Bulk antisymmetric tensor fields in a Randall-Sundrum model
aa r X i v : . [ h e p - t h ] S e p Bulk antisymmetric tensor fields in aRandall-Sundrum model
Biswarup Mukhopadhyaya Regional Centre for Accelerator-based Particle PhysicsHarish-Chandra Research InstituteChhatnag Road, Jhusi, Allahabad - 211 019, India
Somasri Sen Centre for Theoretical PhysicsJamia Milia UniversityNew Delhi 110 025, India
Soumitra SenGupta Department of Theoretical PhysicsIndian Association for the Cultivation of ScienceCalcutta - 700 032, IndiaPACS Nos.: 04.20.Cv, 11.30.Er, 12.10.Gq
Abstract
We consider bulk antisymmetric tensor fields of various ranks in a Randall-Sundrumscenario. We show that, rank-2 onwards, the zero-modes of the projections of thesefields on the (3+1) dimensional visible brane become increasingly weaker as the rankof the tensor increases. All such tensor fields of rank 4 or more are absent from thedynamics in four dimensions. This leaves only the zero-mode graviton to have coupling ∼ /M P with matter, thus explaining why the large-scale behaviour of the universeis governed by gravity only. We have also computed the masses of the heavier modesupto rank-3, and shown that they are relatively less likely to have detectable acceleratorsignals. E-mail: [email protected] Also at Department of Physics, St. Stephen’s College, New Delhi - 110007, India,E-mail: [email protected] E-mail: [email protected] ∼ /M P with all matter, while the massive modes have enhanced couplingthrough the warp factor. It not only accounts for the observed impact of gravity in ouruniverse but also raises hopes for new signals in accelerator experiments [4]. However, thereare various antisymmetric tensor fields which also comprise excitations of a closed string [3],and therefore can be expected to lie in the bulk similarly as gravity. The question we askhere is: can these fields also have observable effects? If not, why are the effects of theirmassless modes less perceptible than the force of gravitation?Bulk fields other than gravitons have been studied earlier in RS scenarios, starting frombulk scalars which have been claimed to be required for stabilisation of the modulus [5].Bulk gauge fields and fermions have been considered, too, with various phenomenologicalimplications [6]. While some of such scenarios are testable in accelerator experiments [7]or observations in the neutrino sector [8], by and large they do not cause any contradictionwith our observations so far.However, the situation with tensor fields of various ranks (higher than 1) is slightlydifferent. For example, as has been already noted, an antisymmetric rank-2 tensor fieldsuch as the Kalb-Ramond excitation [9] can be in the bulk as legitimately as the graviton,and prima facie has similar coupling to matter as gravity. Using a generalised form ofthe Einstein-Cartan action, it has been shown that such a field is equivalent to torsionin spacetime [10], on which the experimental limits are quite severe [11]. This apparentcontradiction has been ameliorated in an earlier work [12] where it has been shown thatthe zero mode of the antisymmetric tensor field gets an additional exponential suppressioncompared to the graviton on the visible brane. This could well be an explanation of whywe see the effect of curvature but not of torsion in the evolution of the universe. Arguments2n this line will however be complete only when we can similarly address the effects ofother, higher rank, antisymmetric fields which occur in the NS-NS or RR sector of closedstring excitations [13]. We address that question in the current study, in the special contextof RS-like models. What we wish to point out as a whole is that the zero mode of anyantisymmetric tensor field undergoes progressive exponential suppression increasing withthe rank of the tensor. Moreover, for most higher rank tensor fields it becomes impossible tohave non-vanishing components on the brane, partly because of the antisymmetric nature ofthe tensor, and partly due to the gauge freedom of these fields, which reduces the availabledegrees of freedom on the brane to zero.In order that a rank-n antisymmetric tensor gauge field X a a ...an can be part of thedynamics, one should be able to write down a rank-(n+1) field strength tensor Y a a ...an +1 = ∂ [ a n +1 X a a ...an ] (1)Since a spacetime of dimension D admits of a maximum rank D for an antisymmetrictensor, one can at most have ( n + 1) = D . Thus any antisymmetric tensor field X can havea maximum rank D −
1, beyond which it will all have either zero components or will becomean auxiliary field with the field strength tensor vanishing identically. Such an auxiliary fieldcan be eliminated via the equations of motion if it has no mass term in the bulk, a featureshared by all antisymmetric tensor excitations of a closed string due to gauge invariance.Now let us consider a 3-brane in an RS-type 5-dimensional anti-de Sitter bulk spacetime,where the extra spatial dimension has been compactified on an S /Z orbifold. There are twobranes at the orbifold fixed points φ = 0 and π , where φ is the angular variable correspondingto the compact dimension. In such a scenario, the 5-dimensional metric can be written as ds = e − σ η µν dx µ dx ν + r c dφ (2)with η µν = ( − , + , + , +), and σ = kr c | φ | . r c is the radius of the compact dimension y ,with y = r c φ . k is on the order of the 5-dimensional Planck mass M . The standard modelfields reside at φ = π while gravity peaks at φ = 0. The dimensional parameters definedabove are related to the 4-dimensional Planck scale M P through the relation M P = M k [1 − e − kr c π ] (3)Clearly, M P , M and k are all of the same order of magnitude. For kr c ≃
12 theexponential factor (frequently referred to as the ‘warp factor’) produces TeV scale massparameters (of the form m = M e − kr c π ) on the visible brane. Thus the hierarchy betweenthe Planck and TeV scales is achieved without fine-tuning.3he closed string modes of excitation pertinent to such a scenario are antisymmetrictensor fields of various ranks, in addition to the graviton. Following the reasoning givenabove, such fields can at most be of rank-4. A rank-5 field has a rank-6 field strength tensorin the kinetic energy term, which, by virtue of its complete antisymmetry, cannot exist in5-dimensions. Thus such a field can be removed using the equations of motion, while fieldsof even higher rank themselves vanish identically.Taking a closer look at the rank-5 field strength tensor Y ABCMN of a rank-4 field, onegets two kinds of terms,namely : Y ABCMN = ∂ [ µ X ναβy ] (4)and Y ABCMN = ∂ [ y X µναβ ] (5)where the Latin indices denote bulk co-ordinates, the Greek indices run over the (3+1)Minkowski co-ordinates and y stands for the compact dimension. The first class of termscan be removed using the gauge freedom δX ABCM = ∂ [ A Λ BCM ] (6)which allows the use of 10 gauge-fixing conditions for an antisymmetric Λ BCM . As a resultone can use X ναβy = 0 (7)The second class of terms do not yield any kinetic energy for X µναβ on the visible brane,and they can thus be removed using the equation of motion. . Thus the rank-4 antisymmetrictensor fields (and of course those of all higher ranks) have no role to play in the four-dimensional world in the RS scenario.Thus all that can matter are the lower rank antisymmetric tensor fields. The case of arank-2 field in the bulk (known as the Kalb-Ramond field) has been already investigated,leading to the rather interesting observation that the warped geometry results in an ad-ditional exponential suppression of its zero mode on the visible brane with respect to thegraviton. This suggests an explanation of why torsion can be imperceptible relative to cur-vature in our four-dimensional universe. In principle, such an auxiliary field can have an interaction term of the form X µναβ B µν B αβ with secondrank antisymmetric tensor fields. If such terms at all exist, they will at most result in quartic self-couplingsof the rank-2 field X MNA , with the corresponding field strength Y MNAB .The action for such a field in 5-dimensions is S = Z d x √− GY MNAB Y MNAB (8)where G is the determinant of the 5-dimensional metric. Using the explicit form of the RSmetric and taking into account the gauge fixing condition X µνy = 0, one obtains S x = Z d x Z dφ [ e σ η µλ η νρ η αγ η βδ Y µναβ Y λργδ + 4 e σ r c η µλ η νρ η αγ ∂ φ X µνα ∂ φ X λργ ] (9)Considering the Kaluza Klein decomposition of the field X, X µνα ( x, φ ) = Σ ∞ n =0 X nµνα ( x ) χ n ( φ ) √ r c (10)an effective action of the following form can be obtained in terms of the projections X nµνα onthe visible brane: S X = Z d x Σ n [ η µλ η νρ η αγ η βδ Y nµναβ Y nλργδ + 4 m n η µλ η νρ η αγ X nµνα X nλργ ] (11)where m n is defined through the relation − r c ddφ ( e σ ddφ χ n ) = m n χ n e σ (12)and χ n satisfies the orthonormality condition Z e σ χ m ( φ ) χ n ( φ ) dφ = δ mn (13)In terms of z n = m n k e σ and f n = e σ χ n , equation (12) can be recast in the form[ z n d dz n + z n ddz n + ( z n − f n = 0 (14)which is a Bessel Equation of order 1.The solution for χ n is given by χ n = e − σ f n = e − σ N n [ J ( z n ) + α n Y ( z n )] (15)where J ( z n ) and Y ( z n ) respectively are Bessel and Neumann functions of order 1. α n and N n are integration constants which can be determined from orthogonality and the continuityconditions at the orbifold fixed points. In addition, the continuity condition for the derivativeof χ n at φ = 0 yields α n = − J ( m n k ) Y ( m n k ) (16)5sing the fact that e kr c π >> m n on the brane is expected to beon the order of TeV scale ( << k ), α n ∼ π ( m n k ) << φ = π gives J ( x n ) = 0 (18)where x n = z n ( π ) = m n k e kr c π . The roots of the above equation determine the masses of thehigher excitation modes. As x n is of order unity, the massive modes lie in the TeV scale.Furthermore, the normalisation condition yields N n = e kr c π √ kr c J ( x n ) (19)and the massive modes can be obtained from the equation χ n ( z n ) = q kr c e σ e kr c π J ( z n ) J ( x n ) (20)The values of the first few massive modes of the rank-3 antisymmetric tensor field arelisted in Table 1, where we have also shown the masses of the graviton as well as the rank-2antisymmetric Kaluza-Klein modes. It can be noticed that the rank-3 field has higher massthan the remaining two at every order, and, while the Kalb-Ramond massive modes canhave some signature at, say, the Large hadron collider (LHC), that of the rank-3 massivetensor field is likely to be more elusive. n m gravn ( T eV ) 1 .
66 3 .
04 4 .
40 5 . m KRn ( T eV ) 2 .
87 5 .
26 7 .
62 9 . m Xn ( T eV ) 4 .
44 7 .
28 10 .
05 12 . Table 1: The masses of a few low-lying modes of the graviton, Kalb-Ramond (KR) andrank-3 antisymmetric tensor (X) fields, for kr c = 12 and k = 10 Gev.
Finally, and most crucially, we examine the massless mode, whose strength on the braneneeds to be compared to that of the graviton and the rank-2 field. The solution for thismode is given by χ = − C kr c e − σ + C (21)6equiring the continuity of dχ dφ at φ = π , one obtains C = 0. The normalisationcondition finally gives χ = q kr c e − kr c π (22)This shows that the zero mode of the rank-3 antisymmetric tensor field is suppressedby an additional exponential factor relative to the corresponding rank-2 field which alreadyhas an exponential suppression compared to the zero mode of the graviton. Using the sameargument as in reference [12], one can translate this result into the coupling of the field Xto matter, and show that the interaction with, say, spin-1/2 fields is suppressed by a factor e − kr c π . Thus the higher order antisymmetric field excitations have progressively insignificantroles to play on the visible brane, with the fields vanishing identically beyond rank 3.In conclusion, the graviton seems to have a unique role among the various closed stringexcitations in a warped geometry. This is because the intensity of its zero mode on the 3-brane leads to coupling ∼ /M P with matter fields, which is consistent with the part playedby gravity (or more precisely the curvature of spacetime) observed in our universe. On thecontrary, while bulk antisymmetric tensor fields upto rank-3 can still have non-vanishing zeromodes in four-dimensional spacetime, their strength is progressively diminished for ranks-2and 3. This may well serve as an explanation of why the evolution of our universe is solelycontrolled by gravitation. In addition, the masses of the higher modes also tend to increasewith rank, making them less and less relevant to accelerator experiments. Acknowledgement:
The work of BM was supported by funding from the Departmentof Atomic Energy, Government of India, through the XIth Five Years Plan. SS and SSGacknowledge the hospitality of the Regional Centre for Accelerator-based Particle Physics,Harish-Chandra Research Institute.
References [1] L. Randall and R. Sundrum, Phys. Rev. Lett. , 3370 (1999); ibid , 83, 4690 (1999).[2] For a general discussion see, for example, Y. A. Kubyshin, arXiv: hep-ph/0111027; V.A.Rubakov, Phys.Usp. ,(2001); C. Csaki, arXiv:hep-ph/0404096.[3] See, for example, String Theory , J.Polchinski, Cambridge University Press., Cambridge,(1998).[4] See, for example, D. K. Ghosh and S. Raychaudhuri, Phys.Lett.
B495 , 114, (2000);John F. Gunion, arXiv: hep-ph/0410379; T. Rizzo, Phys.Lett.
B647
43, (2007), M.7rai et al. , Phys. Rev.
D75 ,095008 (2007); CDF Collaboration (T. Aaltonen et al.),arXiv:0707.2294 [hep-ex].[5] W. D. Goldberger, M. B. Wise, Phys.Rev.Lett. , 4922, (1999) ;A. Dey , D.Maityand S. SenGupta, Phys.Rev. D75
B643 , 348, (2006).[6] H. Davoudiasl, J.L. Hewett and T.G. Rizzo ,Phys.Rev.Lett. ,2080,(2000);Phys.Lett. B473 et al. , Phys.Rev.
D62 ,084025 (2000); R. Kitano, Phys.Lett.
B481 . 39 (2000); J.-P. Lee, Eur. Phys. J.
C34 , 237 (2004); M. Guchait, F. Mahmoudi,K. Sridhar, JHEP , 103 (2007); T. Rizzo, Phys.Lett.
B647
43 (2007).[8] Y.Grossman, M. Neubert, Phys.Lett.
B474 , 361 (2000).[9] M. Kalb and P. Ramond , Phys.Rev. D9 , 2273 (1974).[10] .P. Majumdar , S. SenGupta, Class.Quant.Grav. , L89 ,(1999).[11] R.T. Hammond, Phys.Rev. D52 ,6918 (1995); S.SenGupta and S.Sur, Eur.Phys.Lett. ,601 (2004) and references therein.[12] .B. Mukhopadhyaya, S. Sen and S. SenGupta, Phys.Rev.Lett. , 121101,(2002;Erratum-ibid.89