Neutrino, Photon Interaction in Unparticle Physics
aa r X i v : . [ h e p - ph ] J a n Neutrino, Photon Interaction in Unparticle Physics
Sukanta Dutta ∗ and Ashok Goyal † Department of Physics and Electronics, SGTB Khalsa College,University of Delhi. Delhi-110007. India. Department of Physics and Astrophysics,University of Delhi. Delhi-110007. India.
Abstract
We investigate the impact of unparticle physics on the annihilation of relic neutrinos with theneutrinos identified as primary source of ultra high energy (UHE) cosmic ray events, producing acascade of photons and charged particles. We compute the contribution of the unparticle exchange tothe cross-sections ν ¯ ν → γ γ and ν ¯ ν → f ¯ f scattering. We estimate the neutrino photon decouplingtemperature from the reaction rate of ν ¯ ν → γ γ . We find that inclusion of unparticles can in factaccount for the flux of UHE cosmic rays and can also result in the lowering of neutrino - photondecoupling temperature below the QCD phase transition for unparticle physics parameters in a certainrange. We calculate the mean free path of these high energy neutrinos annihilating themselves withthe relic neutrinos to produce vector and tensor unparticles. PACS numbers: 14.80.-j, 13.15+g, 13.85.Tp, 98.70.SaKeywords: unparticle, neutrino-photon, ultra high energy, cosmic rays ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION High energy neutrino interactions are of great interest in astrophysics, cosmology and inhigh energy cosmic ray physics. Neutrinos have been considered as possible candidates of UltraHigh energy (UHE) cosmic rays as opposed to protons and photons on account of their abilityto travel cosmic galactic distances without significantly degrading their energy. In comparisonto protons and photons, neutrinos have relatively weaker interaction cross - section with relicneutrinos and CMBR. By the same token they present difficulty in initiating air showers. Weiler[1] proposed that if their energy could correspond to Z − resonance i.e. , E ν ≈ m Z m ν ≃ eV (1)they would have significant annihilation cross -section with relic neutrinos of mass consistentwith the oscillation data. The difficulty in realizing this scenario is to identify the source ofUHE neutrinos with their energy close to Z − resonance. Decay of super heavy relic particles M χ ≥ GeV has been proposed [2] to be the source of these highly energetic neutrinos thatcan explain cosmic ray events above the GZK cut off. There however, remains the problem ofconfining the production of these neutrinos in a spherical shell at red shift Z = (cid:20) M χ − (cid:21) = (cid:20) M χ m ν m − (cid:21) (2)so that this energy near earth is close to Z − resonance energy for the ν ¯ ν annihilation crosssection on relic neutrinos to be large. If such a scenario is not realized in nature, we wouldrequire large neutrino hadron cross section in the milli barn (mb) region for the neutrinosto initiate showers high in the atmosphere. Current estimate of UHE neutrino - hadron crosssection in the standard Model (SM) by Gandhi et. al. [3] put the cross section in the 10 − − − mb range for E ν ∼ eV.We thus require anomalously large high energy neutrino interactions. In this context theoriesof n − extra dimensions with large compactification radius and TeV scale gravity [4] provide thepossibility of enhancing neutrino interactions through the exchange of a tower of massive spin2 bulk gravitons (Kaluza Klien excitations). Contribution of Kaluza Klien excitations and itsimpact on UHE cosmic ray physics has been discussed by several authors [5] in the literature.The high energy neutrino photon interactions are also of great interest in astrophysics andcosmology. Scattered photons on neutrinos through γν → γν are predominantly circularlypolarized due to the left handed nature of neutrinos [6]. In the early universe, the photons andneutrinos decouple i.e. the process ν ¯ ν → γγ ceases to occur at a temperature T ∼ . γγ → ν ¯ ν has been calculated in the SM [8] and shows a s behaviorupto W ± pair production threshold beyond which it starts falling. The cross section is givenby σ (cid:0) ν ¯ ν → γγ (cid:1) = s π (cid:18) A g α em π m W (cid:19) where A = 14 . . (3)The contribution of Kaluza Klien excitations to these processes has been computed by Dicuset. al. [9], who have shown that the contribution is not large enough to allow high energyneutrinos to scatter from relic neutrinos through ν ¯ ν → γγ but photon neutrino decouplingtemperature may in fact be lowered.Recently Georgi [10, 11] has proposed that scale invariance, which has been a powerful toolin physics, may indeed exist at a scale much above the TeV scale. He argued that a scaleinvariant sector with non trivial infrared fixed point which couples to SM may appear. Atlow energy scale this gives rise to what has been called an unparticle operator O U with a nonintegral scale dimension d U having a mass spectrum which looks like a d U number of masslessparticles. The unparticles have continuous mass spectrum. The unparticle operators can havedifferent Lorentz structures and couple to the SM fields below a large mass scale through aneffective non- re-normalizable Lagrangian L eff . = O SM O U M d U + d SM − (4)where Λ U is energy scale of the order of 1 TeV. It is related to high mass scale M U through κ = C U (cid:18) Λ U M U (cid:19) d BZ + d SM − (5)where C U is the dimensionless coupling constant, d SM is dimension of SM operator, and d BZ is the dimension of Banks-Zaks ( BZ ) scale invariant [12] sector interacting with SM fieldsthrough the exchange of high mass particles M U induced by the Lagrangian given in equation(4). Unparticle operators with different Lorentz structure corresponding to scalar, vector, tensorand spinor operators have been considered in the literature.This unparticle sector can arise as stated in [10] from the hidden sector or from stronglyinteracting magnetic phase of a specific class of supersymmetric theories [13] or from hiddenvalleys model [14]. However, we also note that under a specfic conformal invariance [15] thepropagators for vector and tensor are modified.3n this paper we study the attenuation of high energy neutrinos through interaction withthe present density of relic neutrinos through ν ¯ ν → U , ν ¯ ν → γ γ and ν ¯ ν → f ¯ f processes.The last two processes will proceed through the exchange of U unparticles and would directlyproduce a cascade of high energy photons and hadrons which could account for the flux of UHEcosmic rays. We also estimate the photon neutrino decoupling temperature. In section II wegive the unparticle interactions with the SM fields and calculate the cross sections for the aboveprocesses. In section III we give an estimate of neutrino - photon decoupling temperature andmean free path of neutrinos through intergalactic journey. This is followed by the discussion ofour results in section IV. II. NEUTRINO ANTINEUTRINO ANNIHILATION
The effective interactions consistent with SM gauge symmetry for the vector and tensorunparticles with SM fields are given by κ V Λ d U − U ¯ f γ µ f O U µ ; and κ A Λ d U − U ¯ f γ µ γ f O U µ (6)and for tensor unparticles the interactions are − i κ T Λ d U U ¯ f (cid:0) γ µ ↔ D ν + γ ν ↔ D µ (cid:1) ψ f O µν U and κ T Λ d U U F µα F αν O U µν (7)where f stands for a SM fermion doublet or singlet, F µν is electromagnetic field tensor andthe dimensionless coupling constants κ i ’s are related to the coupling constant C U and the massscale M U through κ V , A Λ d U − U = C V , AU Λ − d U U M U and κ T Λ d U U = C U T Λ − d U U M U . (8)The neutrinos being left handed the scalar operator does not couple to them and therefore weconsider only the vector and tensor operators. The unparticle propagator for the vector andtensor fields are given by [10, 16] (cid:2) A F ( P ) (cid:3) µν = A d U (cid:0) d U π (cid:1) (cid:0) − P (cid:1) d U − π µ ν ( P ) where π µ ν ( P ) = − g µν + P µ P ν P ; (9) (cid:2) A F ( P ) (cid:3) µν, ρσ = A d U (cid:0) d U π (cid:1) (cid:0) − P (cid:1) d U − T µ ν, ρσ ( P ) where T µ ν, ρσ ( P ) = 12 (cid:20) π µ ρ ( P ) π ν σ ( P ) + π µ σ ( P ) π ν ρ ( P ) − π µ ν ( P ) π ρ σ ( P ) (cid:21) . (10)They satisfy the conditions P µ π µν ( P ) = 0 and P µ T µν, ρσ ( P ) = 0. Further the unparticleoperator O U and O U µν are taken to be Hermitian and transverse and the tensor unparticle4 ¯ ν → γ γ Λ U = 1 TeV d U = 1 . d U = 1 . d U = 1 . √ s in GeV σ i n f b − − − − − − − − − − FIG. 1: SM + Unparticle contribution to the total cross section in fb for the photon pair productionfrom neutrino pair annihilation via tensor unparticle for √ s varying from 0.1 GeV - 160 GeV operator is also traceless. A d U is the normalization factor for the two point unparticle operatorand is given by A d U = 16 π / (2 π ) d U Γ (cid:0) d U + (cid:1) Γ ( d U −
1) Γ (2 d U ) (11)The spin averaged cross section induced by the vector and tensor unparticle operators for theprocess ν (cid:0) p (cid:1) + ¯ ν (cid:0) p (cid:1) → U (12)are calculated to be σ V av . (cid:0) ν ¯ ν → U (cid:1) = (cid:12)(cid:12) κ V (cid:12)(cid:12) (cid:18) s Λ U (cid:19) d U − A d U s (13) σ T av . (cid:0) ν ¯ ν → U (cid:1) = 132 Λ U (cid:12)(cid:12) κ T (cid:12)(cid:12) (cid:18) s Λ U (cid:19) d U − A d U (14)For the scattering process ν (cid:0) p (cid:1) + ¯ ν (cid:0) p (cid:1) → γ (cid:0) k (cid:1) + γ (cid:0) k (cid:1) (15)we only have the contribution from tensor unparticle operator. The SM contribution to theabove process has been given in reference [9], we can easily compute the total contribution tothe spin averaged cross-section and we get 5 ¯ ν → f ¯ f Λ U = 1 TeV d U = 1 . d U = 1 . d U = 1 . d U = 1 . d U = 1 . √ s in TeV σ i n f b − − − − − FIG. 2: SM + vector unparticle + tensor unparticle contribution to the total Cross Section in fb forthe charged fermion pair production from neutrino pair annihilation for √ s varying from 0.1 GeV - 1TeV σ av . (cid:0) ν ¯ ν → γ γ (cid:1) = 120 π h A s + 2 A SM A Unp . cos (cid:0) π ( d U − (cid:1) s d U +1 + A . s d U − i (16)where A SM = (cid:20) . g W α em π m W (cid:21) ; A Unp . = " (cid:12)(cid:12) κ T (cid:12)(cid:12) Z d U d U U and Z d U = A d U (cid:0) d U π (cid:1) (17)The cross section for ν (cid:0) p (cid:1) + ¯ ν (cid:0) p (cid:1) → f (cid:0) p ′ (cid:1) + ¯ f (cid:0) p ′ (cid:1) (18)get contribution from both the vector and tensor unparticles operators and we get σ av . (cid:0) ν ¯ ν → f ¯ f (cid:1) = 2 G F s π (cid:12)(cid:12) R ( s ) (cid:12)(cid:12) "(cid:0) C f V + C f A (cid:1) + 2 cos (cid:0) π d U + Φ (cid:1) C f V B VU + (cid:0) B VU (cid:1) + (cid:0) B TU (cid:1) where B VU = " (cid:12)(cid:12) κ V (cid:12)(cid:12) Z d U √ (cid:12)(cid:12) R ( s ) (cid:12)(cid:12) Λ − G F (cid:19) (cid:16) s Λ (cid:17) d U − ; Φ = tan − (cid:20) − m Z Γ Z s − m z (cid:21) ; R ( s ) = m z ( s − m Z ) − i m Z Γ Z ( s − m Z ) + m z Γ Z ; B TU = " √ (cid:12)(cid:12) κ T (cid:12)(cid:12) Z d U √ (cid:12)(cid:12) R ( s ) (cid:12)(cid:12) Λ − G F (cid:19) (cid:16) s Λ (cid:17) d U − C f V = T f − Q f sin θ W ; and C f A = T f . (19)In equation (19) the second term within the square bracket is the interference term involvingthe SM vector and axial current with the vector unparticle current. The interference term ofthe vector and the tensor contribution vanishes identically after angular integration.6or a ν of energy E ν annihilating a relic neutrino of mass m ν we get σ V av . (cid:0) ν ¯ ν → U (cid:1) ≃ (cid:12)(cid:12) κ V (cid:12)(cid:12) A d U − d U (cid:18) m ν E ν eV (cid:19) d U − (cid:18) Λ U (cid:19) − d U pb ; (20) σ T av . (cid:0) ν ¯ ν → U (cid:1) ≃ (cid:12)(cid:12) κ T (cid:12)(cid:12) A d U − d U (cid:18) m ν E ν eV (cid:19) d U − (cid:18) Λ U (cid:19) − d U pb ; (21) σ av . (cid:0) ν ¯ ν → γ γ (cid:1) ≃ . × − (cid:18) m ν E ν eV (cid:19) × " . × (cid:12)(cid:12) κ T (cid:12)(cid:12) Z d U (cid:18) m ν E ν eV (cid:19) d U − (cid:18) Λ U (cid:19) − d U +5 . × − d U ) (cid:12)(cid:12) κ T (cid:12)(cid:12) Z d U (cid:18) m ν E ν eV (cid:19) d U − (cid:18) Λ U (cid:19) − d U pb . (22) σ av . (cid:0) ν ¯ ν → f ¯ f (cid:1) ≃ . × (cid:18) m ν E ν eV (cid:19) "(cid:0) C f V + C f A (cid:1) |D ( m ν , E ν ) | + 6 . × − d U C f V (cid:12)(cid:12) κ V (cid:12)(cid:12) Z d U (cid:18) m ν E ν eV (cid:19) d U − (cid:18) Λ1 TeV (cid:19) − d U × |D ( m ν , E ν ) | cos { d U π + Θ } + 9 . × − d U (cid:12)(cid:12) κ V (cid:12)(cid:12) Z d U (cid:18) m ν E ν eV (cid:19) d U − (cid:18) Λ1 TeV (cid:19) − d U + 1 . × − − d U (cid:12)(cid:12) κ T (cid:12)(cid:12) Z d U (cid:18) m ν E ν eV (cid:19) d U − (cid:18) Λ1 TeV (cid:19) − d U where D ( m ν , E ν ) = (cid:20) . × − (cid:18) m ν E ν eV (cid:19) + 0 . i (cid:21) − and Θ = Arg . (cid:2) D ( m ν , E ν ) (cid:3) . (23)The cross-section given in equations (22) and (23) for Λ U = 1 TeV are depicted in Figures 1and 2 respectively. III. ν MEAN FREE PATH AND ν γ
DECOUPLING TEMPERATURE
From the expression of the cross sections given above, we find that the mean free path ofthe neutrinos in their intergalactic journey is dominated by the annihilation of UHE neutrinoson relic neutrinos ( relic density n ν ≃
56 cm − ) through the production of unparticles. In the7bsence of any leptonic asymmetry, we have λ Vν ¯ ν →U = (cid:20) σ V (cid:0) ν ¯ ν → U (cid:1) n ν (cid:21) − ≃ . × d U +1 (cid:12)(cid:12) κ V (cid:12)(cid:12) − A d U (cid:20) m ν E ν eV (cid:21) − d U (cid:20) Λ U (cid:21) d U − Mega Parsec . (24)and λ Tν ¯ ν →U ≃ . × d U +6 (cid:12)(cid:12) κ T (cid:12)(cid:12) − A d U (cid:20) m ν E ν eV (cid:21) − d U (cid:20) Λ U (cid:21) d U Mega Parsec . (25)The mean free path calculated from the production of the vector unparticle increases withthe with d U and are much less than the tensor unparticle production as the vector unparticlecross-section dominates over the tensor one. In table III we present the mean free paths forvarious values of d U . The corresponding value in the SM, namely λ ν ¯ ν → Z ⋆ ≃ . × MegaParsec. At resonance √ s = m Z , the Z exchange gives the largest cross-section with mean freepath λ ν ¯ ν → Z ⋆ ≃ . × Mega Parsec. d U λ V in (cid:12)(cid:12) κ V (cid:12)(cid:12) − M pc. 1.28 × × × × × TABLE I: Mean free Path of UHE ν due to annihilation with relic ν to produce vector unparticle. The temperature at which the reaction ν ¯ ν → γγ ceases to occur can be obtained from thereaction rate per unit volume i.e.Γ = 1 (cid:0) π (cid:1) " Y i =1 Z d ~p i e E i /T + 1 σ av . (cid:12)(cid:12) ~v (cid:12)(cid:12) (26)Substituting the cross sections from equation (22)Γ = 1640 π " A T F (5) + 2 A SM A Unp . cos (cid:0) π ( d U − (cid:1) d U +3 d U + 3 T d U +8 F ( d U +3 ) + 4 d U +1 d U + 1 A . T d U +4 F (2 d U +1 ) where F ( n ) = Z x n e x + 1 dx = ζ ( n + 1) Γ( n + 1) (cid:18) − n (cid:19) (27)The interaction rate R is obtained by dividing the reaction rate Γ by the neutrino numberdensity at temperature T namely n ν = 3 ζ (cid:0) (cid:1) T π . (28)8 x. Dim. d U = 1 . d U = 1 . d U = 1 . d U = 1 . d U = 1 . d U = 1 in TeV T i n M e V − FIG. 3: Contours depicting the decoupling Temperature T in MeV as a function of Λ U for a fixeddimensionless coupling (cid:12)(cid:12) κ i (cid:12)(cid:12) = 1 and various values of d U varying between 1 and 2. Thus the reaction rate R is given as R ν ¯ ν → γγ = 1 . × − (cid:18) T (cid:19) ⊗ " − d U d U cos (cid:0) π ( d U − (cid:1) d U + 3 (cid:12)(cid:12) κ T (cid:12)(cid:12) Z d U F ( d U +3 ) (cid:18) T (cid:19) d U − (cid:18) Λ U (cid:19) − d U +6 . × − d U d U (cid:12)(cid:12) κ T (cid:12)(cid:12) Z d U (2 d U + 1) F (2 d U +1 ) (cid:18) T (cid:19) d U − (cid:18) Λ U (cid:19) − d U sec − (29)Using the relation between the age of the universe and the temperature during this eranamely t = 2 . (cid:20) T (cid:21) − sec , (30)9he condition that at least one interaction takes place gives R ν ¯ ν → γγ × t = 1. The solution ofthis equation gives the decoupling temperature and is shown in Fig. 3 IV. RESULTS AND DISCUSSION
In order to access the importance of the contribution of the unparticles to ultra high energyneutrino annihilating on cosmic neutrino background, we have plotted the average cross-sections σ av (cid:0) ν ¯ ν → γ γ (cid:1) and σ av (cid:0) ν ¯ ν → f ¯ f (cid:1) in Figures 1 and 2. These cross sections for the highestenergy neutrinos have to be large in the vicinity of µ barns if the high energy photons andcharged fermion pairs produced in these reactions which can then fragments into protons, haveto account for the flux of the ultra high energy cosmic rays. From these figures we observethat the neutrino annihilation into fermion pairs can indeed be large and may even surpassthe cross section at Z resonance which has been considered in the literature [1] as a possiblemechanism to explain UHE cosmic ray flux for 10 eV events. For the highest energy neutrinos E ν ≈ − eV and neutrinos of the super Kamiokande motivated mass ≃ − eV, we findthe cross-section σ av (cid:0) ν ¯ ν → f ¯ f (cid:1) to vary between 10 fb to 10 fb for the unparticle dimension d U varying from 1.1 to 1.9 respectively. This cross cross-section is enough for at least one scatteringto occur i. e , they satisfy the condition σ av (cid:0) ν ¯ ν → f ¯ f (cid:1) n ν c t = 1 for n ν c t ≃ cm − where n ν is the relic density (56 cm − ) and t = 13 . × years, the age of the universe. Thecross section for the production of a cascade of γ rays through σ av (cid:0) ν ¯ ν → γ γ (cid:1) is not enough toaccount for the UHE cosmic ray events. However, from Figure 3 we see that this process can stillgive enough contribution to significantly lower the neutrino - photon decoupling temperature.In conclusion, the present study show that unparticle physics can keep alive the hope ofidentifying UHE cosmc ray events with the highest energy neutrinos and the possibility oflowering the neutrino - photon decoupling temperature below the QCD phase transition albeitfor low unparticle operator dimensions and coupling of the order one for Λ U ≈ [1] T. J. Weiler, Astropart. Phys. , 303 (1999).[2] G. Gelmini and A. Kusenko, Phys. Rev. Lett. , 1378 (2000).[3] R. Gandhi, C. Quigg, M. H. Reno and I. Sarcevic, Astropart. Phys. , 81 (1996); ibid, Phys. Rev.D , 093009 (1998)[4] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B , 263 (1998); ibid, Phys.Rev. D , 086004 (1999); I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, hys. Lett. B , 257 (1998).[5] S. Nussinov and R. Shrock, Phys. Rev. D , 105002 (1999); P. Jain, D. W. McKay, S. Pandaand J. P. Ralston, Phys. Lett. B , 267 (2000); A. Goyal, A. Gupta and N. Mahajan, Phys.Rev. D , 043003 (2001).[6] D. A. Dicus and W. W. Repko, Phys. Rev. D , 5106 (1993); V. K. Cung and M. Yoshimura,Nuovo Cim. A , 557 (1975).[7] C. N. Yang, Phys. Rev. , 242 (1950).[8] A. Abbasabadi, A. Devoto, D. A. Dicus and W. W. Repko, Phys. Rev. D , 013012 (1999);D. A. Dicus, C. Kao and W. W. Repko, Phys. Rev. D , 013005 (1999).[9] D. A. Dicus, K. Kovner and W. W. Repko, Phys. Rev. D , 053013 (2000).[10] H. Georgi, Phys. Rev. Lett. , 221601 (2007).[11] H. Georgi, Phys. Lett. B , 275 (2007).[12] T. Banks and A. Zaks, Nuc. Phys. B , 189 (1982).[13] P. J. Fox, A. Rajaraman and Y. Shirman, Phys. Rev. D , 075004 (2007).[14] M. J. Strassler, hep-ph/0801.0629, (2008).[15] B. Grinstein, K. Intriligator and I. Z. Rothstein, hep-ph/0801.1140 (2008).[16] K. Cheung, W. Y. Keung and T. C. Yuan, Phys. Rev. D , 075015, (2007)., 075015, (2007).