Note on Unparticle Decays

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 impp, 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 p 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.i, 11.25.Hf, 14.80.j

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 U. 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 U) SM operator which can provide a channel for unparticle 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 1 2 2 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 UU= 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 4m2 ¼js 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 ð Þ¼ U where R ¼ 1ð1Þ corresponds to solutions in the first iPU s ð Þ ; 0 ð Þ 2 sinðd Þ ðs þ m2 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 ðd 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 ðdU 1Þð2dUÞ 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 (p 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 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 masses of the standard model particles the unparticle ð0Þ Ad ð Þ¼ U ð 2ÞdU 2 ð 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 ðsÞ 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 ! 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 ðsÞþðsÞ give an analytical formula for this lower bound on the mass U U gap. The polarization ðsÞ is given in the MS-renormalization For mg > 2m the isolated unparticle pole is complex 3 scheme by [14] (p > 0) and appears in the second Riemann sheet, and this now corresponds to the decay of a resonance. This case 2 2 1 þ ðsÞ ð Þ¼ U U þ ð Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is exhibited in Fig. 2, where m and are plotted vs d s 2 log 2 2 2 s log ; p p U 32 m 2ðsÞ1 ¼ 2 for the case mg 4m (thick solid lines). Finally, since U 2 2 (7) is a global factor in the polarization, the values of mg mp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and exhibit an approximate scaling behavior with 2 . where ðsÞ¼ 1 4m2=s and we have set the renormal- p U We want to emphasize here that there are complex pole ization scale equal to the cutoff U. (For numerical work solutions for all values of dU in the considered range we fix U ¼ 100m.) The location of the unparticle resonances will be deter- 2 2For d very close to 2, one can also have m >m , but in mined by the propagator poles s ¼ mp impp in the U p g s m such cases the mass difference between the pole and the mass complex plane (with p the pole mass and p its width). gap is infinitesimal. The polarization ðsÞ has a branch cut that we take from 3In all cases we also found the corresponding shadow pole [16] the threshold at s ¼ 4m2 to infinity along the real axis with in the unphysical sheet, as required by Hermitian analyticity.

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2 FIG. 1 (color online). Left panel: Plot of mp as a function of dU for U ¼ 5, mg ¼ m, and ¼ U ¼ 100m. Right panel: Plot of mp vs mg for U ¼ 5 and dU ¼ 1:2. All masses are in units of m.

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FIG. 2 (color online). Left (right) panel: Plot of mp (p) as a function of dU for U ¼ 5 and mg ¼ 4m (thick solid lines). Corresponding results based on the analytical approximation of Eq. (11) are plotted in thick dashed lines. All masses are in units of m.

1 dU < 2, unlike what was claimed in Ref. [13]. In our Figure 2 compares the values for mp and p obtained using 2 case, nothing special happens for dU > 3=2, and mp and the analytic approximation in (11) (thick dashed lines) with 2 p smoothly approach mg and zero, respectively, when the full numerical results (thick solid lines), showing that dU ! 2. the analytical approximation is excellent. We can use this It is easy to understand our results analytically. For approximation to write down analytically the lower bound 2 values of s close to the resonance region, one can approxi- on mg to avoid a tachyon. It is given by mate the complex polarization by the constant ð Þ ð0ÞA 1= 2 dU 2 dU ð Þ’ ð 2Þ mg > ; (14) s mg 2j sinðd Þj U 2 2 ¼ U log U þ 2 ðm2Þ with ð0Þ¼2 =ð162Þ logð =mÞ. 322 m2 g U U We can gain further insight into the unparticle spectrum 1 þ ðm2Þ g by calculating the spectral function for the one-loop cor- log 2 i ; (10) 1 ðmgÞ rected propagator, 1 and a simple calculation yields an analytic approximation ðsÞ¼ Im½iPð1Þðs þ iÞ: (15) for the pole mass and width as U

2 2 2 2 2 mp ’ mg m cos; mpp ’ ðmgÞm j sinj; As we show below, this spectral function will faithfully (11) reproduce the main features of the pole structure discussed previously, also giving information on the unparticle con- where tinuum above the mass gap. The expression we ﬁnd for this spectral function is the following: 2 ð Þ jðm ÞjA 1= 2 dU 2 g dU m (12) ð 2 Þ 2j sinðd Þj 1 Im½ðsÞ s mp U ðsÞ¼ þ ð4m2 m2 Þ U j ð1Þð Þj2 p ð1Þð Þ and DU s dDU s =ds 2ð Þ ðs m2Þ2 dU 2 2 2sin dU g 1 Im½ðm Þ þ ðs mgÞ : (16) ¼ arctan g : (13) A j ð1Þð Þj2 ð Þ 2 dU DU s 2 dU Re½ðmgÞ

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2 FIG. 3 (color online). Left panel: Plot of UðsÞ for mg ¼ m, dU ¼ 1:25, U ¼ 5, and ¼ U ¼ 100m. Right panel: Plot of K ðdUÞ for the same values of mass parameters. All masses are in units of m.

The first term of UðsÞ is proportional to the imaginary It can be easily calculated that the height of the peak is 2 5 part of ðsÞ [which contains a factor ðs 4m Þ], and thus independent of dU and given by the simple expression 2 2 for mp > 4m it corresponds, through the Cutkosky rules, 32 to a width for the unparticles which decay beyond the max ’ : (18) 2 2 U 2 ð 2Þ threshold. The second term (for mp < 4m ) corresponds U mg to a real pole in the first Riemann sheet, and should be interpreted as a stable (un)particle of mass mp. Finally, the Therefore, as the width goes to zero we do not recover a ð Þ 0 ð Þ4 Dirac delta function at mp, and the resonance will be very third term is proportional to the imaginary part of DU s and does not correspond to any unparticle decay, but gives difficult to detect experimentally over the continuous back- rise to the familiar continuous contribution to the spectral ground starting at mg. 2 2 function above the mass gap (a similar continuum appears Notice that for mg > 4m the resonant (‘‘on-shell’’) in the Higgs spectral function when the Higgs is coupled to production of unparticles would dominate the amplitude an unparticle operator [17]). ’’ ! ’’, as it happens with an ordinary exchange of When the decay OU ! ’’ occurs it should give rise to particles in the s channel. Here the presence of unparticles a resonant structure in the spectral function UðsÞ through should be detected through a peak in the invariant mass the term proportional to Im½ðsÞ, with an approximate distribution of the final state, similar to the case of a new 2 0 Breit-Wigner distribution centered around mp of width p. particle resonance (e.g. the production of a Z ). For the case 2 2 This should correspond to the structure of the poles of the mp < 4m the resonance is located below the production propagator Pð1ÞðsÞ in the complex s plane, i.e. to the zeroes threshold and the spectral function is dominated by the of the function Dð1ÞðsÞ which we have previously studied. continuous contribution, which does not provide any de- In the left panel of Fig. 3 we have plotted the spectral cay. In that case there is no resonant production and the function for the case of Fig. 1, in which there is no resonant production of the final state ’’ will be as if induced ‘‘off- interpretation, but instead a real pole appears. We see a shell.’’ The presence of unparticles in the intermediate state delta function corresponding to that pole and a continuous should be detected by the continuous enhancement of the 2 corresponding cross section. This situation is reminiscent component for s>mg. In the right panel of Fig. 3 we have plotted the strength of the isolated pole, K2ðm2 ;d Þ, de- of the familiar case of exchange of graviton Kaluza-Klein p U modes in Arkani-Hammed–Dimopoulos–Dvali theories of fined as extra dimensions, where the excess of the cross section is 1 2ð Þ¼ used to put bounds on the value of the fundamental scale. K s; dU ð Þ : (17) dD 1 ðsÞ=ds The formalism to be used for any realistic standard U y model channel, e.g. AA, c L c R,orH H, is similar to The fact that for mg > 2m the pole width is sharpening the one used in the toy model considered in this paper. In 2 for increasing values of dU (as shown by the right plot in every case, for the particular channel OU ! AB,ifmp > Fig. 2) is also shown in Fig. 4, in which we plot the spectral ðm þ m Þ2 the unparticle should be detected in the cor- ¼ A B function for values of the dimension dU 1:25 (left panel) responding cross section through a peak in the invariant ¼ and dU 1:5 (right panel). In both cases we see a clear mass distribution of the final state, which should recon- resonant contribution that overwhelms the continuous one. struct the resonant pole, much like the reconstruction of a In this region and for values of s close to the value of m2 , 0 2 2 p Z resonance. On the contrary, if mp < ðmA þ mBÞ then the unparticle behaves as a resonance.

5 This statement is true up to values of dU very close to 2, for 4 Notice that this imaginary part is only different from zero for which the width of the resonance is zero and mp ¼ mg for all 2 2 s>mg, and thus it contains a factor ðs mgÞ. practical purposes.

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FIG. 4 (color online). Left (right) panel: Plot of UðsÞ for dU ¼ 1:25 (dU ¼ 1:5), mg ¼ 4m, U ¼ 5, and ¼ U ¼ 100m. All masses are in units of m. the only indirect detection of the unparticle should be by an We are indebted to Rafel Escribano for useful discus- excess of events with respect to the corresponding standard sions about the pole structure of the propagator. J. M. N. model cross section. thanks the Department of Physics of the University of In the standard model the only relevant (unsuppressed) Notre Dame for hospitality during the last stage of this operator which can give rise to unparticle two-body decays work. The work is supported in part by the European 2 is UjHj OU, an interaction which has been thoroughly Commission under the European Union through the analyzed in Refs. [4,17]. In that case the mixing in the Marie Curie Research and Training Networks ‘‘Quest broken phase provided by the Lagrangian term UvhOU for Uniﬁcation’’ (MRTN-CT-2004-503369) and gives rise to a tree-level mixing between the Higgs and ‘‘UniverseNet’’ (MRTN-CT-2006-035863); by the Oð 2 2Þ unparticles which is Uv . Since this mixing is of the Spanish Consolider-Ingenio 2010 Programme CPAN Oð 2 Þ same order as the one-loop polarization, U , it can be (CSD2007-00042); by a Comunidad de Madrid project resummed in the unparticle propagator along with , (P-ESP-00346); and by CICYT, Spain, under Contract leading to [4] No. FPA 2007-60252 and No. FPA 2008-01430.

v2 APPENDIX: NORMALIZATION OF THE D ½s; pðsÞ ! D ½s; pðsÞ þ 2 ; (19) U U U 2 þ SPECTRAL FUNCTION s mh i In this appendix we address the issue of the normalization of the spectral function ðsÞ. The integral of a where m is the tree-level (unmixed) Higgs mass. The U h spectral function along the real axis is determined by the analysis should follow similar lines to those presented in normalization of the state being considered. For instance, if this paper (after including the extra mixing term) by just ð Þ we take the tree-level spectral function 0 , we would replacing m ! mh. We have checked that the qualitative U results found in this paper do not change after the inclusion write of the Higgs-unparticle mixing. Finally, for other channels Z 1 ð0Þ ¼ ð0Þð Þ ¼h j i corresponding to standard model particles which do not NU U s ds U U ; (A1) acquire any vacuum expectation values, the qualitative 0 results should be similar to those presented in this paper. where jUi represents the unparticle operator OU. This To summarize our results, we have studied unparticle integral turns out to be UV divergent which corresponds decays into SM particles, and showed that this possibility is to a non-normalizable state jUi. Strictly speaking, one controlled by the relation between the unparticle mass gap should cut off this integral at a scale of order , beyond þ U mg and the production threshold mA mB (the latter are which the theory leaves the conformal regime. The nor- þ the masses of the decay products). When mg >mA mB malization integral scales as there is enough phase space, unparticles can decay into SM Z 2 particles, and that decay is accounted for by the appearance ð0Þ U ð0Þ ð Þ¼ ð Þ ð 2 ÞdU 1 NU U U s ds U : (A2) of a complex pole on the unparticle one-loop resummed 0 propagator. If, in turn, mg

055011-5 DELGADO, ESPINOSA, NO, AND QUIRO´ S PHYSICAL REVIEW D 79, 055011 (2009) After including one-loop polarization effects the shape One can also show that of the spectral function is affected, but it keeps the same normalization as before. In fact, deﬁning Z Z 2 2 ð Þ¼ U ð Þ ð Þ ð0Þð Þ¼ U ½ ð Þ ð0Þð Þ NU U U s ds; (A3) NU u NU U U s U s ds 0 0 ð 2 Þ2dU 3 one ﬁnds U ; (A5) 2 2 NU m ¼ þ O g þ O U 2 ð0Þ 1 logU ; (A4) 2 ð 2 Þ2 dU NU U U which tends to 0 for dU < 3=2, a case in which the equality ð Þ 2 2 ð 2 Þ1= 2 dU which tends to 1 for U mg, U . of the normalizations holds also in this stronger sense.

[1] H. Georgi, Phys. Rev. Lett. 98, 221601 (2007); Phys. Lett. Senoguz, Phys. Lett. B 661, 276 (2008). B 650, 275 (2007). [9] M. J. Strassler, arXiv:hep-ph/0801.0629. [2] P.J. Fox, A. Rajaraman, and Y. Shirman, Phys. Rev. D 76, [10] G. Cacciapaglia, G. Marandella, and J. Terning, J. High 075004 (2007). Energy Phys. 01 (2008) 070; arXiv:hep-ph/0804.0424; [3] B. Grinstein, K. A. Intriligator, and I. Z. Rothstein, Phys. D. Stancato and J. Terning, arXiv:hep-ph/0807.3961. Lett. B 662, 367 (2008); K. Cheung, W. Y. Keung, and [11] A. Rajaraman, AIP Conf. Proc. 1078, 63 (2009). T. C. Yuan, Phys. Rev. Lett. 99, 051803 (2007). [12] M. A. Stephanov, Phys. Rev. D 76, 035008 (2007). [4] A. Delgado, J. R. Espinosa, and M. Quiro´s, J. High Energy [13] A. Rajaraman, Phys. Lett. B 671, 411 (2009). Phys. 10 (2007) 094. [14] B. A. Kniehl, Phys. Rep. 240, 211 (1994). [5] J. McDonald, arXiv:hep-ph/0709.2350; arXiv:hep-ph/ [15] R. Escribano, A. Gallegos, J. L. Lucio, M. G. Moreno, and 0805.1888. J. Pestieau, Eur. Phys. J. C 28, 107 (2003). [6] T. G. Rizzo, J. High Energy Phys. 10 (2007) 044; J. High [16] P. V. Landshoff, Nuovo Cimento 28, 123 (1963). Energy Phys. 11 (2008) 039. [17] A. Delgado, J. R. Espinosa, J. M. No, and M. Quiro´s, [7] M. Bander, J. L. Feng, A. Rajaraman, and Y. Shirman, J. High Energy Phys. 04 (2008) 028; J. High Energy Phys. Rev. D 76, 115002 (2007). Phys. 11 (2008) 071. [8] V. Barger, Y. Gao, W. Y. Keung, D. Marfatia, and V.N.

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