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Unparticle Example in 2D Unparticle Example in 2D The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Georgi, Howard, Yevgeny Kats. 2008. Unparticle Example in 2D. Physical Review Letters 101, 131603. Published Version http://dx.doi.org/10.1103/PhysRevLett.101.131603 Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:2794951 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#OAP week ending PRL 101, 131603 (2008) PHYSICAL REVIEW LETTERS 26 SEPTEMBER 2008 Unparticle Example in 2D Howard Georgi* and Yevgeny Kats+ Center for the Fundamental Laws of Nature, Jefferson Physical Laboratory, Harvard University, Cambridge, Massachusetts 02138, USA (Received 27 July 2008; published 25 September 2008) We discuss what can be learned about unparticle physics by studying simple quantum field theories in one space and one time dimension. We argue that the exactly soluble 2D theory of a massless fermion coupled to a massive vector boson, the Sommerfield model, is an interesting analog of a Banks-Zaks model, approaching a free theory at high energies and a scale-invariant theory with nontrivial anomalous dimensions at low energies. We construct a toy standard model coupling to the fermions in the Sommerfield model and study how the transition from unparticle behavior at low energies to free particle behavior at high energies manifests itself in interactions with the toy standard model particles. DOI: 10.1103/PhysRevLett.101.131603 PACS numbers: 11.15.Tk, 11.10.Kk, 14.80.Àj The term ‘‘unparticle physics’’ was coined by one of us Uð1Þ charge. In the infrared, the resulting interaction to describe a situation in which standard model physics is flows to a coupling of two charged scalars to an unparticle weakly coupled at high energies to a sector that flows to a field with a nontrivial anomalous dimension. We apply scale-invariant theory in the infrared [1,2]. In this class of the operator product expansion to the solution of the models, one may see surprising effects from the production Sommerfield model to find the correlation functions of of unparticle stuff [3] in the scattering of standard model the low-energy unparticle operator. Finally, we study the particles. Studying such models forces us to confront some simplest unparticle process shown in Fig. 1 in which two interesting issues in scale-invariant theories and effective toy standard model scalars ‘‘disappear’’ into unparticle field theories. stuff. Because we have the exact solution for the unparticle It is important to remember that unparticle physics is not correlation functions, we can see precisely how the system just about a scale-invariant theory. There are two other makes the transition from low-energy unparticle physics to important ingredients. A crucial one is the coupling of the high-energy physics of free particles. The answer is the unparticle fields to the standard model. Without this simple and elegant. The ‘‘spectrum’’ of the model consists coupling, we would not be able to ‘‘see’’ unparticle stuff. of unparticle stuff and massive bosons. As the incoming Also important is the transition in the Banks-Zaks theory energy of the standard model scalars is increased, the [4] from which unparticle physics emerges from perturba- unparticle stuff is always there, but more and more massive tive physics at high energies to scale-invariant unparticle bosons are emitted, and the combination becomes more behavior at low energies. This allows us to find well- and more like the free-fermion cross section. controlled perturbative physics that produces the coupling The Sommerfield(-Thirring-Wess) model [6,7] is the of the unparticle sector to the standard model. Without this Schwinger model [8] with an additional mass term for transition, the coupling of the standard model to the un- particle fields would have to be put in by hand in a completely arbitrary way, and much of the phenomeno- logical interest of the unparticle metaphor would be lost. In this Letter, we explore the physics of the transition from unparticle behavior at low energies to perturbative behavior at high energies in a model with one space and one time dimension in which the analog of the Banks-Zaks model is exactly solvable. This will enable us to see how the transition takes place explicitly in a simple inclusive scattering process (Fig. 1). We begin by describing our analog Banks-Zaks model and its solution. It is a 2D model of massless fermions coupled to a massive vector field. We call it the Sommerfield model because it was solved by Sommerfield [5] in 1963 [6]. Next, we describe the high-energy physics that couples the Sommerfield model to our toy standard model, which is simply a massive scalar carrying a global FIG. 1. A disappearance process. 0031-9007=08=101(13)=131603(4) 131603-1 Ó 2008 The American Physical Society week ending PRL 101, 131603 (2008) PHYSICAL REVIEW LETTERS 26 SEPTEMBER 2008 the vector boson [9]: a power of x2, and the 2-point function is proportional to an unparticle propagator [13] 1 m2 L ¼ ði@6 À eA6 Þ À FF þ 0 AA : (1) 4 2 1 iÁOðxÞ!iÁUðxÞ¼ ; (9) 2ð Þ2að 2 þ Þ1þa We are interested in this theory since it is exactly solvable 4 m x i and (unlike the Schwinger model) has fractional anoma- where lous dimensions. In particular, we are interested in the e2 1 composite operator a À ¼ : (10) 2 þ 2 2 O à m 1 m0=e 2 1 (2) Thus at large distance and low energies, the composite because, in the low-energy theory, it scales with an anoma- operator O scales with dimension 1 þ a, corresponding lous dimension. to an anomalous dimension of a for à .For0 <m < The solution for all fermion Green’s functions in the 2 1 0 1, a is fractional, which leads to unparticle behavior. In model can be written down explicitly, in terms of propa- momentum space gators for free fermions, and for massless and massive scalar fields with mass m: iAðaÞ 2 a iÁUðpÞ¼ 2a ðp ÀiÞ 2 2ðmÞ sinðaÞ 2 2 e m ¼ m0 þ : (3) Z 1 AðaÞ 2 2 a i ¼ 2a dM ðM Þ 2 2 ; (11) The physical mass m plays the role in this model of the 2ðmÞ 0 p ÀM þi unparticle scale ÃU from Ref. [1], setting the scale of the where the function transition between free particle behavior at high energies and unparticle behavior at low energies. Explicitly, the sinðaÞÀðaÞ AðaÞ (12) scalar propagators are [10] 21þ2aÀð1 þ aÞ Z d2p eÀipx is positive in the range relevant to our model ( À 1 <a< ÁðxÞ¼ ð2Þ2 p2 À m2 þ i 0). Since pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 AðaÞ 2 2 a ¼ K0ðm Àx þ iÞ; (4) Im ÁUðpÞ¼ ðp Þðp Þ ; (13) 2 2ðmÞ2a Z d2p eÀipx i Àx2 þ i the unparticle phase space is DðxÞ¼ 2 2 ¼ ln 2 : (5) ð2Þ p þ i 4 x0 AðaÞ 0 2 2 a ÈUðpÞ¼ ðp Þðp Þðp Þ : (14) The n-point functions for the O and Oà fields can then ðmÞ2a be constructed using the operator product expansion. We To generate a coupling to a toy standard model, we will describe all of this in detail in a separate publication assume that the very high-energy theory includes the in- [11,12]. Here we will simply write down and use the result teraction for the 2-point function in position space L 5 à 5 BðxÞ int ¼ ½ ð1 þ Þ þ ð1 À ÞþH:c: O Oà 2 iÁOðxÞh0jT ðxÞ ð0Þj0i¼ 2 2 ; (6) 4 ðx þ iÞ Ã Ã Ã ¼ ð 21 þ 12ÞþH:c: (15) where that couples the fermion of the unparticle sector to a 4e2 neutral complex scalar with mass m m that plays BðxÞ¼exp i f½ÁðxÞÁð0ÞÀ½DðxÞDð0Þg m2 the role of a standard model field. The interaction is 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mediated by the heavy fermion with mass M 2e 2 2 2 ¼ exp 2 ½K0ðm Àx þiÞþlnðm Àx þiÞ ; m; =m and the same coupling to A as . The theory m à à has a global Uð1Þ symmetry with charge þ1 for , 1, (7) à and 2, À1 for , 2, and 1, and 0 for . Integrating out with , we obtain ¼ e E =2: (8) h 22 L ¼ ðOÃ2 þ OÃ2Þ;h : (16) int 2 M In the short-distance limit (jx2j1=m2), BðxÞ!1, and one obtains free-fermion behavior. In the large-distance The composite operator O defined in (2) has charge À2 2 2 limit (jx j1=m ), K0 does not contribute, so BðxÞ is just under the global Uð1Þ symmetry. 131603-2 week ending PRL 101, 131603 (2008) PHYSICAL REVIEW LETTERS 26 SEPTEMBER 2008 X1 n Z 2 In a 4D unparticle theory, the interaction corresponding ð4aÞ d pU to (16) would typically be nonrenormalizable, becoming iÁOðPÞ¼ 2 iÁUðpUÞ n¼0 n! ð2Þ more important as the energy increases. That does not Yn 2 Xn happen in our 2D toy model. But we can and will study d pi 2 2  iÁðp Þ ð2Þ PÀpU À p : the process in Fig. 1 in the unparticle limit and learn 2 i j i¼1 ð2Þ j¼1 something about the transition region between the ordinary particle physics behavior at energies large compared to m (22) and the unparticle physics at low energies. To that end, we consider the physical process þ ! This describes a sum of 2-point diagrams in which the Sommerfield stuff shown in Fig.
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