PHYSICAL REVIEW D 79, 055011 (2009) Note on unparticle decays Antonio Delgado,1 Jose´ R. Espinosa,2,3 Jose´ Miguel No,2 and Mariano Quiro´s3,4 1Department of Physics, 225 Nieuwland Science Hall, University of Notre Dame, Notre Dame, Indiana 46556-5670, USA 2IFT-UAM/CSIC, Facultad Ciencias UAM, 28049 Madrid, Spain 3IFAE, Universitat Auto`noma de Barcelona, 08193 Bellaterra, Barcelona, Spain and ICREA, Institucio` Catalana de Recerca i Estudis Avanc¸ats, Barcelona, Spain 4Theory Division, CERN, Geneva 23 CH-1211, Switzerland (Received 30 January 2009; published 16 March 2009) The coupling of an unparticle operator OU to standard model particles opens up the possibility of unparticle decays into standard model fields. We study this issue by analyzing the pole structure (and spectral function) of the unparticle propagator, corrected to account for one-loop polarization effects from virtual standard model particles. We find that the propagator of a scalar unparticle (of scaling dimension 2 2 2 1 dU < 2) with a mass gap mg develops an isolated pole, mp À impÀp, with mp & mg below the unparticle continuum that extends above mg (showing that the theory would be unstable without a mass gap). If that pole lies below the threshold for decay into two standard model particles, it corresponds to a 2 stable unparticle state (and its width Àp is zero). For mp above the threshold, the width is nonzero and related to the rate of the unparticle decay into standard model particles. This picture is valid for any value of dU in the considered range. DOI: 10.1103/PhysRevD.79.055011 PACS numbers: 12.60.Ài, 11.25.Hf, 14.80.Àj Unparticle physics was introduced in Ref. [1] as the decay, along with the associated resonant structure, will effective description of a conformal theory coupled to the depend on the precise relationship between the mass gap standard model (SM). Unparticles have their origin in a mg and the SM threshold of the channel to which the hidden sector that flows to a strongly coupled conformal unparticle operator is coupled. In particular, we will con- theory with an infrared fixed point below some energy sider the decay of unparticles into SM particles via the L ¼ O O O scale ÃU. Since that theory is strongly coupled, the anoma- Lagrangian coupling U SM U, where SM is a lous dimensions can be large and (below the scale ÃU) SM operator which can provide a channel for unparticle À À unparticle operators can have a dimension dU which differs decay and U is a coupling with dimension 4 dU dSM. 2 2 sizably from its (integer) ultraviolet dimension. In this Examples of such SM operators are F, mfff, and jHj . article we consider unparticles not charged under the SM However, instead of focusing on a particular SM opera- gauge group and (in order to enhance their interactions tor, we start by simply considering a toy model with a real with the standard model) with the lowest possible dimen- scalar ’, with bare mass m and zero VEV, coupled to the O 0 sion. Therefore, we will discuss scalar unparticles U with unparticle scalar operator O with scaling dimension d U U 1 dU < 2 [2,3]. through the effective Lagrangian The conformal invariance of the unparticle sector is L ¼ 1ð Þ2 À 1 2 2 À 1 2O explicitly broken by its interactions with the standard eff 2 @’ 2m0’ 2U’ U; (1) model. Moreover, when the Higgs field acquires a vacuum which should capture the main features of more realistic expectation value (VEV), this large breaking of conformal channels. invariance gives rise to a mass gap mg in the unparticle The last term in the Lagrangian above induces a tadpole spectrum that consists of a continuum of states above mg term for the unparticle operator at one loop, which would [4]. The mass gap plays a relevant role in the cosmology trigger an unparticle VEV.1 This is similar to what happens [5] and phenomenology [6–11] of unparticles and it should 2 when the operator OU is coupled to jHj and the Higgs be taken into account when constraining the unparticle field H acquires a VEV (although there the tadpole is a theory from cosmological and experimental data. tree-level effect). Here we see that this tadpole problem is In this paper we consider the issue of the stability of more generic and would appear even without coupling the unparticles coupled to the standard model or, in other unparticles to the Higgs. It was shown in Ref. [4] that in the words, their possible decay into SM particles. (This is a presence of such tadpoles an IR divergence appears, which controversial subject; see [5,9,12,13].) This issue should has to be cut off by an IR mass gap mg. In the context of [4] have a great impact for unparticles in their influence on the mass gap can be introduced in various ways such that early Universe cosmology, in their capability as dark mat- conformal invariance is spontaneously broken along with ter candidates, and in their possible detection at high- energy colliders through their production and subsequent 1This tadpole is quadratically sensitive to UV physics, so one 2 ð 2Þ decay into SM particles. We will see that the possibility of expects it to be of order UÃU= 16 . 1550-7998=2009=79(5)=055011(6) 055011-1 Ó 2009 The American Physical Society DELGADO, ESPINOSA, NO, AND QUIRO´ S PHYSICAL REVIEW D 79, 055011 (2009) electroweak symmetry. Here we just assume that such a the principal Riemann sheet corresponding to 0 mass gap is provided by the theory. Of course, the VEV of 2, where is defined as s À 4m2 ¼js À 4m2jei. The OU in turn induces a one-loop correction to the mass of the second Riemann sheet is reached by shifting ! þ 2. field ’. We assume that this one-loop corrected mass It can be easily seen that a change in the Riemann sheet is squared is positive, m2 > 0, so as to keep h’i¼0.An equivalent to the replacement ðsÞ!ðsÞ. Then, since alternative possibility is to impose the renormalization the complete propagator is a function of , condition of a zero unparticle tadpole at one loop so that Dð1ÞðsÞD½s; ðsÞ; (8) hOUi¼0. As we show later on, a nonzero mass gap will be necessary in any case. the pole equations In the presence of the mass gap mg the unparticle propagator reads [1,2] D ½s; RðsÞ ¼ 0 (9) ð0Þ 1 Ad 1 À ð Þ¼ U where R ¼ 1ð1Þ corresponds to solutions in the first iPU s ð Þ À ; 0 ð Þ 2 sinðd Þ ðs þ m2 À iÞ2 dU DU s U g (second) Riemann sheet [15]. (2) A numerical analysis of the pole equation (9) shows that, besides the unparticle continuum, an isolated pole appears. 2 with Note that the tree-level propagator had no pole (mg is not a 165=2 Àðd þ 1=2Þ pole but a branch point), and therefore the pole appearance A ¼ U ; (3) dU 2d is a purely one-loop effect. Because of the sign of this ð2Þ U ÀðdU À 1ÞÀð2dUÞ 2 2 2 radiative effect, we find that mp is always below mg, but where we have explicitly introduced the mass gap mg that quite close to it, as the polarization is a radiative effect: 2 & 2 ¼ breaks the conformal invariance. In fact, in some scenarios mp mg.Formp 2m this isolated pole is real (Àp 0) this parameter can be related to the VEVof the Higgs field, and located in the first Riemann sheet. Such a pole does not as was shown in Ref. [4]. A spectral function analysis correspond to any decaying unparticle, and it is entirely shows that, at this level, the unparticle spectrum is a due to the fact that Æ Þ 0 below the threshold and could be continuum extending above the mass gap. More precisely, interpreted as an unparticle bound state. We show in Fig. 1 the spectral function, defined as (left panel) a plot of mp vs dU for m ¼ mg. In this plot one can see that indeed mp ! mg for dU ! 2. ð0Þð Þ¼1 ½ ð0Þð þ Þ U s Im iPU s i ; (4) An immediate consequence of the negative mass shift 2 2 responsible for mp <mg is that it yields a lower bound on is given by the scale of conformal breaking mg. That bound is related to the masses of the standard model particles the unparticle ð0Þ Ad À ð Þ¼ U ð À 2ÞdU 2 ð À 2Þ U s s mg s mg : (5) operator is coupled to (m in our case). This fact is shown by 2 2 Fig. 1 (right panel), where the pole squared mass mp is The polarization ÆðsÞ induced in the unparticle propa- plotted vs mg for dU ¼ 1:2. We can see that the isolated gator by the one-loop diagram exchanging ’ fields can be unparticle pole becomes tachyonic for small values of mg simply added by a Dyson resummation to give (mg < 0:5m). Moreover, this shows that in the particular ! À ð1Þ ¼ 1 ¼ 1 limit mg 0 the theory becomes unstable. Later on we iPU ð Þ ð Þ : (6) D 1 ðsÞ D 0 ðsÞþÆðsÞ give an analytical formula for this lower bound on the mass U U gap. The polarization ÆðsÞ is given in the MS-renormalization For mg > 2m the isolated unparticle pole is complex 3 scheme by [14] (Àp > 0) and appears in the second Riemann sheet, and this now corresponds to the decay of a resonance.