Effects of Goldstone Bosons on Gamma-Ray Bursts

Gamma-ray bursts (GRBs) are the most energetic explosion events in the universe. An amount of gravitational energy of the order of the rest-mass energy of the Sun is released from a small region, within seconds or longer. This should lead to the formation of a fireball of temperature in the MeV range, consisting of electrons/positrons, photons, and a small fraction of baryons. We exploit the potential of GRB fireballs for being a laboratory for testing particle physics beyond the Standard Model, where we find that Weinberg's Higgs portal model serves as a good candidate for this purpose. Due to the resonance effects, the Goldstone bosons can be rapidly produced by electron-positron annihilation process in the initial fireballs of the gamma-ray bursts. On the other hand, the mean free path of the Goldstone bosons is larger than the size of the GRB initial fireballs, so they are not coupled to the GRB's relativistic flow and can lead to significant energy loss. Using generic values for the GRB initial fireball energy, temperature, radius, expansion rate, and baryon number density, we find that the GRB bounds on the parameters of Weinberg's Higgs portal model are indeed competitive to current laboratory constraints.


Introduction
Gamma-ray bursts (GRBs) are the most energetic explosion events in the universe (for recent reviews, see Refs. [1,2,3,4,5,6,7].) They emit a huge amount of energy of the order of 10 52 erg or higher [8,9,10,11,12], within a short timescale. The initial burst of gamma-ray radiation is usually followed by an "afterglow" at longer wavelengths, ranging from X-ray, optical to radio. First detected by the military Vela satellites in late 1960's [13] properties. It is established that they are of cosmological origin, with the highest redshift recorded so far being z = 9.4 [22]. Following the investigations of Ref. [23], GRBs are commonly classified in two classes according to their T 90 , the time during which 90% of the burst's fluence is accumulated. Long bursts (T 90 > 2 s) may be due to the collapse of massive stars [24], while short bursts (T 90 < 2 s) are speculated to originate from the binary neutron star or neutron star -black hole mergers [1,25]. There are also ultra-long bursts [26], or bursts whose detection requires a new classification scheme [27]. The fireball model [28,29,30,31] is the simplest and most conventional model to explain the observed non-thermal high-energy prompt emission, the variability over short timescales, and the generation of the afterglow of GRBs (see Refs. [4,5,32,33,34,35] for detailed reviews.) In this model, the central engine is a black hole or a neutron star, surrounded by a matter accretion disc, which causes a jet of material blasted outward at relativistic speed. During the course of the fireball expansion, the thermal energy contained in the electrons, positrons and photons are gradually converted into kinetic energy of the baryons, which are accelerated to a high Lorentz factor. The kinetic energy is converted to gamma-ray photons in the collisions between internal shock waves travelling at different speeds. At some large distances away from the central engine where the fireball becomes optically thin, the gamma-ray photons can escape and be observed as the prompt emission. As the shock waves continue to propagate outward, they eventually interact with the interstellar medium, causing the latter to emit radiations. The long duration and the wide electromagnetic spectrum covered by those radiation processes then account for the observed afterglows. The tremendous amount of energy release and the high initial temperature of the GRB fireball makes it an excellent laboratory for particle physics. In the Standard Model (SM), Refs. [8,36] have studied the effects of neutrinos on the GRB initial fireballs. It is found that although neutrino production therein is rapid enough to cool the fireball, the high opacity of the latter to the neutrinos efficiently prevents dramatic energy losses of itself [36]. In Ref. [37], effects of the neutron component were studied in dependence of the final Lorentz factor of the GRB plasma wind. It showed that neutrons can strongly influence a GRB by changing the dynamics of its shocks in the surrounding medium. Beyond the SM, the possibility of using axions and other exotic particles for transferring the gravitational energy of the central collapsing object into the GRB fireball was investigated in Refs. [38,39,40,41], and Ref. [42], respectively. In this work we shall show that another good example is provided by Weinberg's Higgs portal model [43], which was proposed to account for the fractional cosmic neutrinos in the early universe. In this model, Weinberg considered a global U(1) continuous symmetry associated with the conservation of some quantum number, and introduced a complex scalar field to break it spontaneously. The Goldstone bosons arising from the symmetry breaking would be massless or nearly massless, and their characteristic derivative coupling would make them very weakly-interacting at sufficiently low temperatures. The latter property is crucial, since the Goldstone bosons must decouple from the early universe thermal bath at the right moment so that their temperature is a fraction of that of the neutrinos (see e.g. Ref. [44].) We have examined energy losses due to the emission of Weinberg's Goldstone bosons in a postcollapse supernovae core [45], while collider phenomenology has been investigated in Ref. [46]. In this work we scrutinise the production and propagation of Weinberg's Goldstone bosons in the initial fireballs of gamma-ray bursts. In Section 2 we briefly summarise generic properties of the GRB fireball model. We then review Weinberg's Higgs portal model and existing laboratory constraints on it in Section 3. In Section 4 we calculate energy loss rates due to Goldstone boson production by electron-positron annihilation, photon scattering, and nuclear bremsstrahlung processes taking place in GRB initial fireballs. Subsequently in Section 5 we estimate the mean free path of the Goldstone bosons, which is set by their scattering on the electrons/positrons and nucleons. In Section 6 we use relativistic hydrodynamics to study the effects of the Goldstone bosons, and confront the results with existing laboratory constraints. In Section 7 we summarise.

The Fireball Mechanism
From the correlation of the GRB duration with the progenitor environment, it is believed that the long duration GRBs result from a collapsar, and the short GRBs from merger. In either case, relativistic outflows are powered by the central black hole or neutron star, which is surrounded by an accretion disc formed by the inwardly spiraling instellar material. Two most discussed jet production mechanisms are electromagnetic extraction of the black hole rotation energy [47], and pair annihilation of neutrinos that emanate from the accretion disc [48,49,50] (see also Ref. [51] for a combination of both.) The outcome is a large amount (of order of the solar rest mass) of gravitational energy released within a short time, from a small region, which leads to the formation of an e ± -γ fireball. A fraction of the gravitational energy is converted into neutrinos and gravitational waves. The thermal neutrinos are sensitive to the thermodynamics profiles of the accretion disc, while gravitational waves are sensitive to the dynamics of the progenitors. The Super-Kamiokande [52] and Sudbury Neutrino Observatory (SNO) [53] experiments have searched for MeV-neutrinos from the long and short GRBs. From the non-detection they have put upper limits on the GRB neutrino fluence. A much smaller fraction goes into the fireball of temperature in the MeV range, which consists of e ± , photons and baryons, and may contain a comparable amount of magnetic field energy. The initial photon luminosity inferred is many orders of magnitude larger than the Eddington limit, i.e. the radiation pressure far exceeds the gravitational force, so the fireball will expand. For a steady spherically symmetrical flow with four velocity u µ = (u 0 , u R , 0, 0) in the spherical coordinates (t, R, θ, φ), the equations of relativistic fluid dynamics are [37] p + ρ n B u 0 = const. , Here n B is the baryon number density, and p and ρ are the pressure and the total energy density, respectively. All the three quantities are measured in the fluid comoving frame. The components of the flow four-velocity are u 0 = √ −g 00 Γ and u R = βΓ/ √ g RR , with β and Γ its three-velocity an Lorentz factor, Γ = 1/ 1 − β 2 . If the gravitational effects of the wind itself are negligible, the metric is −g 00 = g −1 RR = 1 − R S /R, where R S is the Schwarzschild radius of the central object. The hydrodynamic equations need to be supplemented with an equation of state, e.g. p = ρ/3. As long as the constituents of the fireball plasma are strongly coupled, they are in thermal equilibrium, and the fireball expansion is adiabatic. Combining with the equation of adiabatic process pn −γ B = const., with γ = 4/3, one arrives at the equation for the evolution of the Lorentz factor of the wind: Here R sat and Γ l are the saturation values for the fireball radius and the fireball Lorentz factor, respectively. If magnetic fields are included as an additional component of the GRB fireball, the Lorentz factor evolution is modified to Γ(R) ∝ R µ for R < R sat , and Γ ≃ const. for R sat < R < R dec . Here 1/3 ≤ µ ≤ 1, with µ = 1 corresponding to the baryon dominated jet, and µ = 1/3 to a magnetic field dominated jet [35,54]. In any case, the Lorentz factor first increases with the radius R. When R reaches R sat ∼ 10 9 cm, the fireball enters the coasting phase, with all the fireball thermal energy converted into the kinetic energy of the baryons. The fireball continues expanding at a constant rate until it runs into the external medium and slows down. At the deceleration radius R dec ∼ 10 16 cm, the deceleration of the fireball expansion becomes significant. Correspondingly, the fireball comoving temperature evolves as T ′ ∝ R −( µ+2 3 ) for R < R sat , and ∝ R −2/3 when R sat < R < R dec . The bulk Lorentz factor Γ can be measured [55,56,57], and lower limits on the Lorentz factor have been inferred by requiring that the GRBs be optically thin to high energy photons [58]. Ref. [10] deduced Γ min = 608 ± 15 and 887 ± 21 for GRB 080916C, while Γ min ≃ 1200 for the short gamma-ray burst GRB 090510. The saturation value for the Lorentz factor is determined by the initial raito of radiation energy to rest mass This ratio must be of the order ∼ O(10 2 ), so that the baryons may be accelerated to a Lorentz factor Γ ≈ E/M 0 high enough to produce the observed gamma-rays.
On the other hand, if the ratio is too large, the fireball is radiation-dominated. Depending on its value, there are four types of fireballs. We consider the most interesting case, the relativistic baryonic fireball, which corresponds to the case 1 < η B < (3σ T E/8πm p R 2 0 ) 1/3 ≈ 10 5 (E/10 52 ergs) 1/3 (R 0 /10 7 cm) −2/3 [31,32], where σ T is the Thompson cross section, and m p the proton mass. Within the fireball model, there are many mechanisms proposed to explain the GRB observations. In the internal-external scenario, the prompt emission is produced by the internal shocks [59], and the afterglow by the external shocks. Under the assumption that the central engine produces ejecta shells with a highly variational distribution of Lorentz factor, the internal shocks are formed when the faster shells catch up with the slower ones. The external shocks arises when the fireball expands into external medium. For a summary or review of the GRB fireball model, we refer to Refs. [4,5,32,33,34,35].

Generic GRB Fireball Parameters
In this work we consider the following generic parameters for the GRB fireballs as Ref. [36]: the initial fireball energy is E = 10 52 -10 54 ergs. The initial radius is that of the Schwarzschild radius R S = 3 (M/M ⊙ ) km, or of the neutron star radius ∼ 10 km. The initial wind velocity is about the sound speed, β 0 ≈ c s = 1/ √ 3. In thermal equilibrium, the radiation energy density and the temperature is related by The total number of effective massless degrees of freedom is with g i the internal degrees of freedom of particle species i, and T i its temperature.
In the initial fireball, photons, electrons, positrons, as well as three flavours of neutrinos are in thermal equilibrum, so g * = 43/4. Assuming that the initial fireball is spherical, its temperature can be expressed by where T = T 11 × 10 11 K, E = E 52 × 10 52 erg, and R = R 6.5 × 10 6.5 cm. We therefore follow Ref. [36] to choose E = 10 52 erg , R 0 = 10 6.5 cm , T 0 = 2.1 · 10 11 K = 18 MeV , as our fiducial value for the initial fireball total energy, radius and temperature, respectively. In view of the recent results in Ref. [57], we also consider larger initial radius, e.g. R 0 = 10 7 -10 8 cm, and lower initial fireball temperature, values, such as T 0 = 8 and 2 MeV. It was shown (see e.g. Ref. [60]) that the sonic point of a Schwarzschild black hole should be located at the radius R c = 3 2 R S , if the particles in the in-and outflow are relativistic so that the equation of state is p = ρ/3. In the case that the GRB jets are formed by energy injection from neutrino pair annihilation, the sonic point of the inflow, R c, 3 ), and that of the outflow, R c,2 > 3 2 R S (where β c,2 = 1 √ 3 ), are separate. In Ref. [60] the location of the outer sonic point is shown for several different energy injection profiles, which is pushed out well above that of the adiabatic flow (R c = 3 2 R S ) in all cases (see also Ref. [37].) In this work we choose as the fiducial value for the fireball initial wind velocity.
Since the initial temperature is higher than the nuclear binding energies, the nuclei are dissociated in nucleons. Requiring η B ∼ 1000 for the initial energy to rest mass ratio defined in Eq. (3), the initial comoving baryon number density in the fireballs should be n B,0 = 5 · 10 31 cm −3 , so that the fireball rest mass M 0 = m N n B,0 V 0 ≈ 10 49 ergs. The electron and the positrion number density are with their phase space distribution functions given by , respectively. The e ± chemical potential µ e is determined by the requirement of charge neutrality and beta-equilibrium in the fireball for a fixed lepton fraction Y e . For the reference temperature, it is µ e /T 0 ∼ 2 × 10 −4 [36], i.e. the electrons and positrons are non-degenerate, so with ζ(3) ≈ 1.20206. Neutrinos are created rapidly in the initial fireball, majorly through the electron-positron pair annihilation process e − + e + → ν +ν. The emissivity for this process is [36,61] Q e − e + →ν iνi = 3.6 · 10 33 (T 11 ) 9 erg s −1 cm −3 , much larger than that for the photo-neutrino e ± + γ → e ± + ν i +ν i , the plasma γ → ν iνi , and the URCA processes e − + p → n + ν e and e + + n → p +ν e . Neutrino mean free path (mfp) is set by the elastic scattering on electrons and positrons ν + e ± → ν + e ± . It is [36] λ (e) = 3.7 · 10 6 (T 11 ) −5 cm , λ (µ,τ ) = 1.6 · 10 7 (T 11 ) −5 cm , for the three flavours, respectively. Neutrinos decouple in two stages, when the optical depth (τ ≡ R/λ) for each neutrino flavour, τ (µ,τ ) and τ (e) , drops to 1.
In this work we consider a baryon dominated fireball jet, and neglect the effects of the magnetic fields.

The Model
In this subsection we briefly summarise Weinberg's model [43] following the convention of Refs. [45,46]. Consider the simplest possible broken continuous symmetry, a global U(1) symmetry associated with the conservation of some quantum number W . A single complex scalar field S(x) is introduced for breaking this symmetry spontaneously. With this field added to the Standard Model (SM), the Lagrangian is where Φ is the SM Higgs doublet, µ 2 , g, and λ are real constants, and L SM is the usual SM Lagrangian. One separates a massless Goldstone boson field α(x) and a massive radial field r(x) in S(x) by defining where the fields α(x) and r(x) are real. In the unitary gauge, one is the physical Higgs field. The Lagrangian in Eq. (14) thus becomes where the replacement α(x) → α(x)/ (2 r ) was made in order to achieve a canonical kinetic term for the α(x) field. In this model, the interaction of the Goldstone bosons with the SM particles arises entirely from a mixing of the radial boson with the Higgs boson. The mixing angle is Collider searches for the SM Higgs invisible decay as well as meson invisible decays have already been used to set strong constraints on the coupling g and/or the mixing angle θ, as will be reviewed in the next subsection.
As will be presented in Section 4, Goldstone boson emissivities in the GRB initial fireballs depends strongly on the total decay width of the radial field r. The r field decays dominantly to a pair of Goldstone bosons, with the decay width given by However, if its vacuum expectation value r is very large and the coupling g is not too small, the decay widths into SM fermion pairs can be comparable or even dominant. Here m f and N c are the mass and the colour factor of the fermion, respectively. For m r > 2m π , the r field can also decay to pion pairs through the effective coupling of the SM Higgs to pions π + π − |L int |ϕ . The effective Lagrangian is [62,63,64] where θ µ µ is the trace of the energy-momentum tensor, valid only at low momentum tranfers 0.3 GeV (see e.g. the discussion in Ref. [44].) The decay width is then In Fig. 1 the three decay widths are shown for the case of r = 1 GeV, g = 0.011. The decay widths for other parameter values can be easily obtained by scaling with g 2 or r 2 .
In the GRB fireballs, the Goldstone bosons can also be produced by nuclear processes and undergo elastic scattering with the nucleons through the ϕ−r mixing. The Higgs effective coupling to nucleons, f N m N / ϕ , has been calculated for the purpose of investigating the sensitivities of the dark matter direct detection experiments [65,66,67,68]. Following the Shifman-Vainshtein-Zakharov approach [69] to evaluate the contributions from the heavy quarks, it can be written in the form In this work we use the estimate of f N = 0.3 for proton and neutron from Ref. [68].

Collider Searches for SM Higgs Invisible Decay
The non-standard decay branching ratio of the SM Higgs is constrained to Γ h→inv.  Tevatron [70]. In Weinberg's Higgs portal model, the SM Higgs can decay into a pair of Goldstone bosons or a pair of the radial field r, with the decay widths given by respectively. The constraint is translated into a bound on the Goldstone boson coupling of |g| < 0.011 .
In the future, the International Linear Collider (ILC) may reach a sensitivity of constraining the branching ratio of SM Higgs invisible decays to < 0.4 − 0.9% [71] in the best scenarios. If this can be realised, the collider bound on the Goldstone boson coupling will be improved by a factor of 5 ∼ 7. In this work we will estimate the effects of the Goldstone bosons on the initial GRB fireballs for the coupling in the range 0.011 > g > 0.0015.

Muon Anomalous Magnetic Moment
There is still a discrepancy between the SM prediction for the muon anomalous magnetic moment [72], a SM µ , and the experimental results from the E821 experiment at Brookhaven National Lab (BNL) [73], a exp µ = 11 659 209(5.4)(3.3) · 10 −10 , where the first errors are statistical and the second systematic. The observed difference of [74] may point to new physics beyond the Standard Model. The contribution from the SM Higgs was first calculated in Ref. [75]. The radial r field can contribute to ∆a µ through its mixing with the SM Higgs [76] ∆a r where G F is the Fermi constant. By demanding ∆a r µ < ∆a µ one obtains a very weak constraint on the mixing angle: θ O(1).

Radiative Upsilon Decays
As first pointed out by Wilczek [77], light Higgs boson can be searched for in the radiative decays of heavy vector mesons. In Weinberg's Higgs portal model, the branching ratio is , where A 0 is a scalar boson. In this work we consider mass of the radial field r below 1 GeV, for which for n = 1 and 3. This is translated into a constraint on Weinberg's Higgs portal model as θ < 0.2.

B Meson and Kaon Decays
As first pointed out in Ref. [81], decays of B mesons to a K meson plus missing energy can be an efficient probe of GeV or sub-GeV scalar dark matter. In Refs. [76,82] this constraint has been applied to Weinberg's Higgs portal model. The BaBar Collaboration has reported an upper limit at the 90% confidence level of B(B + → K + + νν) < 1.3 · 10 −5 , as well as B(B 0 → K 0 + νν) < 5.6 · 10 −5 [83]. The CLEO Collaboration also reported a 90% C. L. upper limit of 4.9 · 10 −5 and 5.3 · 10 −5 on the branching ratio for the decays B ± → K ± X 0 and B 0 → K 0 S X 0 , respectively, where X 0 is any neutral massless weakly-interacing particle [84]. In the SM, the branching ratio for the total B(B → Kνν) branching ratio is estimated to be (4.5 ± 0.7) · 10 −6 . In Weinberg's Higgs portal model, this branching ratio is [76,82] with the form factor given by and τ B is the B-meson lifetime, V tb and V ts the CKM matrix elements, and m t , m b , m s , m B and m K the corresponding quark and meson masses, with m ± ≡ m B ± m K . We follow Ref. [82] and use the most stringent constraint which imposes a constraint on the ϕ − r mixing angle that θ < 0.0016, for m r < m B − m K . If the radial field r is lighter than 354 MeV, the decay of K meson to a π meson plus missing energy is a more powerful probe. The E787 and E949 experiments at the BNL has used stopped kaons to study the rare decay K + → π + νν [85]. The branching ratio B(K + → π + νν) = 1.73 +1.15 −1.05 · 10 −10 determined with the observed seven events and background estimation is consistent with the SM prediction of 7.8(75)(29) · 10 −11 [86,87], where the first error summarises the parametric, and the second the remaining theoretical uncertainties. For K + meson decay into the radial r field, the branching ratio can be calculated similarly as in Eq. (29), using the form factor (see e.g. Ref. [82]), In this work we follow Refs. [76,82] and use the constraint which imposes a very stringent constraint on the mixing angle as θ < 8.7 · 10 −5 , for m r < m K − m π = 354 MeV. Laboratory constraints from muon anomalous magnetic moment, radiative upsilon decays, as well as B + and K + invisible decays are plotted in Fig. 8, in terms of upper limits on g r , the product of the Goldstone boson coupling times the vev of the r field, versus its mass m r (cf. Eq. (17)).
The energy loss rate due to Goldstone boson production in the GRB fireball comoving frame is where f e + ( p 1 ) and f e − ( p 2 ) are the electron and positron distribution function, respectively, as given below Eq. (10). The energy of the two Goldstone bosons in the final state are denoted by ω 1 and ω 2 , while the energy of the positron and electron in the initial state by E 1 and E 2 . A symmetry factor of 1/2! is included for the two identical particles in the final state. One can perform the d 3 q 1 d 3 q 2 integral analytically analogous to the Lenard's Identity [88] for the e + e − → νν process, We use the Maxwell-Boltzmann statistics for the electron and positron distribution function, and make a change of integration variables from E 1 , E 2 , and cos θ, to Defining x ≡ E + / √ s, z ≡ s/T 2 , z r ≡ m 2 r /T 2 , z Γ ≡ Γ 2 r /T 2 , and z 0 ≡ 4m 2 e /T 2 , the energy loss rate is reduced to the simple form Q e + e − →αα = T 7 16 (2π) 5 which we evalulate numerically using the VEGAS Monte Carlo integration subroutine [89]. In the resonance region z ∼ z r , we simplify the dz integral by taking limit of the Poisson kernel lim ǫ→0 1 π ǫ a 2 + ǫ 2 = δ(a) .
Therefore the dzdx integral part can be approximated by for z ∼ z r and m r Γ r ≪ T 2 . The results for T 0 = 18 MeV and various m r , r values are shown in Fig. 2. In the resonance region, One sees that for a given m r , the Goldstone boson emissivity is enhanced significantly due to the resonance effect as Q e + e − →αα ∝ r 2 , as long as Γ r→ff ≪ Γ r→αα . In Fig. 3 we show the Goldstone boson emissivity Q e + e − →αα for other GRB initial fireball temperatures than the fiducial value T 0 = 18 MeV, such as T 0 = 8 and 2 MeV. In the resonance region, the T -dependence arises solely from the dx integral. For very large m r values away from the resonance region, the Goldstone boson emissivity depends very sensitively on the GRB fireball temperature as As will be presented in Section 5, the opacity of the GRB fireballs to the Goldstone bosons depends strongly on the Goldstone boson energy. The Goldstone boson pairs are emitted with an average energy of where ω ≡ ω 1 + ω 2 , and v M is the Møller velocity. The results for T 0 = 18 MeV and r = 1, 10, and 100 GeV are shown in Fig. 4  <r> = 1 GeV <r> = 10 GeV <r> = 100 GeV Q neutrino Figure 2: Energy loss rate due to Goldstone boson production from e − e + → αα divided by the Goldstone boson coupling g 2 vs. the radial boson mass m r . The GRB initial fireball temperature is set at the fiducial value T 0 = 18 MeV, and the vacuum expectation value of the radial boson is assumed to be r = 1, 10, and 100 GeV (from bottom to top). Also shown is the energy loss rate for neutrino production, Q e − e + →νν , at the same temperature T 0 .
where α and G F are the fine-structure constant an the Fermi constant, respectively. The form factor F γ enters through the amplitude for the SM Higgs decay to two photons, in this case a function of the centre-of-mass (cm) energy √ s in the photon collision. The cm energies attainable at the typical temperature of the initial GRB fireballs correspond to the mass of the light (sub-GeV) Higgs boson studied in Refs. [90,91]. For simplicity, we use a constant value of |F γ | 2 = 4 to approximate the result of Ref. [91]. The energy loss rate is then where x, z, z r , and z Γ are defined as in last subsection. In the resonance region, it can also be expressed in the form of Eq. (41), with Γ r→e + e − replaced by Γ r→γγ . Since the branching ratio for r → γγ is smaller than 10% of that for r → e + e − for m r ≤ 200 MeV, and becomes comparable only for m r ≃ 500 MeV, this process is always subdominant in the parameter space we consider in this work.

Nuclear Bremsstrahlung Processes
In the one-pion exchange (OPE) approximation (see e.g. Ref. [92]), there are four direct and four exchange diagrams, corresponding to the Goldstone boson pairs being emitted by any one of the nucleons. Summing all diagrams and expanding in powers of (T /m N ), the amplitude for the nuclear bremsstrahlung processes N(  with q ≡ q 1 + q 2 , and k ≡ p 2 − p 4 and l ≡ p 2 − p 3 are the 4-momenta of the exchanged pion in the direct and the exchange diagrams, respectively. Here, α π ≡ (2m N f π /m π ) 2 / (4π) ≈ 15, with f π ≈ 1 being the pion-nucleon "fine-structure" constant. Goldstone boson pairs can also be emitted from the exchanged pion, and this contribution is of the same order as Eq. (46) in the (T /m N ) expansion. We calculate the energy loss rate in the fireball comoving frame where ω 1 , ω 2 are the energy of the Goldstone bosons in the final state, and the distribution functions of the nucleons in the initial and the final state are given by f j ( p j ) = (n B /2)(2π/m N T ) 3/2 e −| p j | 2 /2m N T . The symmetry factor S is 1 4 for nn and pp interactions, whereas for np interactions it is 1. We perform the integral over the Goldstone boson momenta first where ω = ω 1 + ω 2 , and the dimensionless integral is Hereω ≡ ω 1 /ω, and θ is the angle between the two emitted Goldstone bosons. We evaluate Eq. (49) numerically using the VEGAS subroutine, and then evaluate the integral in Eq. (47) over the nucleon momenta following Ref. [93]. In the nonrelativistic limit the nucleon energies are just E j = m N + | p j | 2 /2m N . To simplify the nucleon phase space integration, one introduces the centre-of-mass momenta P , so that p 1,2 = P ± p i and p 3,4 = P ± p f , as well as z ≡ p i · p f /| p i || p f |, the cosine of the nucleon scattering angle. The integral over d 3 P can be done separately. After that one makes a change to dimensionless variables u ≡ p 2 i /m N T , v ≡ p 2 f /m N T , x ≡ ω/T , and y ≡ m 2 π /m N T . For simplicity we neglect the pion mass m π inside the curly bracket in spins |M N N →N N αα | 2 , Eq. (46), in comparison with the momentum transfer k and l. The energy loss rate is then where we have defined the integral I 0 by and the β term by With the initial comoving baryon number density in the fireball set at the fiducial value n B = 5 · 10 31 cm −3 , we find that the energy loss rate due to nuclear bremsstrahlung processes is always ∼ 10 −8 times that due to electron-positron annihilation process.

Goldstone Boson Mean Free Path in the GRB Fireball
In this section we estimate the fireball's opacity to the Goldstone bosons. The Goldstone boson mean free path in the initial GRB fireball is set by the elastic scattering on electrons and positrons α + e ± → α + e ± , as well as on nucleons α + N → α + N.

Scattering on Electrons and Positrons
The amplitude for Goldstone boson scattering on electrons and positrons α(q 1 ) e ± (p 1 ) → α(q 2 ) e ± (p 2 ) is where t = (q 2 − q 1 ) 2 = (p 1 − p 2 ) 2 . We follow Ref. [94] to calculate the reaction rate Using the polar angle cos θ ≡ p 1 · q 1 /| p 1 || q 1 | and the azimuthal angel φ ′ which is measured from the ( p 1 , q 1 )-plane, the 9-dimensional integral can be simplified to with the dimensionless variables ǫ 1 ≡ E 1 /m e , ǫ 2 ≡ E 2 /m e , and u 1 ≡ ω 1 /m e . The functions in the above equation are defined as and respectively, and the limits for the dǫ 2 integration are determined to be ǫ max, min To evaluate q 1 · p 2 , we need to know the angle where with ∆ 1 + ∆ 2 = θ. We evaluate Eq. (55) numerically using the VEGAS subroutine. In Fig. 6 we plot the αe → αe scattering rate divided by the Goldstone boson coupling g 2 , for an incident Goldstone boson energy of ω 1 = 540, 180, and 90 MeV, assuming the fiducial initial GRB fireball temperature T 0 = 18 MeV. The rates for T 0 = 8 MeV are also displayed, for Goldstone boson incident energy ω 1 = 320, 160, and 40 MeV. We find that for all Goldstone boson energies attainable in the GRB initial fireballs and all m r values, R αe→αe 4g 2 s −1 .

Scattering on Nucleons
The interaction rate for α(q 1 )N(p 1 ) → α(q 2 )N(p 1 ) can be calculated similarly as R αe→αe in Eq. (55) by replacing m e with m N and using the non-relativistic Maxwell- The amplitude squared is where g N is the effective coupling of the Goldstone bosons to nucleons. For low incident Goldstone boson energies ω 1 ≪ m N , the nuclear recoil effects can be neglected, and so the interaction rate can also be easily estimated by We found that the results from this method agree with those from the full calculation within 10% for ω 1 40 MeV. The results are shown in Fig. 7, where we assume the baryon number density in the GRB fireball is n B = 5 · 10 31 cm −3 . Although the baryon number density is four orders of magnitude smaller than that of the electrons and positrons, due to the large nucleon mass m N , this channel dominates over the scattering on electrons and positrons. The figure indicates that there is an upper bound on the scattering rate, R αN →αN 4 · 10 4 (f N g) 2 s −1 . With f N ∼ 0.3, the Goldstone boson mean free path in the initial GRB fireball is then for all m r and ω 1 values. Taking into account the current collider constraint of |g| < 0.011, we find that λ α 7.9 × 10 10 cm ≫ R 0 , for all m r values. We conclude that the Goldstone bosons produced in the initial fireball of GRBs cannot be trapped therein, i.e. the GRB initial fireballs are transparent to the Goldstone bosons. The consequence will be discussed in the next Section.

Hydrodynamics of GRB fireballs in the Presence of Goldstone Boson Production
We apply the relativistic hydrodynamics for describing quark-gluon plasma anticipated at the LHC or the Relativistic Heavy Ion Collider (RHIC) [95] (see e.g. Ref. [96] for review articles on this topic) to study the GRB fireballs.

Hydrodynamics of GRB Fireballs with Dissipation
The evolution of GRB fireballs is governed by the equations for the conservation of (baryon) particle number and for the conservation of energy and momentum, respectively. Here j ν represents an effective source term, with a negative (positive) j 0 term denoting an energy sink (source). The baryon number flux is N µ = n B u µ . For each particle species i in the fluid, one expands its phase space distribution function around the equilibrium value, as f = f 0 + δf . The deviation from the equilibrium value is related to a characteristic relaxation time. With such a correction, the stress-energy tensor is then modified to (see e.g. Ref. [97]) with u µ the four-velocity, and ∆ µν = g µν + u µ u ν the project tensor to the subspace orthogonal to the fluid velocity. Here we choose the signature of the metric to be (−, +, +, +), and the fluid four-velocity u µ is specified using the definition by Landau and Lifshitz. Following this definition, the tensor equation π µν u ν = 0 must be satisfied. The shear tensor and the bulk viscosity pressure in the lowest order of the velocity gradients are of the form respectively, with η and ζ denoting the shear and the bulk viscosity coefficient. However, as mentioned in Ref. [98], to avoid the acausality problems, the dissipative fields should be regarded as independent dynamical variables. The shear viscosity can be estimated using the Green-Kubo relation [99] (see also, e.g. Ref. [100] for a recent numerical study.) From kinetic theory, the shear viscosity coefficient is (see e.g. Ref. [101]) i.e. it is determined by particle species j in the fluid with number density n j transporting an average momentum p j over a momentum transport mean free path λ j . To solve the equation for the conservation of energy and momentum, one projects it in the direction of the fluid velocity and that orthogonal to the fluid velocity, obtaining (see e.g. Ref. [102]) and with ǫ = ǫ 0 + δǫ. These conservation equations need to be supplemented with an equation of state for the fireball plasma. When the GRB fireball expansion reaches the coasting phase, i.e. the Lorentz factor Γ is constant, one can transform to the Milne coordinates as in e.g. Refs. [103,104]. The effects of the dissipation fields are to transfer the kinetic energy into heat, while the energy source (sink) increase (decrease) the total energy. In the initial fireball of GRBs, we can assume that all particle species -the electrons and positrons, photons, as well as the protons and neutrons -are strongly coupled and thus are all in thermal equilibrium. Now consider the case that from their interactions some exotic particle species are copiously created. If they are not fully thermalised, they lead to a slower expansion of the fireball. However, in the last section we found that the Goldstone boson mean free path λ α exceeds the size of the initial fireball R 0 (cf. Eq. (63)). The Goldstone bosons produced therein are not trapped and therefore are not thermalised at all. In this case Eq. (68) is not applicable, since its validity requires λ α ≪ R 0 . The effects of the Goldstone bosons can still be estimated by transforming to the fireball comoving frame. Following the definition by Landau and Lifshitz, in this frame the terms involving π µν or π b completely vanish.

The GRB Fireball Energy Loss Criterion
In the fireball comoving frame, we demand that the Goldstone bosons transport away an amount of energy comparable to the initial fireball radiation energy before their emissivity decreases significantly with the temperature. In the GRB fireball comoving frame where the four-velocity is u ν = (1, 0, 0, 0) in spherical coordinates (t ′ , R ′ , θ ′ , φ ′ ), the hydrodynamic equations Eq. (69) and (70) are simply where the coordinates in the comoving frame and in the observer frame are related by the Lorentz factor, i.e. t ′ = t/Γ and R ′ = ΓR. Here the baryon number density n B , the energy density ǫ and the the pressure p, as well as the energy loss or creation per unit volume per unit time Q, are all comoving quantities. The Goldstone bosons are emitted isotropically in the fireball comoving frame, so the net momentum flux herein is j 1 = j 2 = j 3 = 0. One can regard the Goldstone bosons as an energy sink. Using the equation for energy conservation in Eq. (72), we can derive a constraint on the Goldstone boson emissivity in the GRB initial fireball as Choosing ∆R 0 ∼ R 0 , this criterion is equivalent to the comparison of the cooling timescale t c with the fireball expansion timescale t e in Ref. [36] In Fig. 8 we plot the upper limits on g r , the Goldstone boson coupling times the vev of the r field, versus its mass, m r , obtained by using the criterion in Eq. (73). The GRB initial fireball temperature, radius, and energy, are chosen at the fiducial value T 0 = 18 MeV, R 0 = 10 6.5 cm, and E = 10 52 erg, as well as a lower initial temperature T 0 = 8 MeV. If the temperature of the GRB initial fireball is as low Υ -> γ + r B + -> K + + r K + -> π + + r ∆a µ Υ -> γ + r B + -> K + + r K + -> π + + r ∆a µ Υ -> γ + r B + -> K + + r K + -> π + + r ∆a µ Υ -> γ + r B + -> K + + r K + -> π + + r ∆a µ Υ -> γ + r B + -> K + + r K + -> π + + r ∆a µ Υ -> γ + r B + -> K + + r K + -> π + + r ∆a µ Υ -> γ + r B + -> K + + r K + -> π + + r ∆a µ Υ -> γ + r B + -> K + + r K + -> π + + r ∆a µ . Also shown are the upper limits from muon anomalous magnetic moment ∆a µ , radiative Upsilon decays Υ(nS) → γ + r, B + invisible decay B + → K + r, as well as K + invisible decay K + → π + r (dash-dotted lines, from top to bottom.) as T 0 = 2 MeV, no constraint on the parameters of Weinberg's Higgs portal model can be obtained. In fact, the GRB bounds on g r have a slight dependence on the Goldstone boson coupling g, which becomes visible when Γ r→ff is no longer negligible compared to Γ r→αα . Here we consider g = 0.011 saturating current collider bounds, as well as g = 0.0015 which might be probed by future collider experiments. For the latter case, the upper limits are less stringent for m r > 240 MeV if T 0 = 18 MeV, or for m r > 70 MeV if T 0 = 8 MeV. An inspection of Fig. 8 indicates that in the mass range m r /T 0 10 − 15, the GRB bounds are indeed competitive to current laboratory constraints reviewed in Section 3.2. They are more stringent than the constraints from muon anomalous magnetic moment and radiative upsilon decays, while weaker than those from the B + and K + meson invisible decays by 1-3 orders of magnitude.

Summary
We aimed to study the effects of the Goldstone bosons in Weinberg's Higgs portal model on the initial fireballs of gamma-ray bursts. We first calculated the energy loss rates therein due to Goldstone boson production in different channels, including electron-positron annihilation, photon scattering, and nuclear bremsstrahlung processes. We found that resonance effects significantly enhance the energy loss rate for the electron-positron annihilation process, even for the mass of the radial field r approaching 30 − 40 times the initial GRB fireball temperature. On the other hand, in the calculation of the Goldstone boson mean free path, there is no such effect present in the processes of Goldstone boson scattering on nucleons and electrons or positrons. Interestingly, we found that although nuclear bremsstrahlung processes are of no importance in Goldstone boson production, the scattering on nucleons dominates over scattering on electrons and positrons by four orders of magnitude in setting the Goldstone boson mean free path in the GRB fireballs. However, for all Goldstone boson energies attainable in the GRB initial fireballs and all m r values, the Goldstone boson mean free path always exceeds the initial fireball radius. Thus the Goldstone bosons do not couple to the GRB fireball plasma. The initial GRB fireballs are transparent to the Goldstone bosons, so that they freely transport the fireball energy away and act as an energy sink. We obtained constraints on g r , the Goldstone boson coupling times the vacuum expectation value of the r field, by using the energy loss rate criterion derived from the hydrodynamic equations in the GRB fireball comoving frame. Assuming generic values for the GRB initial fireball temperature, radius and energy, we found that in the mass range of m r /T 0 10 − 15, the GRB bounds are indeed competitive to current laboratory constraints. They are more stringent than the constraints from muon anomalous magnetic moment and radiative upsilon decays, while weaker than those from the B + and K + meson invisible decays by 1 − 3 orders of magnitude.