A Note on Unparticle in Lower Dimensions

Physics Letters B 668 (2008) 336–339

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Physics Letters B

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A note on unparticle in lower dimensions ∗ Patricio Gaete a, Euro Spallucci b, a Departamento de Física, Universidad Técnica Federico Santa María, Valparaíso, Chile b Dipartimento di Fisica Teorica, Università di Trieste and INFN, Sezione di Trieste, Italy article info abstract

Article history: Using the gauge-invariant but path-dependent variables formalism, we examine the effect of the space– Received 2 July 2008 time dimensionality on a physical observable in the unparticle scenario. We explicitly show that long- Received in revised form 19 August 2008 range forces between particles mediated by unparticles are still present whenever we go over into lower Accepted 25 August 2008 dimensions. Available online 2 September 2008 © 2008 Elsevier B.V. All rights reserved. Editor: M. Cveticˇ

The physical consequences of non-trivial scale invariance in son loop approach. It is interesting to observe that from expression high-energy interactions have been extensively discussed at the (1), when dU = 0, the Coulomb potential correction became linear present time [1–14]. As is well known, the interest in studying leading to the confinement of static charges. It should, however, this new scale invariant sector is mainly due to the possibility of be noted that the dU = 0 case is not allowed because the gamma obtaining unusual properties of matter with non-trivial scale in- function is not analytic for this case. It is worth mentioning at this variance occurring in the infrared regime [15]. This new sector has stage that a range for dU of 1 < dU 2 has been considered in the been called as the unparticle sector [16,17]. We further note that literature. In this connection it becomes of interest, in particular, recently a novel way to describe unparticles has been considered to recall that for dU < 1 there is a non-integrable singularity in the [18]. The crucial ingredient of this development is to introduce differential decay rate into unparticles as EU → 0 [16].Aswasob- continuous mass spectrum objects, which permits to interpret un- served by Georgi [16], this is in accord with a theorem due to Mack particle as a field with continuously distributed mass. [21] where it is shown that in an unitary theory fields with dU < 1 In this context it may be recalled that one of the most interest- are not allowed. Thus, from a physical point of view, the above re- ing of the phenomena predicted by unparticle physics is the exis- mark (dU = 0 case) on the static potential (1) may be considered tence of long-range forces between particles mediated by unparti- as another manifestation of the arguments claimed in [16].Here cles [2,19]. More specifically, it was shown that the corresponding it is important to emphasize that the foregoing observations are modified Coulomb potential may be written as restricted to three space dimensions only, and it naturally raises the question of its generalization in lower dimensions. In fact, it is q2 1 V = − not quite evident that the same phenomenon will be repeated in 4π L two and one space dimensions. The present work specifically deals − + − 2dU 2 with this problem, where we examine the effect of the space–time × + 2 (dU 1/2)(dU 1/2) l 1 − , (1) dimensionality of the problem under consideration on a physical 2dU 1 (2d ) L π U observable. 1 − −1 It is worth recalling at this point that two-dimensional models where l is a scale factor, given by l = 2 2dU 2 .Hered is ( ΛU ) U have been an extraordinary theoretical laboratory to test ideas in a non-integral scale dimension of the unparticle field, and de- ΛU quantum field theory. Of particular interest are non-perturbative fines a critical energy scale where the standard model particles can issues like confinement and spectrum of models. Of these, the interact with unparticles. We further note that a different method Schwinger model [22] has probably enjoyed the greatest popularity for arriving at the same static potential profile (1),basedonthe due to several features that it possesses. For example, the spectrum gauge-invariant but path-dependent variables formalism, was also contains a massive mode, the charge is screened and confinement developed in [20]. An important feature of this methodology is is satisfactorily addressed [23,24]. We further note that recently that it provides a physically-based alternative to the usual Wil- the unparticle stuff in one space dimension has been studied [26]. In particular it was considered the Sommerfield model [27], that is, the exactly soluble two-dimensional theory of a massless fermion * Corresponding author. E-mail addresses: [email protected] (P. Gaete), [email protected] coupled to a massive vector boson. Notice that this model is the (E. Spallucci). Schwinger model with an additional mass term for the vector bo-

0370-2693/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2008.08.055 P. Gaete, E. Spallucci / Physics Letters B 668 (2008) 336–339 337 son. As was explained in [26], the Sommerfield model is an inter- now compute the interaction energy for a single mode in Eq. (4). esting analog of a Banks–Zaks model [15], approaching a free the- The canonical Hamiltonian can be worked as usual and is given by ory at high energies and a scale invariant theory with non-trivial − m2 1 anomalous dimensions at low energies. It is worth recalling at this 2 1 i k HC = d x − Π 1 + Πi 2 Δ + a2 stage that Banks and Zaks investigated the unusual properties of n matter with non-trivial scale invariance in infra-red regime. Inter- 1 m2 + i + + k ij estingly, this new kind of stuff has no definite mass at all. As men- Π ∂i A0 Fij 1 F , (5) 4 Δ + a2 tioned before, this new sector has been called as the unparticle n m2 sector [16,17]. The above remark opens up the way to a stimulating i =− + k k0i where Π (1 + 2 )F are the canonical momenta. Here, we discussion on the existence of long-range forces between particles Δ an have simplified our notation by setting Δ + ≡ Δ. mediated by unparticles in the two-dimensional case. On the other (2 1) Following our earlier discussion [20],theresultingstaticpoten- hand, also it is important to recall here that three-dimensional tial for two opposite charges located at y and y takes the form: theories have been studied by various authors in the last few N ∞ years [28–30]. As is well known, they are interesting because of its 1 q2 q2a2 V = − K (M L) + n L , (6) connection to the high-temperature limit of four-dimensional the- 2 0 k,n e 2π 4Mk,n ories [31–33] as well as for their applications to condensed matter k=1 n=0 k physics [34]. Thus, as already mentioned, the main purpose here where M2 ≡ m2 + a2, L ≡|y − y | and K (M L) is a modified is to examine the effects of the space–time dimensionality on a k,n k n 0 k,n Bessel function. physical observable for the three and two-dimensional cases. To In Eq. (6) the first terms is the one which will account for the do this, we will work out the static potential for the theories un- “unparticle” corrections, while the second one is the “Coulombic der consideration by using the gauge-invariant but path-dependent interaction” in 2 + 1 dimensions. It is worth to remark contrary variables formalism along the lines of Ref. [20]. As a result, there to expectation it is not a logarithmic potential, rather we find are two generic features that are common in the four-dimensional that massive KK-modes produce a linear term. This difference can case and its lower-dimensional extensions studied here. First one, be traced back to the different definition of static potential. In- the existence of long-range forces between particles mediated by stead of using the standard one, we use the gauge-invariant/path- unparticles. The second point is related to that in an unitary theory dependent approach developed in [24,25].However,thesumover fields with d < 1 are not allowed. U n is ill-defined as it is linearly divergent. The “string-tension” σ We turn now to the problem of obtaining the interaction en- k for each field is given by the divergent sum ergy between static point-like sources for the lower-dimensional ∞ extensions under consideration. To do this, we shall compute the 2 2 = q 1 n expectation value of the energy operator H in the physical state σk , (7) 4 R 2 2 2 | n=0 n + m R Φ describing the sources, which we will denote by H Φ .Inor- k der to introduce some notation for our subsequent work we start and needs some regularization prescription. It follows that this from the four-dimensional space–time Lagrangian density [18]: part of the result is necessarily ambiguous and cannot be taken too seriously as a candidate to a confinement potential. As an example, N 2 1 m one can see that in the decoupling limit mk R 1, zeta-function L = − kμν k + k k − k 2 F Fμν Aμ ∂μϕ , (2) regularization gives a negative string tension. Since our main mo- 4e2 2e2 k=1 k k tivation is to compute the correction to the static potential for the three-dimensional case, we drop out this term and consider the zero mode only. Thus, it follows that where mk is the mass for the kth scalar field. Following our earlier procedure [20], to compute the interaction energy we need to 2 N q 1 2 carry out the integration over the ϕ-fields. Once this is done, we Vn=0 =− K0 m L . (8) 2π e2 k arrive at the following effective theory for the gauge fields: k=1 k Before switching-on unparticle effects, we introduce a further sim- N 1 1 m2 plification by assuming that the test charges are at distance L > L(3+1) = − k + k kμν Fμν 1 F . (3) eff 2 + 1/mk for any k. As we are looking for long-range forces this is a = e 4 Δ(3 1) k 1 k fairly reasonable choice. In this case, the Bessel function can be ap- proximated with exp(− m2 L) and the static potential (8) may be Next, in order to obtain the corresponding effective Lagrangian k written as density in 2 + 1 dimensions, we compactify one spacelike dimen- N sion by using a sort of Kaluza–Klein approach [35]. It follows that q2 1 −mk L Vn=0 − e . (9) the expression (3) can be rewritten as 2π e2 k=1 k

N ∞ Now we are ready to include unparticle effects in the potential (9). 1 1 m2 LKK = − k + k kμν By following [18,20], we go into the continuum mass spectrum eff Fμν 1 F , (4) e2 4 Δ + + a2 →∞ k=1 n=0 k (2 1) n limit, N , and replace the sum over k by an integral ∞ 2 √ q Ad − − 2 2 2 → = − U dU 2 tL with a ≡ n /R , and R is the compactiﬁcation radius. In the limit Vn=0 V U − t e dt, (10) n 2πe2 Λ2dU 2 R →∞ the difference between nearby energy levels vanish and U 0 the (3 + 1)-dimensional continuum spectrum is recovered. In the − where t = m2, ρ(t ) ≡ tdU 2 is the spectral density, and A is a opposite case, R → 0, the massive modes becomes more and more k dU normalization factor which is given by heavy and decouple from the physical spectrum. The only surviv- = 5/2 ing mode corresponds to n 0 and describes the dimensionally 16π (dU + 1/2) A ≡ , (11) reduced model in 2 + 1 dimensions. With this at hand, we can dU 2d (2π) U (dU − 1)(2dU ) 338 P. Gaete, E. Spallucci / Physics Letters B 668 (2008) 336–339

where dU is the scale dimension of the unparticle field. We also Thus, in 1+1 dimension we recover the correct confining potential 2 = 2 note here that in Eq. (10) we have assumed e e . A direct com- 2 k = 2πq putation on the t-variable yield V dU 1 = σ L, σ = . (21) U 2 2 − e l q2 √ l 2dU 2 V = − A 2 (2d − 5/2) . (12) From the above result (18) it is meaningful to ask whether a sim- U 2 dU π U 2πe l L ilar thing happens in the case of the Sommerfield model studied

Using (11), we see that V U reads in [26]. The Sommerfield model is a Lagrangian field theory which describes massless spinors interacting with a massive vector bo- q2 1 1 = − son, in 1 + 1 dimensions. The Lagrangian density reads: V U − − 2πe2l (π)2dU 3 (2)2dU 9/2 − 1 m2 + − 2dU 2 ¯ μν 0 μ (dU 1/2)(2dU 5/2) l L = ψ(i∂/ − e/A)ψ − Fμν F + Aμ A , (22) × , (13) 4 2 (dU − 1)(2dU ) L where m0 is the mass for the vector boson Aμ.In[26] authors where we have introduced the scale factor l ≡ 1 .Itmaybe /ΛU study the transition between unparticle behavior at low energy noted that for dU = 1/2, expression (13) reduces to and free particle behavior at high energy from the vantage point q2 π 229/2 of the exactly solvable model (22) mimicking Banks–Zaks model in V U = L. (14) 2πe2l 3l lower dimensions. Here, we are interested to determine the interaction energy in the unparticle phase and the eventual presence of From this, one infers the key role played by the scale dimen- a linear confining term. sion (dU ) in transforming the long-range potential into the con- Following our earlier adaptation of the Stueckelberg procedure fining one. In this way, for the three-dimensional case, unparticles [20], we first restore gauge invariance by means of a suitable com- with dU < 1 would be allowed. However, by unitarity considera- pensating field for the sake of consistency with our gauge-invariant tions the obstruction for unparticles (dU < 1) present in the four- definition of interaction potential. Then, we integrate out both dimensional case is still present whenever we go over into three fermions and compensator field to obtain an effective theory for dimensions. the gauge vector A . Once this is done, we arrive at the following + μ The (1 1)-dimensional case may be studied in the same way effective Lagrangian density: as we did in the (2 + 1)-dimensional counterpart. In such a case, the theory under consideration is given by 1 m2 L =− F 1 + F μν, (23) μν N ∞ 2 4 Δ 1 1 k mk kμν L = − F 1 + F , (15) 2 2 2 2 4 μν + 2 where m = m + e /π . As a consequence, the static potential is = = ek Δ κn,m 0 k 1 n,m 0 given by [24]: where κ2 = n2/R2m2/R2. The situation here is analogous to that − n,m 1 2 q2 e mL encountered in the Schwinger model [24]. This allows us to write V = − . (24) the static potential as 2 m ∞ Again, by considering unparticle as a field with continuously dis- N 2 2 2 2 1 q κn,m − q κn,m V = 1 + 1 − e λL + L , (16) tributed mass, we can write Eq. (24) as e2 2λ λ2 2 λ2 k=1 n,m=0 k N − m2 L q2 e k 2 ≡ 2 + 2 V = − (25) where λ mk κn,m. Since we are interested in estimating the . 2 = 2 long-range correction to the static potential, we will retain only k 1 mk the zero mode contribution in the expression (16). Thus the static potential simplifies to In the same way as was done in the previous case, one finds 2d −3 2 2 U N − m L q l q2 1 e k V = − A (2d − 3)2l (26) V = − . (17) U dU U 0,0 2 2 L 2 e 2 k=1 k m k in agreement with Eq. (18). In this way, both the Proca–Maxwell Following our earlier procedure, we see that the unparticle po- and the Sommerfield model with continuously distributed mass tential corresponding to (17) takes the form leads to the same static potential.

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