacetime coordinates can plies that the scattering amplitude is a Lorentz scalar. Moreover, for A μ a thus be written as z = (x , y ) with μ = (0, 1, 2, 3) and a = a scattering process to be physical the initial and final 4-momenta (4, ..., D − 1). The spacetime symmetry group factors as have to be on the mass-shell. We further demand that the scatter- ing process respects energy–momentum conservation. Taking into P(1, 3) ⊗ Gd , (3) account all these kinematic restrictions for a 2 → 2scattering pro- cess only leaves the famous Mandelstam variables s and t as free where P(1, 3) is the 4-dimensional Poincaré group and Gd is the + parameters in d 1dimension with d > 1. If we would demand symmetry group of d. We further assume that the space de- that the scattering process respects an additional conserved charge scribed by Gd is translational invariant such that the (4 + d)- which is a function of the momenta, then only discrete scattering dimensional momentum angles would be allowed. This is however in conflict with the as- A 3 d 0A sumptions since scattering should be non-trivial for most energies. P = d x d yT , (4) We therefore can conclude that further restrictions on the scatter- AB ing amplitudes should be independent of the 4-momenta of the with A = (0, 1, ..., D − 1) and energy–momentum tensor T is particles. We would call such a symmetry an internal symmetry A conserved, ∂0 P = 0. We also assume that since its generator would commute with the spacetime generators. 2 † A This implies for the general symmetry structure of the S-matrix = = − m P A P with A (0, 1,...,D 1), (5) G → P(1, 3) ⊗ “internal symmetries” . (1) commutes with all group generators and that m2 is a constant for all irreducible representations. The particles momenta in the extra A more detailed mathematical treatment can be found in [3,8,9] dimension thus contribute to the energy–momentum relation but the essence is that Lorentz invariance severely restricts the possible symmetries of the S-matrix. 2 = 2 +||2 + 2 +···+ 2 E m p p4 pD−1 , (6) However, if we extend the underlying 4-dimensional Poincaré invariant spacetime by d space-like dimensions, where we assume although the generators P a with a ∈ (4, ..., D − 1) commute with that the symmetry generators commute with the 4-dimensional all generators of the Poincaré group P(1, 3) and would thus Poincaré group, the scattering process is allowed to respect con- naively account for internal symmetries. served charges which depend on the momenta in the d space-like The assumed spacetime structure gives rise to additional con- dimensions without discretizing the 4-dimensional Mandelstam served charges connected to the particle momenta in the extra variables. We will implement this in section 3. dimensions P a with a ∈ (4, ..., D − 1). Scattering processes will Until now we only considered scattering of scalar degrees of then have to respect additional conservation laws. The schematic freedom which transform trivially with respect to Lorentz trans- scattering process formations. However, introducing particles with spin, we introduce degrees of freedom which transform non-trivially with respect to (p A , pD ) + (pB , 0) → (p A , 0) + (pB , 0), (7) Lorentz transformations. We can now ask if there is a conserved would for example be forbidden. Note that the new conserved charge of a scattering process which depends on the spin of the charges will not discretize the 4-dimensional scattering process. particles. Such a conserved charge would belong to symmetry Moreover, from a 4-dimensional point of view the scattering pro- transformations which relate particles in different representations cess respects additional internal symmetries.
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