Impact of magnetic field on neutrino-matter interactions in core-collapse supernova

We explore the impact of magnetic field on neutrino-matter interactions in core-collapse supernova. We first derive the modified source terms for neutrino-nucleon scattering and neutrino absorption and emission processes in the moment formalism. Then we perform full relativistic three-dimensional, magnetorotational core-collapse supernova simulations of a 20 $M_\odot$ star with spectral neutrino transport. Our simulations treat self-consistently the parity violation effects of magnetic field on the lepton number, energy, and momentum exchanges. The result shows a clear global asymmetry with respect to the equatorial plane in each exchange rate. The asymmetric property arises from two factors: the angle between the neutrino flux and magnetic field and the term, which is parallel to the magnetic field and is also proportional to the deviation of distribution function of neutrinos from thermal equilibrium. The typical correction value amounts to $\sim1$ % relative to the total neutrino-matter interaction rate for the magnetic field strength of $\sim 10^{15-16}$ G. Although these asymmetric properties do not immediately affect the explosion dynamics, our results imply that they would be significant once the neutrinos diffuse out the proto-neutron star core carrying those asymmetries away. We also show that, during our simulation time of $\sim370$ ms after bounce, our results indicate that the correction value due to the modified inelastic scattering process dominates over that of the modified neutrino absorption and emission process.


INTRODUCTION
Massive stars heavier than ∼ 8 M ⊙ terminate their lives with a catastrophic collapse of their central core and subsequent diverse phenomena. Some of them are accompanied by a huge explosion called core-collapse supernova (see Janka et al. 2016;Müller 2016;Radice et al. 2018, for recent reviews). There are currently two major explosion mechanisms suggested. The first one relies on the complex interplay between neutrinos and stellar mantle and is considered to account for CCSNe with a canonical explosion energy of ∼ 10 51 ergs(≡ 1 Bethe, 1B in short). The other is called magnetorotational (MR) explosion that takes place if the progenitor star rotates sufficiently fast and is also magnetized (Bisnovatyi-Kogan 1970;LeBlanc & Wilson 1970;Meier et al. 1976;Müller & Hillebrandt 1979).
The MR explosion is intrinsically an asymmetric phenomenon.
It is characterized by collimated bipolar outflows (Ardeljan et al. 2000;Burrows et al. 2007;Takiwaki et al. 2009;Scheidegger et al. 2010;Winteler et al. 2012;Obergaulinger & Aloy 2020a;Bugli et al. 2020). However the recent full three-dimensional (3D) magnetohydrodynamics (MHD) simulations report also less collimated, slightly weaker outflows (Mösta et al. 2014;Obergaulinger & Aloy 2020a;Kuroda et al. 2020). The bipolar structure is originated from magnetic field amplification taking place mainly along the rotational axis. Strongly amplified magnetic fields eventually eject matter toward the rotational axis. Another aspect of the MR explosion is its relatively high explosion energy compared to the aforementioned neutrino heating-driven explosion. Depending on the available differential rotational energy of proto-neutron star (PNS), the magnetic field can be very efficiently amplified up to the equipartition level, i.e., roughly the same as the differential rotational energy. Due to the efficient conversion, typically one order of magnitude larger explosion can be achieved in the previous MHD CCSN models (Burrows et al. 2007;Takiwaki et al. 2009;Obergaulinger & Aloy 2020a;Kuroda et al. 2020).
Combination of these two aspects makes the MR explosion as a plausible mechanism of a subclass of CCSNe called hypernova (HN), which presents high explosion energy of ∼ 10 K B (see, e.g., Nomoto et al. 2006, and references therein). Observationally, HNe are also often accompanied by a bipolar structure (e.g., Ezzeddine et al. 2019) that further supports the MR explosion as a mechanism of HNe. The asymmetric property of MR explosion also attracts considerable attention as a kick mechanism of compact object. All CCSNe leave behind a compact object either a neutron star (NS) or a black hole and these compact objects are occasionally observed with a proper motion. While observational results of the BH kick velocities are still ambiguous (see, e.g., Repetto et al. (2012) for high-velocity cases or Mandel (2016) for counterarguments), those of NSs show significant velocities of several hundred to 1000 km s −1 (Hobbs et al. 2005;Winkler & Petre 2007). Such a significant proper motion is likely produced during CCSNe via two possible mechanisms (see, e.g., Lai 2001, for a review). The first one is via asymmetric matter motion and the other is via asymmetric neutrino emission. Regarding the former mechanism, most of the recent multi-D simulations of CCSNe indicate that the explosion takes place asymmetrically (for both 2D and 3D models, see, e.g., Bruenn et al. 2016;Takiwaki et al. 2016;Müller et al. 2017;O'Connor & Couch 2018;Pan et al. 2018;Summa et al. 2018;Ott et al. 2018;Nagakura et al. 2019;Vartanyan et al. 2019). This is also the case for MHD simulations (Scheidegger et al. 2010;Winteler et al. 2012;Mösta et al. 2014;Obergaulinger & Aloy 2020a;Bugli et al. 2020;Kuroda et al. 2020). In addition, there are observational supports for these asymmetric CCSN models (e.g., Katsuda et al. 2018). The asymmetric explosion in these numerical simulations is often dominated by low-order spherical harmonics modes. Then the central NS is considered to recoil from conservation of linear momentum and the estimated kick velocity reaches a few 100 to ∼ 1000 km s −1 (Wongwathanarat et al. 2013;Janka 2017;Chan et al. 2018;Müller et al. 2019;Nakamura et al. 2019).
Some progenitor stars, particularly low-mass stars, however, show nearly symmetric ejecta which are inadequate to account for the observed kick velocity (Gessner & Janka 2018). In such cases, asymmetric neutrino emission can be another possible origin of the kick velocity. Analogous to the anisotropic matter ejection, neutrinos exchange momentum with matter and, if the net exchange does not vanish, the PNS is accelerated. While some previous studies report the acceleration due to the anisotropic neutrino emission is minor and can explain the kick velocity of only up to a few 100 km s −1 (Wongwathanarat et al. 2013;Tamborra et al. 2014;Gessner & Janka 2018;Müller et al. 2018), Nagakura et al. (2019) report that its effect is moderate due to self-sustained partial distribution of the electron fraction (Y e ).
The net momentum that the central compact object gains depends on the degree of asymmetry of neutrino emis-sion and also on its duration time. Regarding the former, the lepton-number emission self-sustained asymmetry (Tamborra et al. 2014;O'Connor & Couch 2018;Glas et al. 2019;Vartanyan et al. 2019) is currently one of the relevant mechanisms to induce the asymmetric neutrino emission. According to Tamborra et al. (2014), the LESA can explain the kick velocity of 100-200 km s −1 . The degree of asymmetry of neutrino emission can still increase if we take into account the magnetic field. In the presence of magnetic field, a global asymmetry could appear as a consequence of parity violation effect in the neutrino-matter interactions, e.g., neutrino scattering on slightly polarized free nucleons due to magnetic field or Landau quantization of free electrons.
There have been several works studying the parity violation effects of magnetic field on the neutrino-matter interactions in CCSNe. Bisnovatyi-Kogan (1993) investigated the influence of magnetic field on β-process and derived kick velocity between ∼ 100 and ∼ 3000 km s −1 . Horowitz & Li (1998) explored the effect of elastic scattering of neutrinos on slightly polarized free neutrons due to magnetic field and estimated the magnetic field strength B of ∼ 10 14 G to account for the kick velocity of ∼ 250 km. Later Arras & Lai (1999) (hereafter AL99) pointed out the importance of inelasticity in the scattering process on free nucleons and also of correct treatment of detailed balance of neutrinos in thermal equilibrium. Taking these modifications into account, they formulated the scattering cross section as well as the absorption one and showed that a dipole magnetic field of 10 15−16 G is required to generate a kick velocity of a few hundred km s −1 . Kotake et al. (2005) was the first to implement the effect of parity violation on β-process into the MHD simulation of CCSN and showed the excess/reduction of heating rate of ∼ 0.5 %. Maruyama et al. (2011Maruyama et al. ( , 2012 calculated the neutrino absorption and cross sections in the context of relativistic mean field theory and derived relatively high magnetic field of ∼ 10 17 G to explain the kick velocity of ∼500-600 km s −1 . In addition to the NS kick, the parity violation effect could also contribute to the pulsar spin evolution (Maruyama et al. 2014;Suwa & Enoto 2014).
In this paper, we aim to implement the self-consistent parity violation effects of magnetic field on the lepton number, energy, and momentum exchanges to full relativistic twomoment (M1) neutrino transport code. For that purpose, we first rewrite the absorption and cross sections described in AL99 in moment formalism. Then we apply those modified source terms to full 3D-GR, MR core-collapse simulations of a 20 M ⊙ star with spectral neutrino transport. We calculate three models: nonrotating nonmagnetized, rotating strongly magnetized, and rotating ultra-strongly magnetized models. We also update the neutrino opacities following Kotake et al. (2018). Our results clearly show fingerprints of parity violation effect and exhibit a global asymmetry with respect to the equatorial plane, which is due to the initially dipole like magnetic field employed. Although these asymmetric features do not immediately affect the explosion dynamics, our results imply that they would be significant in the diffusion time scale of neutrinos. In addition, we also show the importance of the modified inelastic scattering process, which has been often omitted in previous literature, relative to the modified neutrino absorption and emission process. This paper is organized as follows. Section 2 starts with a concise summary of our GR MHD neutrino transport scheme. In Section 3, we shortly explain the scattering and absorption cross sections in the presence of magnetic field and also rewrite them suitable for a moment formalism. We describe the initial setup of the simulation together with the updated neutrino opacities in Section 4. The main results and detailed analysis of the effects of parity violation are presented in Section 5. We summarize our results and conclude in Section 6. In Appendix A, we prove that the modified scattering term does not violate the lepton number conservation. Note that the geometrized unit is used in Section 3 unless otherwise stated, i.e., the speed of light, the gravitational constant, and the Planck constant are set to unity, c = G = h = 1, and cgs units are used in Section 5. The metric signature is (−, +, +, +). Greek indices run from 0 to 3 and Latin indices from 1 to 3, except ν and ε that denote neutrino species and energy, respectively.

BASIC ν-GRMHD EQUATIONS
In our full GR radiation-MHD simulations, we solve the evolution equations of metric, MHD, and energy-dependent neutrino radiation. Each of metric and radiation-MHD parts is solved in an operator-splitting manner, but the system evolves self-consistently as a whole satisfying the Hamiltonian and momentum constraints. In this paper, we omit to show the evolution equations for the metric and MHD parts, which can be found in our former papers Kuroda et al. (2016Kuroda et al. ( , 2020. We only recapitulate our basic neutrino transport equations below. We solve the evolution equations for the zeroth E ε and first order radiation moments F α ε measured by an Eulerian observer, with ε representing the neutrino energy measured in the comoving frame. The evolution equations read (Shibata et al. 2011) and (2) Here, α, β i , γ ij , and K ij are the lapse function, shift vector, three metric, and extrinsic curvature, respectively. γ ≡ det(γ ij ) is the determinant of the three metric, P ij is the second order radiation moment measured in the Eulerian frame and is given by an analytic closure relation (see Kuroda et al. (2016)), and S µ ε is the source term for neutrino matter interactions. According to Shibata et al. (2011), the four source term S µ ε in the moment formalism can be expressed as where B(ε, Ω) is the source term for the distribution function of neutrinos f (ε, Ω) with Ω representing the angular dependence. l µ is a unit normal four vector orthogonal to the four velocity u µ and is also used to obtain the radiation moments by angular integration of the distribution function as follow J ε , H α ε , and L αβ ε are the zeroth, first, and second order radiation moments measured in the comoving frame, respectively. In the Doppler and red-shift term,M µ ε is defined bỹ denotes the third rank moment of neutrino distribution function (see Shibata et al. 2011, for more detailed expression). In the next section, we explain how B(ε, Ω) is modified in the presence of magnetic field and then rewrite it using Eq. (3) to obtain the modified four source term which is used in the M1 neutrino transport equations (1)-(2).

NEUTRINO SOURCE TERMS IN THE PRESENCE OF MAGNETIC FIELD
We begin with a brief description of the modified neutrino source term B(ε, Ω) in the presence of magnetic field. We refer the reader to AL99 for more detailed background. In the followings, we focus on the impacts of magnetic field on the two dominant neutrino matter interactions inside the PNS: the neutrino-nucleon scattering and the absorption and emission processes on free nucleons. We now explain each process one by one.

Modified neutrino-nucleon scattering rate
The neutrino scattering rate on free nucleons B sc (ε, Ω) can be written as Here T is the matter temperature and dΓ/(dε ′ dΩ ′ ) is the differential cross section. Variables with ′ denote those of outgoing neutrinos.

K
Modification of the scattering rate due to the magnetic field appears in the differential cross section dΓ/(dε ′ dΩ ′ ) in terms of µ B b. Here µ B and b are the magnetic moment and magnetic field strength, respectively. We use the value b = √ b α b α for the magnetic field strength with b α being the comoving four magnetic field. The typical value of |µ B b| in CCSNe is O(10 −2 ) MeV, even when one considers a strongly magnetized neutron star with b ∼ 10 15 G (magnetar: Duncan & Thompson 1992). Such a value is significantly smaller than the matter temperature T 10 MeV and we can safely neglect the second or higher order terms of µ B b in the expansion of dΓ/(dε ′ dΩ ′ ). Then it reads (see AL99) for electron type neutrino scattering ν e N → ν e N , and for electron type anti-neutrino scatteringν e N →ν e N .
Here A 0 and δA ± are the zeroth and first order terms of Taylor expansion of dΓ/(dε ′ dΩ ′ ) about b.b α is a unit vector parallel to b α . µ ′ is the scattering angle between the incoming and outgoing neutrinos. From Eqs. (6)-(7), one can rewrite the scattering rate B sc (ε, Ω) as a sum of normal scattering rate B b=0 sc (ε, Ω), which is independent on the magnetic field, and correction term B b =0 sc (ε, Ω), which is proportional to b, as B sc (ε, Ω) = B b=0 sc (ε, Ω) + B b =0 sc (ε, Ω).
For reference, the correction terms δA ± (ε, ε ′ , µ ′ , b) are expressed in cgs units as (AL99) for scattering on N , where N takes either n(neutron) or p(proton). In Eq. (9), G F is the Fermi constant with its explicit value being G 2 F = G 2 F c/( c) 4 = 1.55 × 10 −33 [cm 3 MeV −2 s −1 ]. h N V and h N A are the neutral nucleon current form factors and we adopt the same values used in Bruenn (1985); Kuroda et al. (2016). µ N B = g N e /(2m N c) is the nucleon magnetic moment with g n = −1.913 and g p = 2.793, − µN T , and z = q 0 /T . Hereafter, we omit the arguments (ε, ε ′ , µ ′ , b) in δA ± for simplicity.

Moment formalism of the modified neutrino-nucleon
scattering rate In the moment formalism, the scattering term reads S α ε,sc = S α,b=0 ε,sc where For the scattering term without the contribution of magnetic field S α,b=0 ε,sc , we take into account only the isoenergetic scattering. Here, we note that scattering on free nucleon is not a perfect elastic system, especially when it is not in thermal equilibrium. However, the inelastic correction term is still minor in CCSNe (Wang & Burrows 2020) and, furthermore, to maintain the consistency with our previous studies, we simply use the isoenergetic kernel this time. It is written as where χ iso ε is expressed in terms of the isoenergetic scattering kernel (see Appendix 2 in Kuroda et al. 2016, for the explicit expression). Now we move on to how the source term B b =0 sc (ε, Ω) can be expressed in the moment formalism. We note that, in the following, the modified source term S α,b =0 ε,sc arising from the magnetic field fully considers the inelasticity, which is omitted in the term S α,b=0 ε,sc . This is because that the contribution from inelasticity becomes sometime dominant in the final source term S α,b =0 ε,sc . This is particularly true for neutrinos with energy ε 4T (∼ 10 − 80 MeV for T ∼ 3 − 20 MeV) (see the discussion around Eq. (4.31) in AL99), i.e., most of neutrinos inside the PNS. Therefore the inelasticity is crucial in the parity-violation term.
In the following, we consider the neutrino-nucleon scattering. For the anti-neutrino-nucleon scattering, we simply switch δA + and δA − in the equations below. The first term in the right hand side of Eq. (11), which is parallel to u α , corresponds to the zeroth order moment of the source term. Introducing the following same notations used in AL99 (see their Eqs. 4.14-4.15) and where f eq ε = 1/(1 + exp((ε − µ ν )/T )) represents the Fermi distribution function of neutrino with µ ν being the chemical I -5 potential, the zeroth order term becomes Here, we define and To derive Eq. (15), we approximate the neutrino distribution function after scattering into isotropic and nonisotropic parts as where f 0 ε ′ and f 1,α ε ′ do not have angle dependency. They are related to the zeroth and first order radiation moments in comoving frame as Similarly, the first order moment of the source term can also be expressed in terms of the radiation moments as where In the equation,L and Consequently, the summation of Eqs. (15) and (21) yields the final expression of the source term Eq. (11) expressed as In the Appendix A, we prove that this source term does not violate the lepton number conservation. In our practical calculation, we prepare a table of the following values in advance for ε(ε ′ ), which is the neutrino's energy grid used, and for typical values of chemical potential of free nucleons µ N and matter temperature T in CCSNe. In the above equations, we factored out the magnetic field dependence µ N B b from the tabulated value, which can be incorporated later in the simulation from the local value. During the simulation, we interpolate the values A 0,1 along µ N and T directions at each energy grid ε and ε ′ , multiply them by µ N B b, and evaluate Eqs. (16), (17), (22), and (26).

Modified charged current reactions with free nucleons
In this section, we briefly recapitulate how the magnetic field alter the charged current processes (ν e n ⇋ e − p and ν e p ⇋ e + n) based on AL99. In the presence of magnetic field, the energy of electron and positron is quantized, namely Landau quantization. Due to this quantization, the Fermi-Dirac distribution of final state electron and positron is also affected. According to AL99, the absorptivity 1/λ is modified due to the external magnetic field as Here λ −1 0 is the absorptivity without the influence of magnetic field. ǫ mc is the correction factor due to magnetic field and expressed as for the reaction ν e n → e − p, and for the reactionν e p → e + n. The correction terms ǫ mc (e − ) and ǫ mc (e + ) are originated from electrons and positrons at ground-state Landau level. While, ǫ mc (np) andǭ mc (np) are from polarized neutrons and protons.

K
All these terms have a linear dependence on the magnetic field strength b as follows (in cgs units): and In the above equations, g V = 1 and g A = 1.23 are the nucleon charged current form factors (Bruenn 1985), f x (ε) = [1 + exp((ε − µ x )/T)] −1 represents the Fermi-Dirac distribution function of fermion x with energy ε and chemical potential µ x , and Q = m n − m p = 1.2935 MeV is the rest mass difference of the neutron and proton. From the modified absorptivity 1/λ, the emissivity j can be obtained by with µ νe = −µν e = µ e −µ p +µ n being the chemical potential of neutrinos in thermal equilibrium with matter. Then the collision rate for neutrino absorption and emission processes (nae) becomes as with κ = j + λ −1 being the stimulated absorption opacity.
3.4. Moment formalism of the charged current reaction including magnetic field correction Now, analogously to Sec. 3.2, we evaluate the charged current source term in moment formalism including the correction term due to the magnetic field. After performing the angular integral dΩ of the collision rate for neutrino absorption and emission processes, it results again in a summation of normal absorption term S α,b=0 ε,nae and correction term S α,b =0 ε,nae as Here

INITIAL MODELS AND NEUTRINO OPACITIES
Utilizing our ν-GRMHD code including the parity violation source term described above, we perform full 3D CCSN simulations of a magnetized rotating star. Numerical setup is essentially the same as that of our previous paper Kuroda et al. (2020) other than the neutrino opacity. We study the frequently used solar-metallicity model of the 20 M ⊙ star "s20a28n" from Woosley & Heger (2007). For the nuclear EOS, we use SFHo of Steiner et al. (2013). The 3D computational domain is a cubic box with 3 × 10 4 km width in which nested boxes with 10 refinement levels are embedded. Each box contains 64 3 cells1 and the minimum grid size near the origin is ∆x = 458m. The neutrino energy space ε logarithmically covers from 1 to 300 MeV with 12 energy bins.
We shortly mention the updated neutrino opacities. In Kuroda et al. (2020), we adopted the standard weak interaction set in Bruenn (1985) plus nucleon-nucleon Bremsstrahlung (Hannestad & Raffelt 1998). In this study, however, we use up-to-date neutrino opacities based on Kotake et al. (2018) in addition to the correction terms arising from the external magnetic field described in the previous section. One of differences is that we replace the electron capture rate on heavy nuclei by the most elaborate one following Juodagalvis et al. (2010). Furthermore, we also take into account following corrections: inelastic contributions and weak magnetism corrections (Horowitz 2002), the density-dependent effective nucleon mass (Reddy et al. 1999), the quenching of the axial-vector coupling constant (Carter & Prakash 2002;Fischer 2016), many-body and virial corrections (Horowitz et al. 2017), and strangeness contribution to the axial-vector coupling constant (Horowitz 2002). We therefore employ all opacities listed in Table 1 in Kotake et al. (2018) with excluding set3 and set4. The source term can now be summarized as Here, the subindices "nes", "tp", and "brem" stand for the neutrino-electron inelastic scattering, thermal neutrino pair production and annihilation, and nucleon-nucleon bremsstrahlung, respectively. We note that this paper does not focus on measuring the impact of these up-to-date neutrino opacities currently used, i.e., Eq. (43), which will be reported elsewhere (see Kotake et al. 2018, for a detailed 1D comparison). The original progenitor model "s20a28n" assumes neither rotation nor magnetic field during its evolution phase. We thus artificially add them to the nonrotating progenitor model as follows: for the rotational profile, and for the magnetic field in the form of vector potential. Here u φ ≡ u 2 x + u 2 y and ̟ 0 and Ω 0 indicate the size and angular frequency of a rigidly rotating central cylinder, respectively. B 0 and R 0 represent the magnetic field strength at center and the size of central sphere with uniform magnetic field, respectively. (r, θ, φ) denote the usual coordinates in the spherical polar coordinate system. By defining the vector potential on the numerical cell edge and taking their curl B = ∇ × A, the magnetic field defined on the numerical cell surface automatically satisfies the solenoidal constraint. These rotational and magnetic field profiles are identical to those used in our former paper Kuroda et al. (2020).
We set ̟ 0 = R 0 = 10 8 cm corresponding roughly to the iron core size at the precollapse stage. We calculate three models with changing (Ω 0 [rad s −1 ], B 0 [G]) as (0, 0), (1, 10 12 ), and (1, 10 13 ), hereafter labeled as R0B00, R1B12, and R1B13, respectively. The angular frequency Ω 0 = 1 rad s −1 is very reasonable compared to the one of a rotating 20 M ⊙ model in Heger & Langer (2000) that gives Ω 0 ∼ 3 rad s −1 . Regarding the magnetic field strength inside the iron core at pre-collapse stage, there is currently no constraint from observational and stellar evolution calculation sides. The value B 0 = 10 12 G in model R1B12 is widely used in most of previous MRE simulations (Burrows et al. 2007;Takiwaki et al. 2009;Scheidegger et al. 2010;Mösta et al. 2014;Obergaulinger & Aloy 2020a;Bugli et al. 2020;Kuroda et al. 2020) and can be a reference case. On the other hand, B 0 = 10 13 G is enormously strong and might be unrealistic. We, however, calculate such an ultra-strongly magnetized model to measure the impact of magnetic field on the neutrino-matter interactions more easily, since the source terms Eq. (26) and (41) have a linear dependence on the magnetic field strength b.

RESULTS
In this section, we present the results of our simulations. We begin by a detailed explanation of explosion dynamics of our fiducial model R1B12 and then compare three models including neutrino profiles. Then we discuss our main results on the actual impact of magnetic field on the neutrino matter interaction and its possible effects on the dynamics.

Explosion dynamics and neutrino signals
We explain first the dynamical evolution of our fiducial model R1B12. Its evolution is essentially the same as our previous study employing the same initial condition (model R1B12 in Kuroda et al. 2020) and the bipolar outflow is launched soon after bounce. Figure 1 shows the volumerendered 3D entropy structure for model R1B12 at two different time slices t pb = 183 ms (top-left panel) and 367 ms (topright), where t pb represents the postbounce time, and 2D contours of entropy (bottom-left) and plasma β(≡ P mag /P gas ), which is the ratio of magnetic to gas pressure, in logarithmic scale (bottom-right) at the final simulation time. The white vertical line in the top panels indicates the length scale and is also parallel to the rotational axis (z-axis). There are three minipanels in the bottom two panels. Each minipanel shows a 2D slice on y = 0 (minipanel a), x = 0 (b), and z = 0 (c) planes. From the figure, we see a clear bipolar shock structure which continuously expands without a stall. The size of the shock surface increases from ∼ 1000 km to 4000 km for the time interval of ∼ 180 ms. Model R1B12 is thus considered to be entering the shock runaway phase directly after bounce.
Its explosion morphology exhibits a clear bipolar-like structure with a slight asymmetry with respect to the equatorial plane. From bottom minipanels (a,b), the expansion toward the positive z-axis is more energetic than the negative direction. The bipolar structure consists of the high entropy ejecta and the entropy increases with time. The forefront of the bipolar jet shows the highest entropy of s ∼ 15 k B baryon −1 at t pb ∼ 183 ms, while the vicinity of the base of jets shows the highest value exceeding s ∼ 20 k B baryon −1 at t pb ∼ 367 ms (yellowish region in the top-right panel). Furthermore, the jet barycenter shows a clear displacement from the rotational axis, like a helical structure seen in the top-left panel, indicating the appearance of the kink instability (e.g., Begelman 1998). This is consistent with the previous full 3D MHD CCSN simulations (Mösta et al. 2014;Kuroda et al. 2020). From the bottom-right panel, we see that the magnetic pressure dominates over the gas pressure inside the outflow (red region), indicating that the MR-driven explosion occurs in this model.
On the equatorial plane, a clear m = 1 nonaxisymmetric structure becomes prominent (see the minipanels c). Along the equatorial plane, both a continuous mass accretion, with low entropy s 5 k B baryon −1 , and ejection, with relatively high s ∼ 10 k B baryon −1 , simultaneously take place. Furthermore, from minipanel (c) in the bottom-right panel, we find the gas pressure being the dominant in most of the re-gions, i.e., the magnetic field do not play the leading role for the shock expansion. We thus argue that the shock expansion is significantly aided by rotation in the equatorial region (seee Nakamura et al. (2014); Takiwaki et al. (2016); Summa et al. (2018) for the rotation-supported 3D CCSN models and also Kuroda et al. (2020) with magnetic field).
In Fig. 2, we show the same figure as Fig. 1, but for model R1B13 at t pb = 185 ms and at the final simulation time t pb = 316 ms. The remarkable difference from model R1B12 is the absence of a clear bipolar structure. For instance from 2D contours in the bottom panels, the outermost shock surface I -9 Figure 2. Same as Figure 1, but for model R1B13.
locating at R ∼ 4000 km is more roundish and the axis ratio is closer to unity than that of model R1B12. Inside the shocked region, the high entropy region with s ∼ 15 k B baryon −1 appears, although the value is relatively lower compared to model R1B12 throughout the computation. In addition, the entropy structure shows small scale and fragmented structure that is completely different from a global scale high entropy bipolar structure seen in model R1B12.
Another remarkable feature is that the high entropy blobs (e.g., yellowish region in the top two panels) are randomly oriented and basically not in alignment with the rotational axis in contrast to model R1B12. We consider that the reason of this random orientation is due to an absence of continuous magnetic field amplification, particularly the winding ampli-fication along the rotational axis. As we will explain later, this model shows a less mass accretion rate and the winding mechanism, which requires high mass accretion rate with angular momentum, does not operate. Due to lack of the strong toroidal magnetic field along the rotational axis, the matter is not preferentially ejected along the rotational axis. In addition, there might be another possible reason for the misalignment, which is the tilt of rotational axis. Although it is not the scope of this work to investigate in detail if the rotational axis tilts in model R1B13, from a very recent full 3D MHD models in Obergaulinger & Aloy (2020b), the rotational axis, as wel as the orientation of the outflow, can tilt due to asymmetric matter accretion onto the PNS. We found that the mass accretion in model R1B13 takes place not only K along the equatorial plane but also in all directions. It significantly differs from that of model R1B12, which shows a continuous accretion channel mainly along the equatorial plane. Although Obergaulinger & Aloy (2020b) reported the tilt of rotational axis in their weakest magnetized model, the absence of continuous mass accretion along the equatorial plane is a common feature seen in our model R1B13 and theirs.

Figure 3.
Top: time evolution of maximum (thick lines) and averaged shock radii (thin) for all models. To show more clearly the lines of model R0B00, we multiply them by ten. Bottom: we plot the diagnostic explosion energy. Since Eexp of model R0B00 is essentially zero, we omit to plot it. Figure 3 presents time evolution of the shock radius R shock in the top panel and of the diagnostic explosion energy E exp in the bottom, where we use the same definition for E exp as Kuroda et al. (2020). In the top panel, we plot the maximum (thick line) and averaged shock radii (thin) for all models including model R0B00. We multiply the lines of R shock for model R0B00 (black lines) by ten to show them more clearly. We find that the model R0B00, which assumes neither rotation nor magnetic field at initial, does not exhibit the shock revival during our simulation time of ∼ 500 ms after bounce. This is consistent with our previous study Kuroda et al. (2020) and we thus argue that the up-to-date neutrino opacities do not drastically change the explosion dynamics. Therefore, in the bottom panel, we omit the line for model R0B00 because its explosion energy is essentially zero.
The time evolution of shock radii presents a rapidly expanding shock surface in model R1B13 immediately after bounce (red lines). The maximum shock radius reaches R = 1000 km at t pb ∼ 70 ms. During the same period, E exp increases drastically at initial, reaches its maximum ∼ 6 × 10 50 ergs at t pb ∼ 20 ms, and declines afterward. Contribution of each energy to the total diagnostic explosion energy at t pb = 20 ms is as follows: the magnetic energy ∼ 10 51 ergs, the internal energy ∼ 4 × 10 50 ergs, the radial kinetic energy ∼ 2 × 10 50 ergs, and the rotational kinetic energy ∼ 2 × 10 50 ergs. The prompt explosion is thus mainly supported by the magnetic field. The decline seen in E exp after t pb ∼ 20 ms is mostly due to decrease of the magnetic energy in the unbound region. While in model R1B12 (blue lines), the shock front shows a mild expansion at initial (t pb 120 ms) and subsequently becomes faster than that of model R1B13. Its averaged shock radius (thin blue line) exceeds that of model R1B13 at t pb ∼ 310 ms. The explosion energy E exp in model R1B12 increases at t pb ∼ 100 ms and then plateaus around the value of ∼ 10 50 ergs. The shock and explosion energy evolution in models R1B12 and R0B00 are quantitatively in good agreement with our previous report (Kuroda et al. 2020), which employed the same initial condition except the neutrino opacity set. . We plot the mass inflow (Ṁ < 0) and outflow rate (Ṁ > 0) (solid lines) and total accretion rate (dash-dotted) measured at R = 100 km for models R1B13 (red) and R1B12 (blue). We next explain the different explosion dynamics seen in the two magnetized models, particularly focusing on the reason why the initially less magnetized model R1B12 eventually shows more energetic explosion. Figure 4 exhibits the mass inflow (Ṁ < 0) and outflow rate (Ṁ > 0) by the solid lines and the total mass accretion rate by the dash-dotted line. The color represents model either R1B13 (red) or R1B12 (blue). The rate is measured at R = 100 km so that we can encompass the base of MHD outflow locating at R ∼ a few 10 km. In model R1B12 (blue lines), the absolute values of both the mass inflow and outflow rates are significantly larger than those of model R1B13 (red). Especially after t pb 40 ms, the mass inflow rate in model R1B12 is showing a larger value of ∼ 1Ṁ ⊙ s −1 from the one in R1B13. At the same time, the large amount of mass ejection also takes place in model R1B12 resulting in a moderate difference between the total mass accretion rates of two models (dash-dotted lines). The smaller mass inflow rate in model R1B13 is a consequence of the stronger explosion taking place immediately after bounce. As we have already explained, enormously strong shock wave, which is mainly supported by the magnetic pressure, is launched shortly after bounce. The shock propagates outward in all directions and suppresses the subsequent mass accretion onto the central engine of the MHD outflow. Lower accretion produces less liberation gravitational potential energy and thus the strong bipolar flow is not continuously supported. This is the reason of the weaker shock wave eventually seen in model R1B13.
On the other hand, the shock propagates significantly slower in model R1B12 at initial postbounce phase, that allows the development of nonaxisymmetric matter motion, e.g., onearmed spiral pattern. Indeed, the ratio of rotational to gravitational potential energy reaches a few % in model R1B12, while it is significantly smaller ∼ 0.5 % in R1B13. The spiral pattern then produces a flow channel through which noticeable amount of mass accretion takes place as we showed in the bottom panels of Fig. 1.

North-south asymmetry of neutrino matter interaction
In this section, we discuss the modified neutrino-matter interactions in the presence of magnetic field and their actual impact on the dynamics. We begin with a brief explanation of overall picture of neutrino signals. The nonexplosion model R0B00 shows basically the highest luminosity and mean energy in all flavors of neutrinos, while the rotating magnetized models R1B12 and R1B13, which explode shortly after bounce, show rather lower values. Roughly speaking, such a feature is consistent with our previous study (Kuroda et al. 2020) and also with those in recent studies with detailed neutrino transport (Müller et al. 2017;O'Connor & Couch 2018;Summa et al. 2018;Vartanyan et al. 2019).
To see more precisely how the modified neutrino-matter interactions lead to asymmetric properties, particularly with respect to the equatorial plane, we depict several quantities in Fig. 5. Here, we introduce several quantities as follows: the change rate of electron fraction Γ e (≡ ∂Y e /∂t) independent from Γ b=0 e and dependent on Γ b =0 e the magnetic field where n b is the number density of baryons, the energy deposition rate Q b =0 nae originated from the modified neutrino absorption and emission process the energy deposition rate Q b =0 sc originated from the modified inelastic neutrino-nucleon scattering process the energy deposition rate Q tot originated from the normal neutrino matter interactions (see Eq. 43) the torque τ φ that the fluid element gains due to the modified source term where ̟ = x 2 + y 2 is the distance from the rotational axis, and the z-component of the force acting on the fluid element f z due to the modified source term We also introduce following ratios to assess the relative impact of the modified neutrino matter interactions to the normal ones as We note that we take an absolute value of the denominator for the later convenience. Using these quantities, we show in Fig. 5: (minipanel a) the z-component of magnetic field e , (d) log γ e , (e) log q nae , (f) log q sc , (g) log τ φ , and (h) log f z . All panels are showing the contours on y = 0 plane for model R1B12 at t pb = 65 ms. We also plot the mean gain radius by a red circle in panels (c)-(h) for a reference.
In panel (a), we see that B z is positive inside the PNS with its value being ∼ 10 15 G in both hemispheres. This is mostly due to compression of initial uniform or dipole-like magnetic field aligning with the z-axis inside ̟ 1000 km. B φ shows a clear asymmetry with respect to the equatorial plane and the northern (z > 0) and southern (z < 0) hemisphere possess negative and positive B φ , respectively. Within the PNS, the toroidal magnetic field strength reaches ∼ 10 15 G. Reflecting the nonisotropic magnetic field, the change rate of electron fraction Γ b =0 e also shows a clear north-south asymmetry in panel (c). Here we note that Γ b =0 e is affected only by the modified neutrino absorption and emission process (Eq. 41), since the (modified) scattering process Eq. (26) does not change the electron number at all as proved in the Appendix A. The panel (c) shows that the northern hemisphere delptonizes more. This can be understood from Eq. (41). Using the orthogonality conditionL αβ u α = 0, Γ b =0 e is expressed as ǫ mc (Eq. 30) is usually negative due to g 2 V −g 2 A < 0 and also in the degenerate limit of electrons f e − ∼ 1. Therefore, the sign of Γ b =0 e is determined by the angle between the direction of flux of neutrinos H β and that of magnetic fieldb β . Since H β of ν e is usually orienting radially outward inside the PNS, i.e., H i ∼ (H r , 0, 0), the toroidal magnetic field B φ does not contribute to Γ b =0 e as they are mutually orthogonal. The remaining magnetic field component B z is pointing positive z direction from panel (a). As a consequence, the northern hemisphere, where H βb β > 0, Γ b =0 e becomes negative, while the southern hemisphere analogously shows positive Γ b =0 e .

Panel (d) indicates how large is the contribution of Γ b =0
e to the total deleptonization rate |Γ b=0 e |. Here, we note that the most of the northern hemisphere displays dark blue region simply because we cutoff the negative value to emphasize the north-south asymmetry. However, it actually has a similar value as in the southern hemisphere, but with a different sign. We also mention that the deleptonization rate Γ b=0 e , which is free from the influence of magnetic field, shows a nearly perfect symmetry with respect to the equatorial plane. From panel (d), the value γ e in the southern hemisphere reaches several %, which is also the same for the northern hemisphere. The parity violating effect due to the magnetic field thus can potentially produces a significant partial distribution of Y e in the PNS, that will also be discussed later. As in panel (c) shows ∼ ±0.01 s −1 along the rotational axis, if the parity violation effect lasts ∼ 100 ms, the cumulative change reaches δY e ∼ ±10 −3 . In previous studies, Tamborra et al. (2014) reported the existence of the lepton  2019)), which is originated from a partial distribution of Y e in the PNS convection zone (r ∼ 25 km). In their subsequent paper (Glas et al. 2019), they explained the origin of the partial distribution of Y e , dominated mainly by the l = 1 mode, by the PNS convection. Aside from the PNS convection, our study shows for the first time that the neutrino matter interactions in the presence of strong external magnetic field can be another possible mechanism to produce a partial distribution of Y e . Next we evaluate the impact of heating and cooling rates contributed from the modified charged and neutral current reactions in panels (e) and (f), respectively. Comparing the panels (d) and (e), they show the similar pattern as expected, i.e., Q b =0 nae has basically the same sign as Γ b =0 e . The panel (e) indicates the larger heating and cooling rate in the southern and northern hemisphere, respectively. Employing essentially the same initial magnetic field but with various strengths, Kotake et al. (2005) also reported the excess of neutrino heating in the southern hemisphere. Our result is thus qualitatively consistent with theirs. In addition, Kotake et al. (2005) shows a relative contribution of Q b =0 nae to the total heating/cooling rate Q tot of the order of ∼0.1 %, which is also in good agreement with ours. The excess/reduction of the energy exchange rate above the gain radius is several 0.1 % from panel (e).
Regarding Q b =0 sc (or q sc ), Q b =0 sc again shows a clear asymmetric feature in panel (f). Inside the central region with r 30 km, the northern and southern hemisphere shows the excess of neutrino heating and cooling, respectively. Meanwhile, the sign inverts around r ∼ 40 km. The reason of the asymmetry and sign inversion can be understood as follows. From Eq. (26) and also using the fact that a 1,α is proportional to b α (see Eq. 16), we find that S α,b =0 ε,sc is consisted of two parts. One is proportional to H β b β and the other is to |L| ∼ J − J eq , namely the deviation of neutrino distribution function from the thermal equilibrium, as Here we use the relation h αβ b β = b α . The source term for the zeroth radiation moment Q b =0 sc is then rewritten by taking the norm with n α as Inside the deep PNS core, the radiation field and matter velocity exhibit nearly the isotropic structure, i.e., l = 0 with l denoting the degree of spherical harmonics, while the magnetic field shows a uniform (dipole like) field for our initial condition, i.e., l = 1. As a consequence, both the first and second terms, which are depend on the angle between the magnetic field and H β and u i , respectively, basically change their signs between the northern and southern hemispheres. This is the reason of the asymmetric structure with respect to the equatorial plane. The sign inversion seen at r ∼ 40 km depends simply on which term of the integrand becomes the dominant term and cannot be inferred a priori. It is also noteworthy that Q b =0 sc has a comparable or even a slightly larger value than Q b =0 nae . This fact indicates the importance of the modified neutrino-nucleon scattering term due to the magnetic field that is often omitted in the previous studies on the parity violation effects in CCSNe (Kotake et al. 2005;Suwa & Enoto 2014;Dobrynina & Ognev 2020).
We also discuss the parity violation effects on the angular and linear momentum transfer. In panel (g), we show the torque that the PNS gains. It is obvious that the PNS core r 30 km is subjected to torsional stress and the northern and southern hemisphere spins down and up, respectively. A possibility of such torsional effect has been already discussed in Maruyama et al. (2014), in which a region where the direction of rotation is the same with that of magnetic field shows a spin deceleration. Our result thus supports their discussion. However, since they considered only the effect of absorption process influenced by the magnetic field, our result with both the scattering and absorption processes considered shows a more complicated spin deceleration or acceleration profile. Finally from panel (h), we find that the linear momentum transfer occurs mostly in the PNS core. At there, f z shows a positive value in both hemispheres without asymmetry with respect to the equatorial plane. It indicates that the whole PNS core can be kicked toward north at this time. The symmetric structure seen in f z can be understood like below. Here we mention that the main contribution to the momentum transfer comes from the scattering term S α,b =0 ε,sc and we thus focus on this term. Using Eq. (57), f z can be expressed as where we take only the z component for α when we move from the first line to the second. This is because the dominant magnetic field component is B z ∼ b z (> 0) around the rotational axis in the PNS core. Bearing in mind that the radiation field and matter are roughly isotropic inside the PNS core, H z u z and J − J eq basically show the same sign in both hemispheres. Therefore, f z is proportional to b z and it results in a bulk acceleration of PNS core. Analogously, the angular momentum exchange has a dependence on the direction of toroidal magnetic field. Consequently, the anti-parallel toroidal magnetic field between northern and southern hemi-K spheres seen in our magnetized models lead to the spin-down and -up, respectively. We mention that the sign of f z can change with time, as we will show later, probably depending on the neutrino profiles H µ as well as on u µ in Eq. (59). In Fig. 6, we depict the same values as in Fig. 5, but for model R1B13 to see the effects of one-order of magnitude stronger initial magnetic field. From panels (a) and (b), the magnetic field are not showing a similar structure as those in model R1B12. For instance, B z seems to be consisted of higher order l modes. As already discussed in Sec. 5.1 and also shown in Fig. 2, the less mass inflow along the equatorial plane might hinders the collimation effect of outflow at where it is launched. Consequently, the magnetic field at the base of outflow may not be aligned with the rotational axis compared to that in R1B12. Aside from such difference seen in the magnetic field structure, the effects of modified neutrino matter interactions are qualitatively the same as in R1B12. Reflecting the global magnetic field, whose direction is roughly tilted with the angle of ∼ −45 • from the rotational axis on y=0 plane (see panel a), the region with z x shows, e.g., the excess of deleptonization and cooling rate, which is analogous to what we see in the northern hemisphere of model R1B12. The relative contribution of modified neutrino matter interactions to the total ones is approximately several to ∼ 10 % from panels (d)-(f), which is approximately one order of magnitude larger than that in model R1B12.
In the end of this section, we discuss time evolution of the modified source terms and their cumulative impact on the PNS core. In Fig. 7, we show the deleptonization rate (left panel), energy deposition rate (middle), and acceleration of PNS (right). To plot the left panel, we again introduce several quantities as follows: the total deleptonization rate due to the modified interactionsṄ and its cumulative value number of electrons and the ratio R e , which measures the degree of partial distribution of electrons inside the PNS, The volume integral is performed for each hemisphere z > 0 (northern hemisphere, red lines) and z < 0 (southern hemisphere, blue) and also for the region with ρ ≥ 10 12 g cm −3 .
In the middle panel, the energy gain or loss in each hemisphere is evaluated by the volume integral of Q b =0 nae/sc . In the right panel, we evaluate the z component of PNS acceleration a z by where M PNS = dx 3 ρ * is the PNS mass in each hemisphere.
In both the middle and right panels, the solid and dashed lines indicate the contribution from modified neutrino absorption and emission (nae) and scattering (sc) processes, respectively. From the left panel in Fig. 7, the volume integrated deleptonization rate in each hemisphere exhibits a clear asymmetry throughout the simulation time.Ṅ e in the northern hemisphere reachesṄ e ∼ −10 54 s −1 (red solid line) indicating more deleptonization at there. Its cumulative value reaches R e ∼ −0.2 % of the total electron number inside the PNS at the final time of simulation. Meanwhile, in the southern hemisphere, those values have the opposite sign, but with almost the same absolute values as those in the northern hemisphere. Therefore, the parity violation effect can potentially produce a north-south asymmetry in Y e of the order of a few ‰ in this model. Regarding the energy deposition rate in the middle panel, significant energy transfer occurs mainly during the first hundred milliseconds after bounce. However, from panels (e) and (f) in Fig. 5, the energy transfer due to these terms takes place mostly within the gain radius denoted by the red circle. We thus argue that the modified terms do not have a noticeable impact on the neutrino heating explosion mechanism.
The apparent PNS acceleration a z , that is simply evaluated from the source term of momentum exchange, shows a significant value of ∼ 5 × 10 8 cm s −2 and lasts ∼ 100 ms after bounce. We note that the momentum exchange through the normal neutrino matter interaction processes without influence of magnetic field, i.e., via the term S α,b=0 ε (Eq. 43), shows a nearly, not perfect, asymmetric property with respect to the equatorial plane. For instance at t pb = 50 ms, a z evaluated from S α,b=0 ε reaches ∼ +7.1(−7.3)×10 10 cm s −2 in the northern(southern) hemisphere, resulting in a non-vanishing net acceleration of ∼ −2 × 10 9 cm s −2 . We note that our nonrotating nonmagnetized model R0B00 shows the net acceleration of ∼ 10 7 cm s −2 , which is essentially zero. We thus consider that the non-vanishing acceleration contributed solely from S α,b=0 ε originates from the asymmetric hydrodynamic background with respect to the equatorial plane. Anyway, we argue that the symmetric property seen in a z evaluated from S α,b =0 ε (right panel in Fig. 7) has a comparable influence as that from S α,b=0 ε . We also mention that we do not observe a meaningful PNS core kick during the simulation time. This is due to that the relevant neutrinos are still trapped and do not carry away the momentum during  See text for their definitions. In the middle and right panels, the line style corresponds to the neutrino matter interaction, either modified "nae" (solid) or "sc" (dashed) process. the simulation time. It marginally shows a slight oscillation after t pb ∼ 100 ms with the displacement of several hundred meters, possibly indicating the appearance of SASI. Another remarkable thing is that the modified scattering term is again the main contribution by comparing the dotted (sc) and solid (nae) lines in the middle and right panels.
In Fig. 8, we plot the same figure as Fig. 7, but for model R1B13 which employs one order of magnitude larger initial magnetic field. We see a qualitatively similar trend as that of R1B12. During the first ∼ 100 ms after bounce, most of the lepton, energy, and momentum exchanges occur. However, compared to the values in model R1B12, all values plotted in Fig. 8 show roughly one order of magnitude larger values. This is simply because that the modified source terms employed (Eqs. 26 and 41) have a linear dependence on the magnetic field b µ .

SUMMARY AND DISCUSSION
In this study, we have presented a formalization of the correction term of neutrino matter interaction rates in the presence of external magnetic field. The formalism is based on Arras & Lai (1999) and we took into account the modified interaction rates of two major processes in the SN core: neutrino-nucleon scattering and neutrino absorption and emission processes. We extracted the zeroth and first order angular dependencies of the interaction rates and derived the source term suitable for the M1 neutrino transport in full relativity. The final expression of the source terms is described in terms of the normal radiation moments. Because the magnetic potential energy of free electrons and nucleons, ceb and µ B b, respectively, are significantly smaller than the matter temperature in typical MHD CCSN models, we can safely truncate the second or higher order terms in the magnetic field strength, leading to the source terms having a linear dependence on it. We also proved that the modified scattering term does not violate the lepton number conservation, which is crucial to accurately follow the PNS deleptonization.
Utilizing state-of-the-art general relativistic M1 neutrino transport code with the gravitational red and Doppler shift terms being fully considered, we have conducted MHD CCSN simulations of a 20 M ⊙ star (Woosley & Heger 2007). We calculated three models, nonrotating nonmagnetized (R0B00), rotating magnetized (R1B12), and rotating strongly magnetized (R1B13) models to explore the effects of progenitor's rotation and magnetic field both on the dynamics and the modified neutrino matter interactions. For the nuclear EOS, we used SFHo of Steiner et al. (2013). Other than the correction terms in neutrino matter interactions due to the magnetic field, one of major differences from our previous study (Kuroda et al. 2020) is that we used up-to-date neutrino opacities based on Kotake et al. (2018). We adopted the most elaborate electron capture rate on heavy nuclei following Juodagalvis et al. (2010). Furthermore, we also took into account followings: inelastic contributions and weak magnetism corrections (Horowitz 2002), the density-dependent effective nucleon mass (Reddy et al. 1999), the quenching of the axial-vector coupling constant (Carter & Prakash 2002;Fischer 2016), many-body and virial corrections (Horowitz et al. 2017), and strangeness contribution to the axial-vector coupling constant (Horowitz 2002). Investigating the impact of these up-to-date neutrino opacities will be reported elsewhere.
Concerning the dynamics, while no shock revival was observed in model R0B00 during our simulation time of ∼ 500 ms after bounce, the shock expansion initiated shortly after bounce in two magnetized models. We found essentially the same dynamical features between models R0B00/R1B12 in this study and their corresponding models in Kuroda et al. (2020). The only difference is that the bipolar explosion appeared in model R1B12 in this study did not diminish, although it showed a slight asymmetry with respect to the equatorial plane. In model R1B13, we observed the most rapid shock expansion and largest increment of the explosion energy among the three models. However the expansion speed of model R1B12 eventually seemed to be faster than that of model R1B13 at the final simulation time. The explosion energy in model R1B12 reached the value of ∼ 10 50 ergs, which is consistent with our previous study. The reason why the initially less magnetized model R1B12 eventually exhibited more energetic shock expansion is originated from the larger mass accretion rate. In model R1B12, we witnessed significantly larger mass inflow and outflow rates than those in model R1B13. Compared to the case in R1B13, the prompt shock propagated significantly slower in model R1B12 that allowed nonaxisymmetric matter motion, e.g., one-armed spiral pattern, to fully develop. The spiral pattern could then produce a flow channel through which noticeable amount of mass accretion took place, leading to more liberation of gravitational energy.
One of the aims of this study is to self-consistently assess the actual impact of modified neutrino matter interaction rates on the lepton, energy, and momentum exchanges. In addition, we focused on the global asymmetry that could be induced by the initial dipole-like magnetic field employed in this study. From our results, we found a clear asymmetric feature in both the deleptonization and energy deposition rates with respect to the equatorial plane. As for the asymmetric deleptonization rate, the northern(southern) hemisphere loosed electrons more(less). The energy deposition rate through the modified neutrino absorption and emission process showed basically the same asymmetric feature with that of the deleptonization rate. These features can be understood by the dependence of corresponding source term on the inner product of the diffusion flux of neutrinos and the magnetic field. The sign of the inner product changes between the northern and southern hemispheres if the magnetic field is dominated by a dipole (or more precisely odd l-modes) structure. The cumulative impact of asymmetric deleptonization rate asymptotically reaches ∼ 0.1 − 1 %. Our result also indicates the importance of the modified inelastic scattering process, at least in the explosion phase, that has been often omitted in previous literature (Kotake et al. 2005;Suwa & Enoto 2014;Dobrynina & Ognev 2020). The energy deposition rate from the modified scattering term showed roughly a one order of magnitude larger value than that of absorption process.
On the contrary to the lepton and energy exchanges, the linear momentum exchange showed a symmetric property, e.g., the z component of the PNS acceleration showed the same sign and value in both hemispheres. Roughly speaking, such feature stems from that the source term projected on to the Eulerian frame is parallel to the magnetic field. Initially dipolar-like magnetic field leads to a magnetic field configuration in the proto-magneter composed of a nearly uniform z component and anti-parallel toroidal field in both hemispheres. Therefore, both hemispheres gain the acceleration in the same z direction, while there is a torsional effect between the two hemispheres.
In terms of scattering and absorption cross sections, we can summarize our results as follows: the scattering cross sections for neutrinos are enhanced(reduced) if neutrinos propagate in parallel(antiparallel) with magnetic field, which is consistent with the positive acceleration of PNS core. On the contrary, the absorption cross sections are reduced(enhanced) if neutrinos propagate in parallel(antiparallel) with magnetic field, which is in line with, for instance, the smaller deleptonization rate, i.e., neutrinos are less captured, displayed in the northern hemisphere (see panel (c) in Fig. 5). These trends of enhancement or reduction are consistent with the study of Maruyama et al. (2011Maruyama et al. ( , 2012. We stress that most of the energy, leptons, and linear and angular momentums that are transferred through the modified source terms are not immediately taken away from deep inside the PNS core. We actually investigated if there is a noticeable north-south asymmetry in Y e or neutrino's energy and number fluxes on the PNS surface, though we could not find them with significance. Therefore we conclude that the modified source terms do not play important roles in the dynamical time scale and contribute neither the explosion dynamics nor the natal (t pb 1 s) PNS kick. This basically agrees with previous studies Kotake et al. (2005); Dobrynina & Ognev (2020). However, once those trapped neutrinos diffuse out the PNS core on a time scale of O(1) s, namely in the Kelvin-Helmholtz cooling phase, they can potentially produce a ∼ 0.1 − 1 % asymmetry in the PNS structure. Along with the neutrino diffusion, the partial distribution of Y e may gradually appear leading to more pronounced asymmetric neutrino emission. In addition, the modified scattering and absorption processes themselves contribute to globally asymmetric neutrino emission. The final outcome of the asymmetric neutrino signal depends on these two effects that definitely interact each other. Therefore, the assessment of the degree of final outcome is highly complex and cannot be simply extrapolated from our results. It can be explored only through the long time MHD simulation. Our numerical simulations, however, provide us, for the first time, realistic values of the impact of magnetic field on the neutrino matter interactions in MRE scenario, albeit in the early post bounce phase and for a small number of models.

ACKNOWLEDGMENTS
It is a pleasure to thank Almudena Arcones for her encouragement and support for this work and also for her useful comments and suggestions. I acknowledge Kei Kotake and Tomoya Takiwaki for fruitful discussions and the new neutrino opacities that they provided me. This research was supported by the ERC Starting Grant EUROPIUM-677912. Numerical computations were carried out on Cray XC50 at CfCA, NAOJ.