Unparticle Self-Interactions and Their Collider Implications

In unparticle physics, operators of the conformal sector have self-interactions, and these are unsuppressed for strong coupling. The 3-point interactions are completely determined by conformal symmetry, up to a constant. We do not know of any theoretical upper bounds on this constant. Imposing current experimental constraints, we find that these interactions mediate spectacular collider signals, such as $pp ->U ->UU ->\gamma \gamma \gamma \gamma$, $\gamma \gamma ZZ$, $ZZZZ$, $\gamma \gamma l^+ l^-$, $ZZ l^+ l^-$, and $4l$, with cross sections of picobarns or larger at the Large Hadron Collider. Self-interactions may therefore provide the leading discovery prospects for unparticle physics.


I. INTRODUCTION
In unparticle physics, the standard model is extended by couplings to a conformal sector through interactions of the form OO SM , where O is an operator of the conformal sector, and O SM is a standard model operator [1]. The conformal sector may be weakly-coupled [2] or strongly-coupled [3], but in all cases, the conformality of the new sector leads to effects that cannot be explained in terms of standard particle states.
To date, all unparticle studies are based on two key elements: the unparticle phase space, and the unparticle propagator. Conformal invariance fixes unparticle phase space [1], which enters processes with unparticles in the final state, such as ff → f U. Similarly, conformal invariance dictates the form of the unparticle propagator [4,5], which determines virtual unparticle contributions to processes such as ff → U → ff. The forms of the unparticle phase space and propagator imply that unparticles do not behave as standard particles, but are more aptly interpreted as fractional numbers of massless particles [1] or as collections of particles with a particular distribution of masses [6]. These results are valid if the couplings of unparticles to the standard model are all non-renormalizable. If unparticle couplings include the super-renormalizable operator Oh 2 , where h is the standard model Higgs boson, electroweak symmetry breaking breaks conformal invariance [7,8]. This modifies the unparticle propagator and implies that unparticle physics may be probed only in a narrow conformal window, typically at energies between 10 GeV and 1 TeV [7,9]. Such energies are best probed at high energy colliders, and many studies have investigated the collider implications of unparticles [1,4,5,9,10,11,12]. Models with conformal breaking may also share characteristics with hidden valley models [13,14].
Here we study a qualitatively new effect: unparticle self-interactions. Such 3-and higherpoint interactions are always present in conformal theories, and mediate processes such as gg → U → U · · · U, with two or more unparticles in the final state. In the most interesting cases with strongly-coupled conformal sectors, the creation of additional high p T unparticles in the final state does not suppress the rate. Multi-U production therefore differs from all known examples, such as gg → g → g · · · g and gg → γ · · · γ, where the rate is reduced with the addition of every high p T particle.
In this paper, we focus on 3-point self-interactions. These are the natural starting point for several reasons. First, unlike 4-and higher-point self-interactions, 3-point interactions are completely constrained by conformal invariance, up to a constant. Second, 3-point interactions are the leading order at which induced signals may be nearly background free. For example, although the U propagator induces signals like pp → U → γγ, 3-point U selfinteractions mediate pp → U → UU → γγγγ, γγZZ, ZZZZ, γγl + l − , ZZl + l − , 4l, and many other spectacular signals through subprocesses such as the one shown in Fig. 1. For this reason, the most promising signals for unparticle discovery at colliders may in fact be those that are induced by 3-point unparticle interactions. (Note that, although the production of 3 or more unparticles may also be unsuppressed, the requirement that they convert back to standard model particles to be visible does, in fact, imply that these are sub-dominant.) In the following sections, we start from the 3-point correlation function in position space and convert it to a form useful for Feynman diagram calculations. We then calculate collider rates, taking the 4γ signal as our example. Our final result is determined up to the constant entering the 3-point correlation function. As far as we know, there is no theoretical upper bound on this constant -it is bounded only by current experimental constraints. Applying these constraints, we find that the prospects for signals at the Large Hadron Collider (LHC) are truly spectacular -the four-body final states given above may have cross sections of picobarns or larger, leading to obvious signals visible in the first year of the LHC. We also determine the predicted kinematic distributions from unparticles, which are insensitive to overall rates. These distributions provide testable predictions that could be used to distinguish multi-unparticle production from other possibilities for new physics.

II. 3-POINT CORRELATION FUNCTION
We begin by assuming that the standard model is coupled to the unparticle sector through a scalar operator O. For a scalar, unitarity requires dimension d ≥ 1, but there is no upper bound [15,16]. Motivated by prominent supersymmetric examples [3], we consider the range 1 ≤ d < 2. Modifications are required for vector and tensor operators [15,16,17], but we expect that our primary conclusions apply to these cases as well.
Conformal invariance constrains the 3-point O correlation function to be where C ′ d is a constant determined by the unparticle self-coupling strength. In momentum space, it is then where the last equality makes use of the unparticle 2-point correlation function [4]. C d is determined in terms of C ′ d ; we choose to express our results in terms of C d . The 3-point correlation function is the product of 3 propagators from 0 to x, x to y, and y to 0. It is therefore not surprising that, in momentum space, it takes the form of a loop integral. We may therefore make use of the standard techniques available for simplifying loop calculations. Using Feynman parameters for Eq. (2), the 3-point correlation function becomes where Fig. 2.

III. BOUNDS ON UNPARTICLE INTERACTIONS
To investigate the phenomenological implications of the 3-point correlation function discussed in Sec. II, we must introduce unparticle couplings to standard model particles. Here we present here the relevant couplings, and determine the existing constraints on them and the prospects for probing them at the LHC. The results here are independent of unparticle self-interactions, but are of interest in their own right. In addition, in Sec. IV we will present results for a reference value of Λ 4 , which we choose based on the results we derive here.
A scalar unparticle may couple to two gauge bosons or two fermions through the Lagrangian interaction terms leading to the Feynman rules where Λ 4 is some high scale characterizing these non-renormalizable interactions, v is the Higgs vacuum expectation value, e is the proton charge, and c γ , c g , and c f 4 are constants. Following convention, we choose an O normalization by specifying the unparticle propagator [4,5] scalar unparticle propagator : where Here the modified propagator suggested in Ref. [7] is used to take into account the breaking of conformal invariance at a scale µ by unparticle couplings to the Higgs boson. A more detailed analysis was performed in Ref. [8] by considering a deconstructed version of the unparticle-Higgs coupling. That approach led to a propagator of the form where v and m h are the Higgs vacuum expectation value and mass, respectively, and µ U is a scale related to the Higgs-unparticle coupling.
We will be most interested in unparticle couplings to two photons. This coupling, in conjunction with unparticle couplings to gluons and quarks, mediates processes gg, qq → U → γγ, which have been studied previously in Ref. [11]. With our coupling conventions, these processes have differential cross sections where we have averaged and summed over initial and final state colors and polarizations, but have not yet included the factor 1/2 to account for the two identical photons in the final state.
These processes are bounded by existing data, most stringently by the diphoton invariant mass distribution from the CDF Collaboration at the Tevatron [18], based on an integrated luminosity of 1.2 fb −1 at √ s = 1.96 TeV. The events in this distribution have two central photons with |η| < 1.04 and transverse momenta p 1,2 T ≥ 15 GeV. To obtain a lower bound on Λ 4 , we simulate the signal by adopting the following procedure here and in all analyses described below. We set c g = c γ = 1 and e 2 (c f 4 ) 2 = 2π, following the convention justified in Ref. [9], use the CTEQ5L parton distribution functions with a factorization scale µ f = √ŝ [19], and evaluate the cross section with the Monte Carlo program VEGAS [20]. We then impose the identical η and p T cuts given above, and further require m γγ > M th , where m γγ is the diphoton invariant mass, and M th is a threshold mass chosen to maximize the sensitivity to the signal. At the Tevatron, unparticle production through qq dominates over gg, and the most stringent bounds are typically achieved for M th ≈ 350 GeV. The 95% CL lower bound on Λ 4 is then derived following Ref. [21]; these results are given in Fig. 3. For comparison, in Fig. 3 we also present bounds on Λ 4 given previously in the literature from e + e − → U → µ + µ − from LEP/SLC [9] and the unitarity bound on W W scattering [10].
We have also determined the discovery reach for pp → U → γγ at the LHC, assuming 100 fb −1 of data. We evaluate the prompt diphoton background using PYTHIA 6.4 [22], impose kinematic cuts |η| < 2.5 and p 1,2 T > 25 GeV, and again optimize by varying the threshold energy M th . We estimate a 5σ discovery reach by requiring S/ √ B > 5 and S > 5, where S and B are the number of signal and background events. This discovery reach is also presented in Fig. 3.
These results have omitted important effects. Notably, in determining the LHC reach, we have omitted the background from misidentification of jets and electrons. Nevertheless, we may conclude that present limits from the Tevatron and LEP/SLC are comparable, and imply lower bounds on Λ 4 of ∼ 2 TeV for d = 1.1, dropping to below a TeV for d = 1.9. We will adopt the reference value Λ 4 = 1 TeV below. At the LHC, the sensitivity is much higher, with the γγ signal probing Λ 4 in the multi-TeV range. Note that here, and in the rest of this work, we have taken the conformal breaking scale to be µ = 0 in deriving our results. However, we have found that our results are insensitive to this choice, provided µ < ∼ M th . In general, the modifications have the property that they are small for large q 2 . For example, for q 2 ≫ µ 2 , the propagator we used (Eq. (7)) is of the form (q 2 ) d−2 with corrections of order µ 2 /q 2 . In this limit the propagator obtained from Ref. [8] has corrections of order (µ 2 /q 2 ) 2−d v 2 /(q 2 − m 2 h ). We will assume henceforth that the breaking of conformal symmetry is small in the kinematic region we work in so that the correction terms can be ignored.

IV. FOUR PHOTON EVENTS
We now turn to new processes mediated by unparticle self-interactions. As noted in Sec. I, the 3-point function is particularly interesting, and, depending on the unparticle interactions with the standard model, may mediate a variety of processes leading to spectacular 4-body final states. As an example, in this section we consider the four photon signal shown in Fig. 1.
The cross sections for gg, qq → U → UU → γγγγ are completely specified, given the Feynman rules of Eqs. (3) and (6). With momenta as labeled in Fig. 1, we find where we have averaged and summed over initial and final state colors and polarizations, but have not yet included the factor 1/4! to account for the four identical photons in the final state. To evaluate the parton-level cross sections, we use the results of Sec. II for the 3-point correlation function. Setting c g = c γ = 1 and e 2 (c f 4 ) 2 = 2π as above, and integrating over 4body phase space with the Monte Carlo program VEGAS [20], we find that the parton-level cross sections may be written aŝ where the dimensionless proportionality factors f g d and f q d are given in Table I Upper bounds on this combination of parameters from existing data at the Tevatron are also listed, as are the resulting maximum cross sections possible at the LHC and ILC.

A. Bounds from Tevatron
The 4γ unparticle signal is bounded by searches at the Tevatron. The D0 collaboration has searched for the inclusive production of multi-photon final states [23]. Events are selected with three or more photons in the central calorimeter (|η| < 1.1) and E T -ordered cuts on their transverse energies: E 1,2,3 (,4) T > 30, 20, 15 (, 15) GeV. The dominant backgrounds are diphoton production with additional initial state radiation (ISR) photons and events in which jets or electrons are misidentified as photons. No excess of events above the standard model prediction was found in the Tevatron data with integrated luminosity 0.83 ± 0.05 fb −1 collected during 2002-2005. This search result thus sets upper bounds on combinations of the unparticle selfinteraction strength C d and the energy scale Λ 4 . To derive these bounds, we again set c g = c γ = 1 and e 2 (c f 4 ) 2 = 2π. As with the γγ signal discussed in Sec. III, at the Tevatron, the qq contribution dominates over that from gg, since pp collisions provide a large density of anti-quarks, and at √ s = 1.96 TeV, large Bjorken x is required.
We impose the kinematic cuts on rapidities and transverse energy as given above. Estimating the background event rate to be L σ SM qq→4γ ≈ L ( q Q 2 q ) (α/π) 2 σ SM qq→2γ ∼ 0.83 fb −1 × (0.002) 2 × 0.66 pb ∼ O(10 −3 ), we find that it is negligible. The 95% CL upper limit of 3.04 events for zero background and zero events observed therefore becomes the 95% CL bound where σ ref is the total reference cross section, determined by setting Λ 4 = 1 TeV and C d = 1. The resulting upper bounds on C d /Λ 3d 4 are given in Table I.

B. Prospects for LHC and ILC
We now determine the LHC cross sections for 4γ production mediated by unparticle selfinteractions. Using VEGAS [20], we Monte Carlo simulate 4 to 10 million events for several values of d. We require the photons to have rapidity |η| < 2.5 and transverse energies E 1,2,3,4 T > 30, 20,15,15 GeV. The resulting reference cross sections for Λ 4 = 1 TeV and C d = 1 from gg and qq initial states are given in Table I. Of course, the actual cross sections scale with C 2 d . We do not know of any way to bound C d theoretically, or, indeed, to specify a "typical" value for C d . Lacking theoretical guidance, we simply impose existing experimental bounds. Although bounds on unparticles have been discussed at length in the literature, these constraints, including those derived and presented in Sec. III, do not bound C d . For a direct bound, we must therefore turn to the Tevatron results derived above, which, of course, constrain C d and Λ 4 in the combination identical to that which enters the LHC cross section. Assuming these parameters saturate the bound of Eq. (15), we find the maximal cross sections presented in Table I and plotted in Fig. 4. These cross sections are extraordinarily large, ranging from picobarns for d = 1.1 to nanobarns for d = 1.9. If the unparticle self-interaction is anywhere near the largest values allowed by current experimental constraints, this spectacular signal will be discovered very early at the LHC.
The p T distributions of the leading, next-to-leading, next-next-to-leading, and fourth photons in unparticle 4γ events are given in Fig. 5. We find that the p T spectra, even for the 3rd and 4th photon, are remarkably hard. Given this, the standard model background of diphoton production with two ISR photons can be eliminated with p T cuts with little effect on the signal. The dominant backgrounds will be from events with misidentified jets or electrons, but even these may be reduced significantly with hard p T cuts without large reduction in the signal. It would be very interesting to determine the extent to which this is validated by a realistic detector study. Last, we note that the shapes of the p T distributions, along with other kinematic information, are, of course, independent of C d and Λ 4 . They may Leading γ Next-to-Leading γ Next-Next-to-Leading γ Leading γ Next-to-Leading γ Next-Next-to-Leading γ therefore be used to identify unparticles as the source of these signals, and to distinguish them from anomalies predicted by other frameworks for new physics. If the coupling of scalar unparticles to electrons is of the same size as to quarks, 4γ event rates at the International Linear Collider (ILC) can be calculated with Eq. (14) multiplied by 3. ILC reference cross sections and maximal possible cross sections are presented in Table I and Fig. 4. Interestingly, the 4γ cross section at a 1 TeV ILC is about two orders of magnitude larger than at the Tevatron. This is because the parton densities become less than one for x > ∼ 0.3, so the Tevatron pp → 4γ cross section is dominated by the contribution from √ŝ ∼ 300 GeV.

V. DISCUSSION
In this work, we have investigated a new feature of unparticle physics, namely, the selfinteractions of unparticles. Such interactions are necessarily present in conformal theories, are unsuppressed for strongly coupled conformal sectors, and introduce a large range of new phenomena not studied previously.
As an example, we have investigated the 3-point correlation function in detail. This is completely specified by conformal invariance, 1 up to a constant C d , and mediates a host of processes, such as pp → U → UU → γγγγ, γγZZ, ZZZZ, γγl + l − , ZZl + l − , 4l at the LHC. We know of no upper bound on C d . We have therefore imposed only the existing Tevatron bounds, and have found that, for the example 4γ signal discussed here, the allowed cross sections at the LHC are enormous, ranging from picobarns for unparticle dimension d = 1.1 to nanobarns for d = 1.9. Such signals could emerge very early after the LHC turns on, and imply that the 4-body final states could be by far the most promising modes for unparticle discovery at colliders. Given these observations, it would be very interesting to perform similar analyses for the other final states, as well as to carry out realistic detector simulations to determine the extent to which jet and lepton misidentification masks these a priori spectacular signals.
On the theoretical side, the most pressing issue is to determine what values of C d are possible or natural. Conformal invariance by itself does not determine the constant, and we are unaware of any other consistency conditions that could bound C d . It would be very interesting to see if the AdS/CFT correspondence can shed light on this issue. In the absence of theoretical bounds, C d can be large, and pp → UU → 4γ and related 4-body final states may have cross sections larger than pp → U → γγ and related 2-body states. Such a conclusion is highly counter-intuitive, but perhaps possible, given the other surprising properties of unparticles discovered so far.