Bounds on large extra dimensions from photon fusion process in

arXiv:0805.3429v1 [astro-ph] 22 May 2008

Bounds on large extra dimensions from photon fusion process in SN1987A V H Satheeshkumar1,2 and P K Suresh1 1

School of Physics, University of Hyderabad, Hyderabad 500 046, India. Department of Physics, Sri Bhagawan Mahaveer Jain College of Engineering, Jain Global Campus, Kanakapura Road, Bangalore 562 112, India. 2

E-mail: [email protected], [email protected] Abstract. The constraint on the ADD model of extra dimensions coming from photon annihilation into Kaluza-Klein graviton in supernova cores is revisited. In the two photon process for a conservative choice of the core parameters, we obtain the bound on the fundamental Planck scale M∗ & 1.6 TeV. The combined energy loss rate due to nucleon-nucleon brehmstrahlung and photon annihilation processes is rederived, which shows that the combined bounds add only second decimal place to M∗ . The present study can strengthen the results that are available in the current literature for the graviton emission from SN1987A which puts a very strong constraints on models with large extra dimensions for the case of n = 3 .

PACS numbers: 11.25.-w, 11.25.Wx

Keywords: Large extra dimensions, KK Gravitons, supernovae.

Bounds on large extra dimensions from photon fusion process in SN1987A

2

1. Introduction Stars are potential sources for weakly interacting particles such as neutrinos, gravitons, axions, and other new particles that can be produced by nuclear reactions or by thermal processes in the hot stellar interior. The solar neutrino flux is now routinely measured with such a precision that compelling evidence for neutrino oscillations has accumulated. The measured neutrino burst from supernova SN1987A has been used to derive many useful limits. Even when the particle flux can not be measured directly, the absence of visible decay products, notably x- or γ-rays, can provide important information. The properties of stars themselves would change if they lose too much energy into a new channel. This “energy-loss argument” has been widely used to constrain a long list of particles and there properties. All of this has been extensively reviewed [1, 2] The extra dimensional scenario due to Arkani-Hamed, Dimopoulos and Dvali (ADD) [3], model predicts a variety of novel signals which can be tested using table-top experiments, collider experiments, astrophysical or cosmological observations. It has been pointed out that one of the strongest bounds on models of extra dimensions comes from SN1987A [4]. Various authors have done calculations to place such constraints on the extra dimensions [5]-[10]. In this paper, we calculate the energy loss rate due to graviton emission from SN1987A by photon-photon annihilation and derive the bounds on extra dimensions. We combine the result with that of nucleon-nucleon brehmstralung process and derive the corresponding bound on large extra dimensions. Physically, there are two fundamental types of supernovae (SNe), based on what mechanism powers them: the thermonuclear SNe (Type I SNe) and the core-collapse ones (Type II SNe). The core-collapse SNe are the class of explosions which mark the evolutionary end of massive stars (M & 8 M⊙ ). Such stars have the usual onion structure with several burning shells, an expanded envelope, and a degenerate iron core that is essentially an iron white dwarf. The core mass grows by the nuclear burning at its edge until it reaches the Chandrasekhar limit. The collapse can not ignite nuclear fusion because iron is the most tightly bound nucleus. Therefore, the collapse continues until the equation of state stiffens by nucleon degeneracy pressure at about nuclear density (3 × 1014 gcm−3 ). At this “bounce” a shock wave forms, moving outward and expelling the stellar mantle and envelope. The explosion is a reversed implosion, the energy derives from gravity, not from nuclear energy. Within the expanding nebula, a compact object remains in the form of a neutron star or perhaps sometimes a black hole. The kinetic energy of the explosion carries about 1% of the liberated gravitational binding energy of about 3 × 1053 erg, 99% going into neutrinos. This powerful and detectable neutrino burst is the main astro-particle interest of core-collapse SNe. In core-collapse SNe only 10−4 of the total energy shows up as light, i.e. about 1% of the kinetic explosion energy, hence they are dimmer than SNe-Ia, and are not useful as standard candles. In the case of SN1987A, about 1053 ergs of gravitational binding energy was released in few seconds and the neutrino fluxes were measured by Kamiokande [11] and IMB [12]


3

collaborations. Numerical neutrino light curves can be compared with the SN1987A data where the measured energies are found to be “too low.” For example, the numerical simulation in [13] yields time-integrated values hEνe i ≈ 13 MeV, hEν¯e i ≈ 16 MeV, and hEνx i ≈ 23 MeV. On the other hand, the data imply hEν¯e i = 7.5 MeV at Kamiokande and 11.1 MeV at IMB [14]. Even the 95% confidence range for Kamiokande implies hEν¯e i < 12 MeV. Flavor oscillations would increase the expected energies and thus enhance the discrepancy [14]. It has remained unclear if these and other anomalies of the SN1987A neutrino signal should be blamed on small-number statistics, or point to a serious problem with the SN models or the detectors, or is there a new physics happening in SNe? Since we have these measurements already at our disposal, now if we propose some novel channel through which the core of the supernova can lose energy, the luminosity in this channel should be low enough to preserve the agreement of neutrino observations with theory. That is, Lnew channel . 1053 ergs s−1.

(1)

This idea was earlier used to put the strongest experimental upper bounds on the axion mass [15]. Here, we consider the emission of the higher-dimensional gravitons from the core. Once these particles are produced, they can escape into the extra dimensions, carrying energy away with them. The constraint on the luminosity of this process can be converted into a bound on the fundamental Planck scale of the theory, M∗ . The argument is very similar to that used to bound the axion-nucleon coupling strength [1, 16, 17, 18]. The ‘standard model’ of supernovae does an exceptionally good job of predicting the duration and shape of the neutrino pulse from SN1987A. Any mechanism which leads to significant energy-loss from the core of the supernova immediately after bounce will produce a very different neutrino-pulse shape, and so will destroy this agreement as demonstrated explicitly in the axion case by Burrows, Brinkmann, and Turner [18]. Raffelt has proposed a simple analytic criterion based on detailed supernova simulations [1]: if any energy-loss mechanism has an emissivity greater than 1019 ergs g− 1s− 1 then it will remove sufficient energy from the explosion to invalidate the current understanding of SNe II neutrino signal. 2. Supernovae and constraints on large extra dimensions The most restrictive limits on M∗ come from SN1987A energy-loss argument. If large extra dimensions exist, the usual four dimensional graviton is complemented by a tower of Kaluza-Klein (KK) states, corresponding to new phase space in the bulk. The KK gravitons interact with the strength of ordinary gravitons and thus are not trapped in the SN core. During the first few seconds after collapse, the core contains neutrons, protons, electrons, neutrinos and thermal photons. There are a number of processes in which higher-dimensional gravitons can be produced. For the conditions that pertain in the core at this time (temperatures T ∼ 30 − 70 MeV, densities ρ ∼ (3 − 10) × 1014 g


4

cm−3 ), the relevant processes are nucleon-nucleon brehmstrahlung, graviton production in photon fusion and electron-positron annihilation. In SNe, nucleon and photon abundances are comparable (actually nucleons are somewhat more abundant). In the following we present the bounds derived by various authors using nucleon-nucleon brehmstralung and in the next section we give a detailed calculation for photon-photon annihilation to KK graviton process. 2.1. Nucleon-Nucleon brehmstralung This is the dominant process relevant for the SN1987A where the temperature is comparable to pion mass mπ and so the strong interaction between nucleons is unsuppressed. This process can be represented as N + N → N + N + KK

(2)

where N can be a neutron or a proton and KK is a higher-dimensional graviton. The main uncertainty comes from the lack of precise knowledge of temperatures in the core: values quoted in the literature range from 30 to 70 MeV. For T = 30 MeV and ρ = 3 × 1014 g cm−3 , we list the results obtained by various authors. Cullen and Perelstein [5] n = 2, n = 3,

ǫ˙ = 6.79 × 1025 × M∗−4 erg g −1 s−1 ,

M∗ & 50TeV;

(3)

M∗ & 4TeV.

(4)

ǫ˙ = 6.7 × 1025 × M∗−4 erg g −1 s−1 ,

M∗ & 51TeV;

(5)

ǫ˙ = 6.3 × 1021 × M∗−5 erg g −1 s−1 ,

M∗ & 3.6TeV.

(6)

ǫ˙ = 9.24 × 1024 × M∗−4 erg g −1 s−1 ,

M∗ & 31TeV;

(7)

M∗ & 2.75TeV.

(8)

M∗ & 84TeV;

(9)

M∗ & 7TeV.

(10)

ǫ˙ = 1.12 × 1022 × M∗−5 erg g −1 s−1 ,

Barger, Han, Kao and Zhang [6] n = 2, n = 3, Hanhart et. al. [7, 8] n = 2, n = 3,

ǫ˙ = 1.57 × 1021 × M∗−5 erg g −1 s−1 ,

Hannestad and Raffelt [9, 10] n = 2, n = 3, 3.

ǫ˙ = 4.98 × 1026 × M∗−4 erg g −1 s−1 , 23

ǫ˙ = 1.68 × 10 ×

M∗−5

erg g

−1

−1

s ,

Graviton production through photon fusion and energy loss rate

Our aim is to study the energy loss mechanism of SN1987A by graviton emission by photon-photon annihilation in the ADD framework. For this we need to compute the cross-section for the relevant process. Here we present the general formalism for calculating the cross-section [19] for two particle initial state. The scattering cross section is given by Z Y d 3 pf 1 1 σ= υrel 4E1 E2 (2π)3 2Ef f

Bounds on large extra dimensions from photon fusion process in SN1987A ! X X 4 4 pf |Mf i |2 pi − × (2π) δ i

5 (11)

f

with

p

(p1 · p2 )2 − m21 m22 , (12) E1 E2 where pi and Ei being the 3-momenta and the energies of the initial particles whose masses are m1 and m2 ; pf and Ef are the 3-momenta and the energies of the final particles and Mf i is the Feynman amplitude for the process. For a general reaction of the kind a + b → c, in the center-of-mass frame, the expression (11) takes the form Z 3 ′ 1 dp σ= δ(E1 + E2 − E ′ )|M|2. (13) 2 ′ 64π E1 E2 υrel E We use the center-of-mass frame, where we use the following notions. √ s = E1 + E2 , (14) √ (15) E1 E2 υrel = p s, υrel =

where p = p1 + p2 . Next, we focus on the energy loss due to KK gravitons escaping into the extra dimensions. The energy loss as per unit time per unit mass of SN in terms of the cross-section σa+b→c , is given by [20] hna nb σ(a+b→c) vrel Ec i (16) ρ where the brackets indicate thermal average, na,b are the number densities for a, b and ρ is the mass density and Ec is the energy of the particle c. We calculate the cross section using the helicity method [21]-[36]. We follow the conventions and Feynman rules derived in [37]. In the helicity method, it is more convenient to work with polarizations explicitly. Thus, the polarization vectors [38] of a massive graviton are ǫ˙a+b→c. =

± ± e±2 µν = 2ǫµ ǫν , √ 0 0 ± 2 (ǫ± e±1 µ ǫν + ǫµ ǫν ) , µν = r 2 + − + 0 0 e0µν = (ǫ ǫ + ǫ− µ ǫν − 2ǫµ ǫν ) . 3 µ ν 0 Here ǫ± µ and ǫµ are the polarization vectors of a massive gauge boson; for a massive vector boson with momentum pµ = (E, 0, 0, p) and mass m, 1 (17) ǫ+ µ (p) = √ (0, 1, i, 0) , 2 1 ǫ− (18) µ (p) = √ (0, −1, i, 0) , 2 1 (19) ǫ0µ (p) = (p, 0, 0, −E) . m


6

The graviton polarization vectors satisfy the normalization and polarization sum conditions ′

′

es µν esµν∗ = 4δ ss , X esµν esρσ∗ = Bµν ρσ ,

(20) (21)

s

where Bµν ρσ is given by

kν kσ kµ kρ ηνσ − 2 Bµν ρσ (k) = 2 ηµρ − 2 m~n m~n kµ kσ kν kρ + 2 ηµσ − 2 ηνρ − 2 m~n m~n kρ kσ 4 kµ kν ηρσ − 2 . (22) − ηµν − 2 3 m~n m~n The total squared amplitude, averaged over the initial polarizations z and summed over ′ ′′ final states for the reaction ah (q1 ) + bh (q2 ) → ch (p), is given by 2 1 X h h′ h′′ M a (q ) + b (q ) → c (p) (23) 1 2 z ′ ′′ h,h ,h

′

′′

where h, h , h are the helicities and q1 , q2 , p are the momenta of particles a, b, c respectively. Photons are quite abundant in supernovae. Here we consider photon-photon annihilation to KK graviton and the process is given by, γ(k1 ) + γ(k2 ) → KK(p).

(24)

The vertex function for the process (24) is given by [37] i h ηαβ k1µ k2ν − ηµα k1β k2ν − ηνβ k1µ k2α Xµναβ = 2M4 1 + ηµα ηνβ (k1 · k2 ) − ηµν (ηαβ (k1 · k2 ) − k1β k2α ) 2 i 1 + mn mn−m (ηµα ηνβ − ηµν ηαβ ) + (α ↔ β) . 2 The momentum vectors for this reaction are

(25)

pµ ≡ (mn , 0, 0, p)

(26)

k2µ

(28)

k1µ ≡ (k1 , 0, 0, k1)

(27)

≡ (k2 , 0, 0, k2).

In helicity formalism the reaction (24) can happen in two ways γ ± (k1 ) + γ ± (k2 ) → KK ±2 (p) ±

∓

(29)

0

γ (k1 ) + γ (k2 ) → KK (p).

(30)

Next, consider these two reactions separately and find their corresponding amplitudes. For the reaction described in (29), the helicity amplitude for the KK graviton emission by photon-photon annihilation is M γ ± (q) + γ ± (q) → KK ±2 (p) = X µναβ ǫ± (k)ǫ± (q)e±2 ∗ (p). (31) α

β

µν

Bounds on large extra dimensions from photon fusion process in SN1987A The polarization tensors for gravitons are calculated and they are, 1 e±2 11 = + , 2 i ±2 e±2 12 = e21 = , 2 1 e±2 22 = − . 2 The non-zero components of the vertex function are

7

(32) (33) (34)

X 1111 , X 1212 , X 1221 , X 2112 , X 2121 , X 2222 , −X 2211 , −X 1122 Each of them equal to −iκ k1 · k2 . 2 where k1 · k2 = m~n2 /2. Substituting the various quantities that we have calculated above in equation we get 2 M γ ± (q) + γ ± (q) → KK ±2 (p) = κm~n 2. The helicity amplitude for the reaction (30) is, ∓ 0∗ M γ ± (q) + γ ∓ (q) → KK 0 (p) = X µναβ ǫ± α (k)ǫβ (q)eµν (p).

The polarization tensors for gravitons are given by r 2 e011 = e022 = − 3 0 0 e12 = e21 = 0.

(35) (31),

(36)

(37)

(38) (39)

The non-zero components of the vertex function are X 1111 , X 2222 , −X 2211 and −X 1122 and are equal to (35). Substituting the various quantities that we have calculated above in equation (37), we get M γ ± (q) + γ ∓ (q) → KK 0 (p) = 0. (40)

Thus the total squared amplitude, averaged over the initial three polarizations and summed over final states, is 2 κ2 m4 1 X h ′ ′′ ~ n . (41) M γ (k1 ) + γ h (k2 ) → KK h (p) = 3 ′ ′′ 12 h,h ,h

Substituting this in (13) and using (14) and (15), the cross-section for the process is obtained as √ √ πκ2 s (42) δ(m~n − s) , σ= 16 where s is the center of mass energy, and m~n the mass of the KK state at level ~n.


8

Since for large R the KK gravitons are very light, they may be copiously produced in high energy processes. For real emission of the KK gravitons from a SM field, the total cross-section can be written as X σ(~n) , (43) σtot = κ2 ~ n

where the dependence on the gravitational coupling is factored out. The mass separation of adjacent KK states, O(1/R), is usually much smaller than typical energies in a physical process, therefore we can approximate the summation by an integration which can be performed using KK state density function [37],

Rn m~nn−2 . (44) (4π)n/2 Γ(n/2) The volume emissivity of a supernova with a temperature T through the process under consideration is obtained by thermal-averaging over the Bose-Einstein distribution Z Z 2d3~k2 1 1 2d3~k1 Qγ = 3 ω /T 3 ω /T (2π) e 1 − 1 (2π) e 2 − 1 X s(ω1 + ω2 ) σγγ→kk (s, m~n ), (45) × 2ω1 ω2 ρ(m~n ) =

~ n

where the summation is over all KK states, and the squared center of mass energy s is related to the photon energies ω1 and ω2 and the angle between the two photon momenta θγγ as follows: s = 2ω1 ω2 (1 − cos θγγ ) .

(46)

After carrying out the integrals and the summation over KK states, we find 2n+3 Γ( n2 + 3)Γ( n2 + 4)ζ( n2 + 3)ζ( n2 + 4) T n+7 Qγ = , (n + 4)π 2 MSn+2

(47)

where we have used M∗n+2 Rn Sn = Mpl2 and numerically, these Riemann zeta-functions are close to 1. In this calculation, we have neglected the plasma effect, through which the photons can have different energy dispersion relations from those of free particles. We take the supernova core density ≃ 1015 g cm−3 . Using (16), we compute the energy loss rate for n = 2 and n = 3 extra spatial dimensions and hence the lower limits on M∗ using the conservative upper limits on the energy-loss rate of SN [1] ǫ˙SN ∼ 1019 erg g−1 sec−1 .

(48)

The results are summarized below, n = 2, n = 3,

ǫ˙ = 4.7 × 1023 × M∗−4 erg g −1 sec−1 ,

M∗ & 14.72TeV,

(49)

ǫ˙ = 1.1 × 1020 × M∗−5 erg g −1 sec−1 ,

M∗ & 1.62TeV.

(50)

ǫ˙ = 6.837 × 1025 × M∗−4 erg g −1 s−1 ,

M∗ & 50.13TeV;

(51)

M∗ & 4.08TeV.

(52)

We now combine the energy loss rate due to photon fusion process with that of the nucleon-nucleon brehmstralung and rederive the constraints as follows. Cullen and Perelstein [5] n = 2, n = 3,

ǫ˙ = 1.131 × 1022 × M∗−5 erg g −1 s−1 ,


9

Barger, Han, Kao and Zhang [6] n = 2, n = 3,

ǫ˙ = 6.747 × 1025 × M∗−4 erg g −1 s−1 ,

M∗ & 50.96TeV;

(53)

s ,

M∗ & 3.64TeV.

(54)

ǫ˙ = 9.710 × 1024 × M∗−4 erg g −1 s−1 ,

M∗ & 31.39TeV;

(55)

s ,

M∗ & 2.79TeV.

(56)

ǫ˙ = 4.985 × 1026 × M∗−4 erg g −1 s−1 ,

M∗ & 84.03TeV;

(57)

M∗ & 7.00TeV.

(58)

21

ǫ˙ = 6.410 × 10 ×

M∗−5

erg g

−1 −1

Hanhart et. al. [7, 8] n = 2, n = 3,

21

ǫ˙ = 1.680 × 10 ×

M∗−5

erg g

−1 −1

Hannestad and Raffelt [9, 10] n = 2, n = 3,

ǫ˙ = 1.681 × 1023 × M∗−5 erg g −1 s−1 ,

As we expected, the energy loss rate due to nucleon-nucleon brehmstralung is 1 to 3 orders of magnitude more than that due to photon fusion process. Hence the combined bounds add only the second decimal place to M∗ . 4. Conclusions We have revisited the constraints on ADD model coming from photon annihilation into KK graviton in SN cores. For a conservative choice of the core parameters, we obtain the two photon process bounds on the fundamental Planck scale M∗ & 1.6 TeV. The energy loss rate due to nucleon-nucleon brehmstralung is 1 to 3 orders of magnitude more than that due to photon fusion process. Hence the combined bounds add only the second decimal place to M∗ . Thus the present study can strengthen the results which are available in the current literature for the graviton emission from SN1987A. Our results show that the above processes put a very strong constraints on models with large extra dimensions for the case of n = 3. Notice that the plasmon effects are not considered in our calculations and will be done elsewhere. 5. Acknowledgments We thank Dr. Prasanta Kumar Das for collaboration in a similar project. The first author acknowledges Dr. R Chenraj Jain, Wg. Cdr. K L Ganesh Sharma, Prof. T S Sridhar and Mr. M S Santhosh for the kind hospitality and great facilities while writing this paper. References [1] Raffelt G G, Stars as Laboratories for Fundamental Physics Chicago University Press, Chicago, (1996). [2] Raffelt G G, Ann. Rev. Nucl. Part. Sci. 49, 163 (1999). [3] Arkani-Hamed N, Dimopoulos S and Dvali G, Phy. Lett. B 429 263 (1998). [4] Arkani-Hamed N, Dimopoulos S and Dvali G R, Phys. Rev. D 59, 086004 (1999). [5] Cullen S and Perelstein M, Phys. Rev. Lett. 83 268 (1999).

Bounds on large extra dimensions from photon fusion process in SN1987A [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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