The dispersion measures of fast radio bursts have been identified as a powerful tool for testing the zeromass hypothesis of the photon. The classical approach treats the massive photoninduced and plasmainduced time delays as two separate phenomena. Recently Y.B. Wang et al. suggested that the joint influence of the nonzero photon mass and plasma effects should be considered and proposed a revised time delay for massive photons propagating in a plasma medium denoted as inlineformula inlineformula which departures from the classical dispersion relation  SUP2SUP. Here we discuss the derivation presented by Y.B. Wang et al. and show that the classical dispersion relation remains valid based on Proca equations.
['Radio transient sources', 'Intergalactic medium', 'Particle astrophysics', '2008', '813', '96', 'Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Phenomenology']
The Dispersion Relation of Massive Photons in Plasma A Comment on Bounding the Photon Mass with Ultrawide Bandwidth Pulsar Timing Data and Dedispersed Pulses of Fast Radio Bursts
{"The Dispersion Relation of Massive Photons in Plasma: A Comment on 'Bounding the Photon Mass with Ultrawide Bandwidth Pulsar Timing Data and Dedispersed Pulses of Fast Radio Bursts'": 'BAO WANG 1, 2 AND JUN-JIE WEI 1, 2  \n1  \nPurple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, China 2 School of Astronomy and Space Sciences, University of Science and Technology of China, Hefei 230026, China', 'ABSTRACT': "The dispersion measures of fast radio bursts have been identified as a powerful tool for testing the zero-mass hypothesis of the photon. The classical approach treats the massive photon-induced and plasma-induced time delays as two separate phenomena. Recently, Wang et al. (2024) suggested that the joint influence of the nonzero photon mass and plasma effects should be considered, and proposed a revised time delay for massive photons propagating in a plasma medium, denoted as ∆ t ' m γ ∝ ν -4 , which departures from the classical dispersion relation ( ∝ ν -2 ). Here we discuss the derivation presented by Wang et al. (2024) and show that the classical dispersion relation remains valid based on Proca equations.  \nKeywords: Radio transient sources (2008) - Intergalactic medium (813) - Particle astrophysics (96)", '1. INTRODUCTION': "Since the discovery of fast radio bursts (FRBs), numerous studies have employed the dispersion measures (or equivalently, the frequency-dependent time delays) of FRBs to place upper limits on the rest mass of the photon ( m γ ; Wu et al. 2016; Bonetti et al. 2016, 2017; Shao & Zhang 2017; Xing et al. 2019; Wei & Wu 2020; Wang et al. 2021; Lin et al. 2023; Wang et al. 2023; Ran et al. 2024). In the classical treatment, two separate time delays are considered: the plasma-induced time delay ( ∆ t d ∝ ν -2 ) and the massive photon-induced time delay ( ∆ tm γ ∝ ν -2 ). These two contributions are considered to be directly additive. However, in a recent work, Wang et al. (2024) proposed a revised time delay, denoted as ∆ t ' m γ ∝ ν -4 , which is attributed to the joint influence of the nonzero photon mass and plasma effects. In this comment, we point out the contradictions in the revised time delay proposed by Wang et al. (2024), and derive the dispersion relation of massive photons propagating in plasma based on Proca equations (Proca 1936), thereby proving the validity of the classical time delay ( ∝ ν -2 ).", "2. COMMENT ON TIME DELAY ∆ t ' m γ ∝ ν -4": "In this section, we follow the derivation presented in Wang et al. (2024) to examine the revised time delay. We first define four types of arrival times:  \n- (i) The arrival times of massless photons in vacuum and plasma are t 1 and t 2, respectively.\n- (ii) The arrival times of massive photons in vacuum and plasma are t ' 1 = ∫ 1 υ γ g d l and t ' 2 = ∫ 1 υ new g d l , respectively, where υ γ g and υ new g are the group velocities of massive photons in vacuum and plasma, respectively.  \nAccording to Wang et al. (2024), the observed time delay between massive photons with different frequencies ( ν low, ν high; ν low <ν high) in a plasma medium is expressed as  \n∆ t ' obs = ( t ' 2 , low -t ' 1 , low ) -( t ' 2 , high -t ' 1 , high ) = ∫ ( 1 υ new g , low -1 υ γ g , low ) d l -∫ ( 1 υ new g , high -1 υ γ g , high ) d l ≃ A 1 ( 1 ν 2 low -1 ν 2 high ) DM + A 2 ( 1 ν 4 low -1 ν 4 high ) DM , (1)  \nwhere A 1 = e 2 2 π mec , A 2 = e 2 c 3 m 2 γ 4 π meh 2 , and e and me are the charge and mass of an electron, respectively. Here DM = ∫ ne d l is the dispersion measure contribution from the plasma, which  \nis defined as the integral of the number density of electrons ne along the line of sight. The second term in the last row of Equation (1) is regarded as the time delay arises from the combined effects of plasma and nonzero photon mass, which has the form of ∝ ν -4 that differs from previous results ( ∝ ν -2 ; Wu et al. 2016; Shao & Zhang 2017).  \n̸  \nIt is clear that Equation (1) (see also Equation (3) in Wang et al. 2024) presents a contradiction: when the electron number density ne is equal to zero, the time delay is eliminated. This indicates that there is no dispersion for massive photons in vacuum. Figure 1 illustrates the timelines for calculating the time delays in two scenarios: massless photons and massive photons. In the scenario of massless photons in plasma, the observed time delay between low- and high-frequency photons is ∆ t obs = t 2 , low -t 2 , high = ( t 2 , low -t 1 , low) -( t 2 , high -t 1 , high), where t 1 , low = t 1 , high due to the constancy of the light speed c in vacuum. However, for massive photons, the observed time delay is ∆ t ' obs = t ' 2 , low -t ' 2 , high = ( t ' 2 , low -t ' 1 , low ) -( t ' 2 , high -t ' 1 , high ). The additional subtractions of the arrival times of massive photons in vacuum (i.e., t ' 1 , low and t ' 1 , high ) in Wang et al. (2024) cause the elimination of the ν -2 behavior. If we use the correct formula, then the firstorder delay time term would exhibit a frequency dependence of ν -2 , i.e.,  \n∆ t ' obs = t ' 2 , low -t ' 2 , high = ∫ ( 1 υ new g , low -1 υ new g , high ) d l ≃ A 1 ( 1 ν 2 low -1 ν 2 high ) ( DM + DM γ ) , (2)  \nwhere the group velocity υ new g is expressed as  \nυ new g = c ( 1 -m 2 γ c 4 h 2 ν 2 ) 1 / 2 ( 1 -ν 2 p ν 2 ) 1 / 2 ≃ c ( 1 -1 2 ν 2 p ν 2 -1 2 m 2 γ c 4 h 2 ν 2 ) , (3)  \nwhere ν p = ( nee 2 /π me ) 1 / 2 is the plasma frequency. Analogous to DM, the 'effective dispersion measure' (DM γ ) arising from massive photons is defined as DM γ ≡ π mec 5 m 2 γ h 2 e 2 ∫ d l (Shao & Zhang 2017). Equation (2) shows that the classical dispersion relation ( ∝ ν -2 ) for massive photons propagating in plasma still holds.", '3. THE DISPERSION RELATION DERIVED FROM PROCA EQUATIONS': 'The classical Maxwell equations are founded upon the zero-mass hypothesis of the photon. Proca (1936) first considered the addition of a mass term and modified the Maxwell equations, thereby establishing the Proca equations. The', 'Massless': 'Figure 1. Timeline schematic for calculating the time delays. The blue and red timelines illustrate the scenarios of massless and massive photons, respectively. The definitions of arrival times can be referred to section 2.  \n<!-- image -->  \nProca Lagrangian is given as (Jackson 2021)  \nL Proca = -1 16 π F αβ F αβ -1 c J α A α + µ 2 8 π A α A α . (4)  \nIn this Lagrangian, F αβ is the electromagnetic tensor, A α = ( ϕ, A ) is the 4-vector potential, J α = ( cnee , J ) is the 4-vector current, and µ = m γ c / ¯ h is the parameter related to the photon mass. According to the Proca Lagrangian, the equation of motion can be derived as ∂ β F βα + µ 2 A α =4 π J α / c . Combined with the Bianchi identity ∂ ( µ F αβ ) = 0, the vector form of the Proca equations can be written as  \n∇· E = 4 π nee -µ 2 ϕ, (5)  \n∇× E = -1 c ∂ B ∂ t , (6)  \n∇· B = 0 , (7)  \n∇× B = 4 π c J + 1 c ∂ E ∂ t -µ 2 A . (8)  \nOnce we establish the Proca equations, the dispersion relation of massive photons traveling in plasma can be derived. We first separate the variables into two parts: an equilibrium part indicated by the subscript 0, and a perturbation part indicated by the subscript 1. In a plasma with B 0 = J 0 = E 0 = A 0 = 0, the linear approximation allows Equation (8) to be converted to  \nc ∇× B1 = 4 π J1 + ∂ E1 ∂ t -c µ 2 A1 . (9)  \nBy taking the time derivative of this equation, we obtain  \nc ∇× ∂ B1 ∂ t = 4 π ∂ J1 ∂ t + ∂ 2 E1 ∂ t 2 -c µ 2 ∂ A1 ∂ t . (10)  \nThe subsequent steps are to transform each term into the function of E1 . Taking the curl of Equation (6), we have  \n∇× ∂ B1 ∂ t = -c ∇× ( ∇× E1 ) = c ∇ 2 E1 -c ∇ ( ∇· E1 ) = c ∇ 2 E1 + c µ 2 ∇ ϕ 1 , (11)  \nwhere the condition ∇· E1 = -µ 2 ϕ 1 from Equation (5) is used. Since the current is generated by electron motion, by combining the expressions J1 = nee v1 and me ∂ v1 /∂ t = -e E1 , the time derivative of the current can be expressed as  \n∂ J1 ∂ t = -nee 2 me E1 . (12)  \nWealso need the relation between potentials ( ϕ 1 , A1 ) and E1 :  \n∂ A1 ∂ t = -c E1 -c ∇ ϕ 1 , (13)  \nwhich can be derived from the time derivative of B1 = ∇× A1 and Equation (6). We assumed that the perturbation parts oscillate sinusoidally, i.e., ∝ exp[ i ( kx -ω t )], where k and ω are the wave vector and frequency, respectively. Hence, the time derivative and space gradient can be replaced as ∂/∂ t → -i ω and ∇→ ik . Combining Equations (11)-(13), Equation (10)  \ncan be rewritten as  \n( ω 2 -c 2 k 2 -ω 2 p -c 2 µ 2 ) E1 = 0 , (14)  \nwhere ω p = 2 πν p = (4 π nee 2 / me ) 1 / 2 is the plasma frequency. Requiring the terms in the bracket being equal to zero, one has  \nω 2 = c 2 k 2 + ω 2 p + c 2 µ 2 . (15)  \nThis is the dispersion relation of massive photons propagating in plasma. Furthermore, the group velocity of massive photons can be calculated by  \nυ g = ∂ω ∂ k = c ( 1 -ω 2 p + c 2 µ 2 ω 2 ) 1 / 2 ≃ c ( 1 -1 2 ω 2 p ω 2 -1 2 c 2 µ 2 ω 2 ) . (16)  \nOne can see from Equation (16) that the velocity modifications from the plasma and nonzero photon mass effects are independent of each other, with the first-order expansion terms being ∝ ν -2 . This is the natural consequence based on the Proca theory. Therefore, the classical dispersion relation that is adopted for photon mass limits remains valid.  \nThis work is supported by the NSFC (grant Nos. 12422307 and 12373053). 1 2', 'REFERENCES': 'Bonetti, L., Ellis, J., Mavromatos, N. E., et al. 2016, Physics  \nLetters B, 757, 548, doi: 10.1016/j.physletb.2016.04.035  \n-. 2017, Physics Letters B, 768, 326,  \ndoi: 10.1016/j.physletb.2017.03.014  \nJackson, J. D. 2021, Classical electrodynamics (John Wiley &  \nSons)  \nLin, H.-N., Tang, L., & Zou, R. 2023, MNRAS, 520, 1324,  \ndoi: 10.1093/mnras/stad228  \nProca, A. 1936, J. Phys. 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