Long range forces and limits on unparticle interactions

Couplings between standard model particles and unparticles from a nontrivial scale invariant sector can lead to long range forces. If the forces couple to quantities such as baryon or lepton (electron) number, stringent limits result from tests of the gravitational inverse square law. These limits are much stronger than from collider phenomenology and astrophysics.


I. INTRODUCTION
Georgi [1,2] has proposed that unparticles -a nontrivial scale invariant sector [3] -might couple to standard model particles, leading to novel phenomenological signatures. This idea has been further developed by a number of authors [4]- [49]. In this note, we discuss the possibility of long range forces resulting from such interactions. We assume strict conformal invariance down to low energies, so that the unparticle propagator necessarily has a zero momentum pole.
Our analysis is quite similar to that of Goldberg and Nath [28], who considered the possibility that (exactly scale invariant) unparticles might couple to the ordinary energymomentum tensor T µν . Here, we consider couplings between unparticles and currents such as J µ =ēγ µ e orqγ µ q. Interestingly, the electron number current appears in Georgi's e + e − → µ + µ − example in [2]. We find extremely strong limits on such couplings, much stronger than can be obtained by collider experiments and even astrophysics.

II. LONG RANGE FORCE DUE TO VECTOR UNPARTICLE
We consider first baryon current B µ in terms of quarks The coupling between a vector unparticle state with quark currents can be rewritten as Where Λ U is the scale below which operator O U has dimension d U and λ B is a dimensionless coupling constant. In the static limit this interaction generates the potential where is the coefficient associated with the transverse four-vector unparticle propagator [2] and B 1,2 are the baryon numbers of the two interacting masses. Using a relationship involving Gamma functions, V U can be simplified into Note that for d U = 1, this will produce a 1/r repulsive potential Similarly, the unparticle operator can couple to lepton currents with coupling strength λ L and the above can be directly applied as well. For numerical results, we exhibit three examples: λ B = λ = 0 and λ L = 0; λ B = 0 and λ L = λ = 0; and λ B = −λ L = λ. In these cases, the unparticle operator couples to B, L and B − L currents, respectively.

III. NUMERICAL RESULTS
In the limit of unbroken scale invariance, the force between B 1 and B 2 is a long range force similar to gravity, but it may have different 1/r power dependence. To obtain numerical results, we make the approximation B 1,2 ≈ m 1,2 /u, where m 1,2 are the masses of the two interacting bodies and u is the atomic mass unit. Combined with the gravitational potential, the potential between two objects of mass m 1 and m 2 is where Newton's constant G = 6.7 × 10 −39 GeV −2 and captures the d U dependent part of the coefficient. V can then be written as We compare the second term on the right hand side to the power-law potential in Ref. [50], and derive limits on the coupling λ, for k = 2 and k = 3. Ref. [50] also discussed constraints on Yukawa potentials generated by exchange a scalar or a vector boson. For small enough boson masses m, the experimental distance r is much less than the Compton wavelength h/mc. In this limit, the Yukawa potential is approximately 1/r, and we obtain a limit on the case k ≈ 1. For non-integer value of k, we simply interpolate the limit linearly (the precise bound will be of the same order of magnitude). The range of d U we are interested in is 1 ≤ d U ≤ 2 and it is related to k through equation 2d U − 2 = k − 1. The limits on the scalar or vector coupling can be converted into limits on an effective coefficient β 1 for the case of a 1/r potential. Note that when we consider the B, L and B − L cases, the effective couplings will be multiplied by factors of (Z + N)/A ≈ 1, Z/A and N/A respectively, where Z is the number of protons, A the atomic number, and N is the number of neutrons. For the molybdenum pendulum considered in Ref. [50], Z/A = 0.438 and N/A = 0.563. If λ B and λ L happen to satisfy the effective charge becomes zero and the torsion-balance experiment becomes insensitive to λ, so no limit is obtained. However, the experiment [51] was performed with an aluminum pendulum and copper attractor and arrived at similar bounds, covering as well the exceptional parameter space (11).
We should also note that forces coupling to almost any linear combination of B and L which extend over truly macroscopic (e.g., solar system scale) distances are even more tightly constrained if they deviate from 1/r, since they would affect Newtonian orbits. Table I  For comparison purposes, Table I also displays results from supernova constraints (last column, taking Λ U = 1 TeV). Unparticles can induce too-rapid cooling of supernovae [16,48].
A constraint on λ can be deduced, yielding Note that for our limits to apply scale invariance must hold to length scales as long as a fraction of a millimeter. If scale invariance is broken at a scale intermediate between a millimeter and the thermal wavelength of a supernova (roughly, inverse MeV), the supernova constraints will still apply, while ours will not.

IV. CONCLUSIONS
A sector of particle physics which exhibits nontrivial scale invariance would be an exciting discovery. If, however, the scale invariance is exact, long range forces may result which are already strongly constrained by measurements of the gravitational inverse square law. Given the limits derived here, perhaps the most likely unparticle scenario is one in which the scale invariance is only approximate -i.e., it is broken below some energy scale, thereby screening the long range forces. However, in this case the new particle sector, while exhibiting novel dynamics, is not really an unparticle sector!