Plasmon Annihilation into Kaluza-Klein Graviton: New Astrophysical Constraints on Large Extra Dimensions
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Plasmon Annihilation into Kaluza-Klein Graviton: NewAstrophysical Constraints on Large Extra Dimensions
Prasanta Kumar Das, ∗ V H Satheeshkumar,
2, 3, † and P. K. Suresh ‡ Birla Institute of Technology and Science-Pilani,Goa Campus, NH-17B, Zuarinagar, Goa 403726, India. Department of Physics, Sri Bhagawan Mahaveer Jain College of Engineering,Jain Global Campus, Kanakapura Road, Bangalore 562 112, India. School of Physics, University of Hyderabad,Central University P.O., Gachibowli, Hyderabad 500 046, India. (Dated: November 28, 2018)
Abstract
In large extra dimensional Kaluza-Klein (KK) scenario, where the usual Standard Model (SM)matter is confined to a 3+ 1-dimensional hypersurface called the 3-brane and gravity can propagateto the bulk ( D = 4 + d , d being the number of extra spatial dimensions), the light graviton KKmodes can be produced inside the supernova core due to the usual nucleon-nucleon bremstrahlung,electron-positron and photon-photon annihilations. This photon inside the supernova becomesplasmon due to the plasma effect. In this paper, we study the energy-loss rate of SN 1987A due tothe KK gravitons produced from the plasmon-plasmon annihilation. We find that the SN 1987Acooling rate leads to the conservative bound M D > . .
38 TeV for the case of two andthree space-like extra dimensions. ∗ [email protected] † [email protected] ‡ [email protected] . INTRODUCTION Recently it has been noted that the scale of quantum gravity, the four dimensional Planckscale M P l ( ∼ TeV), is just a conjecture without much experimental support and the onlyexperimentally verified scale of gauge interactions in four dimensions lies within the TeVscale. Therefore, the assumptions that gravitation becomes strong at the TeV scale, whilethe standard gauge interactions remain confined to the four dimensional spacetime, doesnot conflict with the today’s experimental data These ideas solve the hierarchy problemwithout relying on supersymmetry or technicolour and the observed weakness of gravity atlong distances is due to the presence of d new spatial dimensions large compared to theelectroweak scale. This can be inferred from the relation between the Planck scales of the D = 4 + d dimensional theory M D and the four dimensional theory M P l , which, for thetoroidal compactification, is given by M P l = (2 πR ) d M d +2 D , (1)where R is the size of the extra dimensions. Putting M D ∼ R ∼ d − cm . (2)For d = 1, R ∼ cm, this case is obviously excluded since it would modify Newtoniangravitation at solar-system distances. For d = 2, we get R ∼ ∼
100 GeV implies that the SM fields can not feel these extralarge dimensions, that is they are confined to only “3-brane”, in the higher dimensionalspacetime called “bulk”. Summarizing, in this framework the universe is D = 4 + d dimen-sional with Planck scale near the weak scale, with d ≥ M D . Here we have investigated this possibility. This would be similar to Farzan’s treat-ment of the Majoron emission in the supernova cooling process as a source of the upperbound on neutrino-Majoron coupling [7] and Raffelt’s treatment on axion emission in pho-ton photon collision [8]. Several other mechanism for the SN 1987A cooling comprising theNew Physics(beyond the Standard Model Physics) are already available in the literatute.Recently Das [9] and others (see [9] for other works) have explored the unparticle physics asa possible cooling mechanism of the supernovae SN 1987A, in which an unparticle stuff canbe produced in the nucleon-nucleon bremstrahlung, electron-positron and photon-photonannihilations and thus cools down the temparature of SN 1987A. II. SUPERNOVA EXPLOSION AND COOLING
Supernovae come in two main observational varieties: Type II are those whose opticalspectra exhibit Hydrogen lines and have less sharp peaks at maxima (of 1 billion solarluminosities), whereas the optical spectra for the Type I supernovae does not have anyHydrogen lines and it exhibits sharp maxima [10]. Physically, there are two fundamentaltypes of supernovae, based on what mechanism powers them: the thermonuclear supernovaeand the core-collapse ones. Only supernovae Ia are thermonuclear type and the rest areformed by core-collapse of a massive star. The core-collapse supernovae are the class ofexplosions which mark the evolutionary end of massive stars ( M ≥ M ⊙ ). The kineticenergy of the explosion carries about 1% of the liberated gravitational binding energy ofabout 3 × ergs and the remaining 99% going into neutrinos. This powerful and detectableneutrino burst is the main astro-particle interest of core-collapse supernovae.In the case of SN 1987A, about 10 ergs of gravitational binding energy was released infew seconds and the neutrino fluxes were measured by Kamiokande [11] and IMB [12] collab-orations. Numerical neutrino light curves can be compared with the SN 1987A data wherethe measured energies are found to be “too low”. For example, the numerical simulation in[13] yields time-integrated values h E ν e i ≈
13 MeV, h E ¯ ν e i ≈
16 MeV, and h E ν x i ≈
23 MeV.On the other hand, the data imply h E ¯ ν e i = 7 . h E ¯ ν e i <
12 MeV. Flavoroscillations would increase the expected energies and thus enhance the discrepancy [14]. Ithas remained unclear if these and other anomalies of the SN 1987A neutrino signal shouldbe blamed on small-number statistics, or point to a serious problem with the SN models orthe detectors, or is there a new physics happening in supernovae?Since we have these measurements already at our disposal, now if we propose somenovel channel through which the core of the supernova can lose energy, the luminosity inthis channel should be low enough to preserve the agreement of neutrino observations withtheory. That is, L new channel ≤ ergs s − . This idea was earlier used to put the strongestexperimental upper bounds on the axion mass [15]. Here, we will consider the gravitonswhich can carry the energy from the core of the supernovae and escape into the bulk ofthe larger dimensional space. The constraint on luminosity of this process can be convertedinto a bound on the 4+d dimensional Planck scale M D . Any mechanism which leads tosignificant energy-loss from the supernovae core immediately after bounce will produce avery different neutrino-pulse shape, and so will destroy this agreement, which in the caseof axion is explicitly shown by Burrows’s et al. [18]. Raffelt has proposed a simple analyticcriterion based on detailed supernova simulations [19]: if any energy-loss mechanism hasan emissivity greater than 10 ergs g − s − then it will remove sufficient energy from theexplosion to invalidate the current understanding of Type-II supernovae’s neutrino signal.Similar arguments can be applied to other particles. The hypothetical majorons are onecase in point [20]. III. CONSTRAINTS ON EXTRA DIMENSIONS
The most restrictive limits on M D come from SN 1987A energy-loss argument. If largeextra dimensions exist, the usual four dimensional graviton is complemented by a tower ofKaluza-Klein (KK) states, corresponding to new phase space in the bulk. The KK gravitonsinteract with the strength of ordinary gravitons and thus are not trapped in the supernovaecore. During the first few seconds after collapse, the core contains neutrons, protons, elec-trons, neutrinos and thermal photons(plasmons). There are a number of processes in whichhigher-dimensional gravitons can be produced. For the conditions that pertain in the coreat this time (temperatures T ∼ −
70 MeV, densities ρ ∼ (3 − × g cm − ), the4elevant processes are shown below • Graviton( G ) production in Nucleon-Nucleon Brehmstrahlung: N + N → N + N + G• Graviton production in photon fusion: γ + γ → G• Graviton production in electron-positron annihilation process: e − e + → G In the supernovae, nucleon and photon abundances are comparable (actually nucleonsare somewhat more abundant). In the following we present the bounds derived by variousauthors using nucleon-nucleon bremhmstrahlung and in the next section we give detailedcalculation for photon-photon annihilation (including the plasma effect inside supernovae)to KK graviton process. We believe that although the dominant contribution will still followfrom nucleon-nucleon bremsstrahlung, however, because of the large uncertainties involvedin such a process calculation inside the hot plasma, the reliable bound will follow fromplasmon + plasmon → KK graviton process. It is worthwhile to mention here that in thiswork we have not considered the effect of plasmon width in the final continuum KK stateproduction, which we believe if be taken into account will not substantiably change ourbound on M D . We will not discuss the electron-positron annihilation to KK graviton as itdoes not give any significant bounds. A. Nucleon-Nucleon Brehmstrahlung
This is the dominant process relevant for the SN 1987A where the temperature is com-parable to m π and so the strong interaction between N’s is unsuppressed. This process canbe represented as N + N → N + N + G , (3)where N can be a neutron or a proton and G is a higher-dimensional graviton.The main uncertainty comes from the lack of precise knowledge of temperatures in thecore: values quoted in the literature range from 30 MeV to 70 MeV. For T = 30 MeV and ρ = 3 × g cm − , we list the results of various authors.Cullen and Perelstein [3] M D > ∼
50 TeV , d = 2; (4)5 D > ∼ , d = 3; (5) M D > ∼ , d = 4 . (6)Barger, Han, Kao and Zhang [4] M D > ∼
51 TeV , d = 2; (7) M D > ∼ . , d = 3 . (8)Hannestad and Raffelt [5] M D > ∼
84 TeV , d = 2; (9) M D > ∼ , d = 3 . (10) IV. METHODOLOGY OF CALCULATION
Each KK graviton state couples to the SM field with the 4-dimensional gravitationalstrength according to [21] L = − κ X ~n Z d x h µν,~n T µν , (11)where the summation is over all KK states labeled by the level ~n . Here κ = √ πG N and G N = 1 /M P l , the 4-dimensional Newton’s constant. T µν is the energy-momentum tensor ofthe SM and h µν,~n the KK state.Since for large R the KK gravitons are very light (because m ~n ∼ /R ), they may becopiously produced in high energy processes. For real emission of the KK gravitons fromthe collision of SM fields, the total cross-section can be written as σ tot = κ X ~n σ ( ~n ) , (12)where the dependence on the gravitational coupling is factored out. Because the massseparation of adjacent KK states, O (1 /R ), is usually much smaller than typical energies ina physical process, we can approximate the summation by an integration according to X ~n → Z ρ ( m ~n ) d ( m ~n ) , (13)where the density of KK states ρ ( m ~n ) = M Pl M dD d π d/ Γ( d/ ( m ~n ) ( d − / . Here we have used therelation M P l = (2 πR ) d M dD . 6ow for a generic 2 → N body scattering, the scattering cross section is given by σ = 1 F lux
Z Y f d p f (2 π ) E f (2 π ) δ p + p − X f p f |M fi | (14)where F lux = 4 E E υ rel . Here E , E are the energies of the initial particles 1 and 2 whosemasses are m and m , respectively and υ rel is the relative velocity between them.For a general reaction of the kind a + b → c , the above expression takes the form σ = 1 F lux |M fi | πδ ( S − m c ) . (15)In the center-of-mass frame, we use the notation √ S for the total initial energy, √ S = E + E (16) F lux = 4 E E υ rel = 4 | p |√ S, (17)where | p | = | p | = | p | = λ / ( S,m ,m )2 √ S and E and E are the energies of the particles a and b . The function λ ( x, y, z )(= x + y + z − xy − yz − zx ), is the standard K ¨ a llen function.Since we are concerned with the energy loss to gravitons escaping into the extra dimen-sions, it is convenient and standard [19, 22] to define the quantities ˙ ǫ a + b → c which are therate at which energy is lost to gravitons via the process a + b → c where c has a decay width,per unit time per unit mass of the stellar object. In terms of the cross-section σ a + b → c thenumber densities n a,b for a,b and the mass density ρ , ˙ ǫ is given by˙ ǫ a + b → c. = h n a n b σ ( a + b → c ) v rel E cm i ρ (18)where the brackets indicate thermal averaging and E cm (= E a + E b ) is the center-of-mass(c.o.m) energy of the two colliding particles a and b . Note that in the present casethe final state KK graviton, although has smaller decay width but is stable over the size ofthe neutron star because of it’s large life time ∼ (100 M eV /m ) yr (See [21]) and thus itcan escape the supernovae while allowing it to cool. V. GRAVITON PRODUCTION IN PLASMON FUSION
Photons are quite abundant in supernovae. Due to plasma effect inside the supernovae,photons becomes effectively massive. These massive photons(of mass m A ,say) are known asplasmons. Our interest is in the plasmon-plasmon annihilation to KK graviton i.e. γ P ( k ) + γ P ( k ) → KK ( p ) . (19)7he plasmon-plasmon-graviton ( G nµν ( q ) A mα ( k ) A n − mβ ( k )) vertex [21] is given by X µναβ = − iκ (cid:20) ( m A + k .k ) C µν,ρσ + D µν,ρσ ( k , k ) + ξ − E µν,ρσ ( k , k ) (cid:21) , (20)where the symbols C µν,ρσ , D µν,ρσ ( k , k ) , E µν,ρσ ( k , k ) are defined as C µν,ρσ = η µρ η νσ + η µσ η νρ − η µν η ρσ ,D µν,ρσ ( k , k ) = η µν k σ k ρ − (cid:20) η µσ k ν k ρ + η µρ k σ k ν − η ρσ k µ k ν + ( µ ↔ ν ) (cid:21) ,E µν,ρσ ( k , k ) = η µν ( k ρ k σ + k ρ k σ + k ρ k σ ) − [ η νσ k µ k ρ + η νρ k µ k σ + ( µ ↔ ν )] . Here we work in the unitary gauge( ξ → ∞ ). In the c.o.m frame, the momentum vectors forthis reactions are k µ = ( E , , , k ) , (21) k µ = ( E , , , − k ) , (22) p µ = ( E G , , , . (23)It often turns out to be more convenient to keep the polarizations explicitly. The polarizationvectors [21] of a massive graviton are e ± µν = 2 ǫ ± µ ǫ ± ν ,e ± µν = √ ǫ ± µ ǫ ν + ǫ µ ǫ ± ν ) ,e µν = s
23 ( ǫ + µ ǫ − ν + ǫ − µ ǫ + ν − ǫ µ ǫ ν ) . Here ǫ ± µ and ǫ µ are the transverse and longitudinal polarization vectors of a massive gaugeboson. For a massive vector boson( e.g. plasmon) with momentum k µ = ( E, , , k ) and mass m A , ǫ + µ ( k ) = 1 √ , , i, , (24) ǫ − µ ( k ) = 1 √ , − , i, , (25) ǫ µ ( k ) = 1 m A ( k, , , − E ) . (26)The plasmon and graviton polarization vectors satisfy the following normalization and po-larization sum conditions e s µ e s ′ ∗ µ = 4 δ ss ′ , X s =1 e sµ ( k ) e s ∗ ν ( k ) = − η µν + k µ k ν m A , (27) e s µν e s ′ ∗ µν = 4 δ ss ′ , X s =1 e sµν ( p ) e s ∗ ρσ ( p ) = B µν ρσ ( p ) , (28)8here B µν ρσ ( p ) is given by B µν ρσ ( p ) = 2 η µρ − p µ p ρ m ~n m ~n ! η νσ − p ν p σ m ~n ! +2 η µσ − p µ p σ m ~n ! η νρ − p ν p ρ m ~n ! − η µν − p µ p ν m ~n ! η ρσ − p ρ p σ m ~n ! . (29)The total squared amplitude, averaged over the initial three polarizations(since massiveplasmons have three state of polarizations) and summed over final states for the process γ P ( k ) + γ P ( k ) → G KK ( p ) is |M| = (cid:18) (cid:19) X s |M| = κ (cid:16) T + T + T + T + T (cid:17) , (30)where T i ( i = 1 , ..
5) are given in appendix A. Substituting this in (15) and using (16) and(17), the total cross-section σ T for this process is obtained as σ T = X ~n σ γ P γ P → G kk ( S, m ~n ) = 12 S Z ρ ( m ~n ) δ ( S − m ~n ) |M| d ( m ~n )= 19 14 d π z Γ( d/ SM D ! d/ N (31)where z = − d and N = M D (cid:18) S m A + Sm A + m A S + 16 m A S + (cid:19) . While derivingEq. 31, we have used ρ ( m ~n ) = R d m d − ~n (4 π ) d/ Γ( d/ and the Planck scale relation Eq. 1.The volume emissivity of a supernova with a temperature T through this process isobtained by thermal-averaging over the Bose-Einstein distribution. Hence, the energy lossrate ( ˙ ǫ γ P = ρ SN ˙ Q γ P ) due to plasmon plasmon annihilation is given by (similar to that of theenergy loss rate via γγ → ν ¯ ν [24])˙ ǫ γ P = 1 ρ SN π Z ∞ ω dω ω ( ω − ω ) / e ω /T − Z ∞ ω dω ω ( ω − ω ) / e ω /T − S ( ω + ω )2 ω ω σ T , (32)where σ T is given in Eq. 31. Note that N γ P = π R ∞ ω dω ω ( ω − ω ) / e ω/T − is the number densityof thermal photons, or rather of transverse plasmons. In the present case, we treat theplasmon to be transverse(with the dispersion relation given by ω = ω + | k | ), since thecontribution coming from the longitudinal plasmon is typically smaller [23, 25]. Also inabove, ω corresponds to plasma frequency in the supernovae core.9inally introducing the dimensionless variables x i = ω i /T ( i = 0 , ,
2) and taking m A (thetransverse plasmon mass) to be equal to ω , we rewrite the above Equation as˙ ǫ γ P = 1 ρ SN T d M dD π Z ∞ x dx x ( x − x ) / e x /T − Z ∞ x dx x ( x − x ) / e x /T − x + x ) d x x F , (33)where F = 118 14 d π z Γ( d/ " T m A X T + T m A X T + 16 m A T X T + 16 m A T X T + 173 , X T = x + x . VI. NUMERICAL ANALYSIS
The SN 1987A energy loss due to KK graviton emission produced in massless photon-photon annihilation already put some bound on the effective scale of gravity M D for d = 2and 3 (see [4]). Here we study the modification of the above bound in a scenario where theplasma effect on photon is taken into account. In our analysis, the key working formula is theEq. 33 which describes the supernovae energy loss rate due to plasmon( γ P ) + plasmon( γ P ) → KK graviton( G KK ). Now for any kind of cooling mechanism which corresponds to anemissitivity > erg g − s − would invalidate our current understanding of Type-IIAsupernovae’s neutrino signal. So the consistency with the neutrino signal requires the energyloss rate ≤ erg g − s − . This gives rise the lower bound on M D . In Fig. 1 we have shownthe energy loss rate to KK gravitons as a function of the scale M D for different number ofextra dimensions d . The right and left curves respectively stands for d = 2 and 3. In thisplot, the inputs taken are as follows: ω = m A (plasmon mass) = 19 MeV, the supernovaetemperature T = 30 MeV and the supernovae core density ρ ≃ g cm − [19]. Thehorizontal line corresponds to the upper bound on the supernovae energy loss rate. Theintersection of this curve with the other two gives rise the following lower bound on M D :for d = 2 we find M D > . d = 3, M D > .
38 TeV. The bound on M D as obtained here is somewhat stronger( d = 2) and weaker( d = 3) than that obtained in [4]which are 15 TeV and 1 . d = 2 and 3, where the relevant process ofinterest was photon-photon annihilation to KK gravitons. Also note that the bound on M D that we find from plasmon-plasmon annihilation to gravitons is somewhat weaker than theone obtained from the nucleon-nucleon brehmstrahlung which are 51(3 .
6) TeV for d = 2(3),respectively. Finally, note that the present supernovae SN 1987A cooling analysis does notallow us to put any bound on M D for d ≥ d ε / d t ( e r g / ( g s )) M D (GeV) (a) Fig. 1 . The supernovae energy loss rate dǫ/dt ( erg g − s − ) due to KK graviton emissionproduced in plasmon plasmon annihilation is shown as a function of M D (GeV) in Fig. 1.For the right curve d = 2 , whereas for the left d = 3 . The upper horizontal curve correspondsto dǫ/dt ≤ erg g − s − . VII. CONCLUSIONS
In summary, we found that the emission of KK graviton by plasmon-plasmon annihilationfrom SN 1987A puts the conservative bound on the effective scale M D of the large extradimensional model in the case of d = 2 and 3. Taking a conservative estimate of thesupernovae temperature T = 30 MeV and plasmon mass m A = 19 MeV (equal to the coreplasma frequency ω o ), we find M D > . d = 2 and M D > .
38 TeV for d = 3. Nobound on M D follows from the present analysis for d ≥ III. ACKNOWLEDGEMENT
The authors are grateful to Professor Ramesh Kaul of Institute of Mathematical Sciences,Chennai for useful discussions.
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