Unparticle Example in 2D

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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

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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.j

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 unparticle 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 eA6 Þ FF þ 0 AA : (1) 4 2 1 iOðxÞ!iUðxÞ¼ ; (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 ¼ : (10) 2 þ 2 2 O 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 .For0 momentum space gators for free fermions, and for massless and massive scalar ﬁelds with mass m: iAðaÞ 2 a iUðpÞ¼ 2a ðp iÞ 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 M þi unparticle scale U 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 sinðaÞðaÞ AðaÞ (12) scalar propagators are [10] 21þ2að1 þ aÞ Z d2p eipx is positive in the range relevant to our model ( 1

¼ eE =2: (8) h 22 L ¼ ðO2 þ O2Þ;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 2 2 2 limit (jx j1=m ), K0 does not contribute, so BðxÞ is just under the global Uð1Þ symmetry.

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X1 n Z 2 In a 4D unparticle theory, the interaction corresponding ð4aÞ d pU to (16) would typically be nonrenormalizable, becoming iOðPÞ¼ 2 iUðpUÞ 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 iðp Þ ð2Þ PpU 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. 1: Because 2 couples to incoming momentum P splits between the unparticle O, we can obtain the total cross section for this process propagator and n massive scalar propagators. Each is from the discontinuity across the physical cut in the O 2- associated with the propagation of a free massive scalar point function. This is analogous to the optical theorem for field, so this gives the discontinuity ordinary particle production. For momenta P1 and P2, X1 n Z 2 this is AðaÞ ð4aÞ d pU 0 2 2 a ðPÞ¼ 2a 2 ðpUÞðpUÞðpUÞ ðmÞ n¼0 n! ð2Þ ImMðP1;P2 ! P1;P2Þ ¼ ; (17) Yn d2p s i 2 2 0 2 2ðpi m Þðpi Þ i¼1 ð2Þ 2 with s ¼ P , P ¼ P1 þ P2, and Xn 2 2 ð2Þ PpU pj : (23) M 2 i ðP1;P2 ! P1;P2Þ¼ih OðPÞ: (18) j¼1 pffiffiffi pffiffiffi In the unparticle limit ( s m), using (13) or directly the For s

iOðxÞ¼iUðxÞ exp½4iaðxÞ X1 n ð4aÞ n ¼ iUðxÞ ½iðxÞ : (21) n¼0 n!

At distances not large compared to 1=m, the higher terms in the sum in (21) become relevant. Notice that, in (21), we have expanded in a only the terms involving the massive boson propagator. This is critical to our results. It would be a mistake to expand iU in powers of a. This would introduce spurious infrared divergences because the massless boson propagator is sick in 1 þ 1 dimensions [14]. The model describes not massive and massless bosons but rather massive bosons and unparticle stuff. In momentum FIG. 2. Phase space forp theffiffiffi disappearance process in Fig. 1 space, we obtain as a function of the energy s (in units of m) for a ¼0:1.

131603-3 week ending PRL 101, 131603 (2008) PHYSICAL REVIEW LETTERS 26 SEPTEMBER 2008 pffiffiffi pffiffiffi *[email protected] 1 2 m s 2 ¼ a ln pffiffiffi þ ð s mÞ ln þ Oða Þ: + 2 eE s m [email protected] [1] H. Georgi, Phys. Rev. Lett. 98, 221601 (2007). (25) [2] H. Georgi, Phys. Lett. B 650, 275 (2007). pffiffiffi For energies s >m, this expression reduces to [3] We prefer ‘‘unparticle stuff’’ to ‘‘unparticles’’ for the physical states, because it is not clear to us what the 1 O 2 ¼ 2 þ ða Þ; (26) noun ‘‘unparticle’’ is supposed to mean and certainly not clear whether it should be singular or plural. that is, the free-fermion resultpffiffiffi (20pffiffiffi). Thus, for jaj1 there [4] T. Banks and A. Zaks, Nucl. Phys. B196, 189 (1982). is a discontinuity in d=d s at s ¼ m, where a transition [5] Georgi’s Ph.D. advisor and Schwinger’s student. occurs from pure unparticle behavior below energy m to [6] C. M. Sommerfield, Ann. Phys. (N.Y.) 26, 1 (1964). pure free-fermion behavior above m (see Fig. 2)[19]. To [7] W. E. Thirring and J. E. Wess, Ann. Phys. (N.Y.) 27, 331 this order, the free-fermion behavior is a sum of the un- (1964). particle and the massive scalar contributions. [8] J. S. Schwinger, Phys. Rev. 128, 2425 (1962). jaj a [9] We will use conventions in which g ¼ diagð1; 1Þ and For larger values of , higher powers of must be 5 included in (25) to approximate the free-fermion regime. ¼ diagð1; 1Þ. [10] K is the modified Bessel function of the second kind, Since each new massive scalar gives a contribution with 0 and x0 is an arbitrary constant thatpffiffiffiffiffiffiffiffiffiffiffiffi will cancel out in only one additional power of a,ifa is close to 1, the free- y the following. For y !1, K0ðyÞ =2ye ! 0.For fermion behavior is approached very slowly. In fact, the O 2 y ! 0, K0ðyÞ¼lnðy=2ÞE þ ðy Þ, where E ¼ limit a !1 is singular, and it corresponds to the 0ð1Þ’0:577 is Euler’s constant. Note that ð0Þ E Schwinger model (m0 ¼ 0). As is often the case, it is not Dð0Þ¼ði=2Þ lnðe x0m=2Þ is finite. trivial to obtain a gauge theory as the limit of a theory with [11] H. Georgi and Y. Kats (to be published). a massive vector boson. The unparticle stuff is absent since [12] There we will also include more complete references to Að1Þ¼0, and the spectrum includes only a massive the unparticle and Sommerfield model literature. boson with m2 ¼ e2=. The case of the Schwinger model [13] Here and below, we incorporate a dimensional factor of 2a has been studied in Refs. [20,21]. 1=ðmÞ in the unparticle propagator so it matches smoothly onto the O propagator. We find this picture of the unparticle scale U ¼ m in the Sommerfield model very satisfying. There is a close [14] This statement has a long history in the mathematical physics literature, going back at least to Ref. [15]. For analog between the way m enters in the process of Fig. 1 an early summary in English, see [16]. See also [17]. It is and the way the dimensional transmutation scale QCD also worth noting that Ref. [15] introduces the notion of enters in inclusive processes in QCD. In QCD, the physical ‘‘infraparticles’’—an approach to continuous mass repre- states are hadrons, typically with masses of the order of sentations of the Poincare´ group, which of course includes QCD unless they are protected by some symmetry (like the unparticle stuff. See [18], where we learned of this inter- pions). But in the total eþe cross section into hadrons (to esting early reference. pick the simplest and most famous example) at high energy [15] B. Schroer, Fortschr. Phys. 11, 1 (1963). 5 E, the sum over physical states reproduces the ‘‘parton [16] J. Tarski, J. Math. Phys. (N.Y.) , 1713 (1964). 31 model’’ result with calculable corrections of order [17] S. R. Coleman, Commun. Math. Phys. , 259 (1973). [18] B. Schroer, arXiv:0804.3563. ffiffiffi 1= lnðE=QCDÞ. We have shown that the process of Fig. 1 p [19] The linear approximationpffiffiffi (25) is not valid for s m in the Sommerfield model works the same way, with the due to large ln s, but we have the exact expression physical states being the massive boson and unparticle (14). stuff. [20] A. Casher, J. B. Kogut, and L. Susskind, Phys. Rev. Lett. We are grateful to C. Cordova, V. Lysov, P. Petrov, 31, 792 (1973). A. Sajjad, and D. Simmons-Duffin for discussions. This [21] A. Casher, J. B. Kogut, and L. Susskind, Phys. Rev. D 10, research is supported in part by the National Science 732 (1974). Foundation under Grant No. PHY-0244821.

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