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Effective Theory of Collective Neutrino Oscillations in Supernovae

Explore the effect of extended n-production region on collective oscillations in supernovae. Justification: large usual matter density in the production region leads to strongly suppressed mixing and frozen flavor evolution. However, this argument can be flawed if flavor conversion is triggered by the parametric resonance in the effective theory. This study aims to examine the effect of nn interactions in the background and their impact on the development of instabilities.

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Effective Theory of Collective Neutrino Oscillations in Supernovae

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  1. SN neutrinos: effective theory of collective oscillations A. Yu. Smirnov MPIK, Heidelberg ECT* workshop ``SN neutrinos at the crossroads’’ Trento, May 14, 2019

  2. Effect of extended n- production region on collective oscillations in supernova A. Yu. Smirnov MPIK, Heidelberg ECT* workshop ``SN neutrinos at the crossroads’’ Trento, May 14, 2019

  3. Introduction The width of the emission region in SN reff ~ O(km) >> losc~ 1/Ve In most studies - surface emission of neutrinos (infinitezimal width of neutrinosphere, antenna ..) Justification: large usual matter density in the production region  strongly suppressed mixing  flavor evolution is frozen This argument is flawed If flavor conversion is triggered by the parametric resonance (in effective theory) the depth of modulation of potentials is strongly suppressed by integration  strong delay of conversion Here: explore this in presenceof nn interactions in the background  effect on developments of instabilities Complete and consistent description by solving quantum kinetic equations

  4. Outline: 1. Effective theory and averaging 2. Averaging and initial condition 3. Averaging in the presence of neutrino interaction 4. Linear stability analysis Based on Rasmus S.L. Hansen, A. Yu. S. JCAP 1804 (2018) 04,057, 1801.09751 [hep-ph], and papers in preparation

  5. Effective theory of collective condition

  6. Effective theory Eventually we detect at the Earth individual neutrinos . Trace back their evolution Evolution of the flavor field in SN Flavor evolution of individual neutrinos Strong flavor transformations as parametric resonance effects - exponential grow, - bi-polar oscillations, - spectral splits re – interpretation?

  7. Effective theory of collective oscillations Above the neutrinosphere collective effects can be completely described by evolution of individual neutrinos in external potentials produced by usual matter and other neutrinos. Flavor diagonal Vn(t) and flavor changingVn(t) potentials are generated Possible collective oscillations effects are consequences of particular time dependences of the potentials The problem is reduced to determination of time dependences of the potentials Inelastic interactions - in certain approximations

  8. Summation over background neutrinos Production region Usual matter potential Ve= 2GFne interaction point . (x, t) p,wp probe neutrino dx l k, wk background neutrino . (x0, t0) Flavor at the production a = e, t k - 3 momentum l – length of trajectory from production (x0, t0) to interaction point (x, t) Production point - the point of last inelastic collision  (a, l, k) characterize a given mode (x, t), k and l determine (x0, t0)

  9. Total Hamiltonian and potentials ne nt • cos2qwp + Ve + Vn sin2qwp + 2Vn eif • sin2qwp + 2Vn e-if cos2qwp - Ve - Vn H = ½ Potentials Vn = dk dl [Vne (k, l) - Vnt (k, l)] (1 – Pet (k, l) ) Vneif =dk dl [Vne(k, l) - Vnt (k, l)] eif(k,l) Pet (k, l) (1 – Pet (k, l)) where Vna(k, l) = 2 GF nna(k, l) (1 – vpvk ) a = e, t nna (k , l) - number density of neutrinos emitted from (x0, t0) in (x, t) Plus contributions from antineutrinos

  10. Effective theory approach Since it is not feasible to perform explicit computations of these potentials, the approach is To extract certain properties of the potentials from their general expressions Some integrations in the potentials which correspond to averaging over energy or over neutrino production point can be done explicitly (approximations) using general form of potentials Assumptions and conjectures on the time dependence of potentials can be introduced Construct potentials using various limits, existing numerical results, simplified solvable examples

  11. Properties of potentials Potentials are integrals of oscillation amplitudes which have oscillatory dependence on time. Therefore potentials are also expected to be oscillatory functions of time determined by intrinsic frequencies of the system Ve , Vn, wp, wk Furthermore, there is the hierarchy of frequencies Ve > Vn >> w general parametrization…

  12. Removing the phase Off-diagonal term: introduce real V’ and the phasef‘ as sin2qwp + 2Vn eif = V’ e-if’ (*) Transformation of the fields y = Uy’ U = diag (e 0.5 if’, e-0.5if’ ) it does not change flavor probabilities. Hamiltonian for y’ Vr(t) V’ (t) V’(t) - Vr(t) H = ½ V’ (t) is def. in (*) Vr(t) = Ve + Vn - cos2qwp + df‘/dt

  13. Parametric enhancement Potentials are modulated by periodic functions, so that the mixing angle in medium tan2qm= - V’/ Vrvaries with a period Tq Parametric enhancement: if the frequency of modulations coincides with eigenfrequency of the system 1/Tp Tq= Tp 2p <Vr>2 + <V’>2 Tp= <Vr>, <V’> - potentials averaged over modulations Large transition probability develops over many periods

  14. Solvable example Flux of collinear background neutrinos  no nn– interactions in the flux Probe neutrino Background neutrinos still coherent flavor exchange Usual uniform matter with potential Ve wk wp Dm2 2 E ne bk w = ne bp Dm - level splitting Flux neutrinos – usual oscillations in matter Pet = sin22qm sin2½Ab Dmt t – evolution time (distance) of the probe neutrino sinbp sinbk Evolution time of the flux neutrino to the moment of intersection with probe neutrino Ab = tf = Abt

  15. Potentials: time dependence Vr(t) V’ (t) V’(t) - Vr(t) diagonal H = ½ Usual matter potential Ve dominates Ve = 103 wk Neutrino to matter potential: Vn0 Ve off-diagonal x = Vr and V’ as functions of the time for different values of x. Ab= 1.1001, wp= wk

  16. Parametric resonance The dependence of Pet on time for different values of wp/wk . x = 0.1, Ab= 1.1001 Resonance condition: Vr(t) = 0 The upper panel - resonance, the middle and lower panels wp/wk are detuned.

  17. Integrations and averaging Strong transitions are driven by periodic modulations of potentials In simplified example (emission from the plain, fixed l, fixed angle, single energy), the modulation term ½fcosfm = ½sin2q nn(l) [1 – cos(bp - bk)] wk cosfm(l, wk) ne-1 In realistic case one should make integrations over - production region, l, - energy (frequency) of neutrinos wk - angle bk 1/ Vereff These integrations suppress the depth of modulations and therefore increase the scale over which transitions (parametric resonance) will develop

  18. Reconstructed potential For two intersecting beams with nu-nu interactions The off-diagonal term parametrization - iVe t - w cos2(p t/p ) Vneif = ae e w – weak dependence on Vn, Ve , w a, p – strong dependence on Vn, w Diagonal element – weak influence

  19. Averaging and initial condition

  20. Effective production region Production processes p + e- = n + ne n + e+ = p + ne Emissivity ne(r) T (r) ne(E, r) = exp[- E/T(r)] J(E, re) = 0.5 sNnN(re)ne (E, re) Survival probability with respect to absorption r re psurv(r, re) = exp - sNnN(r’)dr’ Probability that neutrino observed at r is emitted at re 1 4pF(r) p(r, re) = J(re) psurv(r, re) where neutrino flux at radius r: r 0 F(r) = p(r, re) dre

  21. Neutrino emission probability Neutrino emission probability as function of distance from the center 1 4pF(r) p(r, re) = J(re) psurv(r, re) flux survival probability emissivity - r/r0 r(r) = r0 e T = T0 r0/r the infinitezimal neutrino sphere t (Rnusph) = 2/3 r0 = 10 g/cm3 inf r r0 = 4 km optical width t (r ) = G (r’) dr’ T0 = 50 MeV

  22. The 1-2 correlation Eigenstates of the Hamiltonian in matter without nn - interactions n’1 , n’2 H = H(w, Ve) Dm2 2 E Ve - usual matter potential w = Define the 1-2 correlation - 1-2 element of the density matrix in the eigenstate basis: r’12(r) = y’1(r) y’2*(r) cosqm(r) sin qm(r) ye(r) = For the ne state r re r’12(r) = 0.5sin2qm(re) exp iwm(r’) dr’ mixing at production Difference of eigenvalues in matter

  23. Averaged correlation Integration over production region leads to about 10 orders of magnitude suppression of correlation in the state produced as ne 1/ Vereff This should be taken into account in initial condition for further collective oscillations r re r 0 |<r’12(r) >| = 0.5 p(r, re) sin2qm (re) exp iwm (r’) dr’ dr Analytic result (neglecting Bolzmann factor) labs = 1/sNnn <r’12(r) > = - 0.5 i sin2qm r02/[r0 Velabs2]

  24. Averaging in the presence of n n - interactions

  25. Two intersecting fluxes of collinear neutrinos Model uniform matter potential Ve F(ne) = F(ne) bL= bR = b z symmetry bp - b L and R moving neutrinos have the same evolution single w Uniform distribution of sources Dz 1/Dz, ze = 0 -Dz 0 ze >Dz pn(ze) = ze bL bR ignored d dt ir(ze,t) = [H(t), r(ze,t)] z(t) 0 H0 = H0 (w, Ve) H = H0 + m [r(z’e, t) - r*(z’e, t)] pn(z’e) dz’e m = 2 GFnn For m = 0 |<r’12(r)>| = 0.5 sb sin2qm /(VeDz)

  26. Evolution in the production region Neutrino density changes with z Fraction of neutrinos emitted by a given z z/Dz, z = 0 -Dz 1 , z >Dz fn(z) = Neutrino potential ~ m(z) = m fn(z) Normalized density matrix as for single state z 0 1 fn(z) rint(z) = r(ze, z) dze integration  averaging

  27. Numerical results Pee(z) = ree (z) |<r12(z)>| Computed for different values ofDz b = p/4 q = 0.15 Ve = 30wp = 10 m Parameters used: r12 ~ - rex = [ree (1 - ree) ]1/2 Relation  two peaks ree = 0.5 corresponds to r12max = 0.5 Specific features depend on the widthDz

  28. Survival probability and 1-2 correlation Narrow production region Dz << z1 Survival probability and the 12-correlation averaged over production region as function of distance Thick lines – analytic solution of the linearized equations Shadowed - emission regions Dzw < 1.5

  29. Survival probability and 1-2 correlation Wide production region Dz > z1 Dzw < 12

  30. General features Collective effects: bi-polar oscillations Effect of integration (averaging) over production region Delay of strong transformation which increases withDz Dynamics is best seen in terms of r12 Two benchmark points: z0 - the point of onset of development of instability - exponential grow of r12 z1 - coordinate of the first maximum of correlation |<r12(z1)>| ~ 0.5 Three regions of z: 0 - z0 - the averaging phase r12(z) decreases z0 - z1 - exponential grow of r12(z) z > z1 regular (quasi-regular) bi-polar oscillations

  31. Narrow vs. wide production regions Narrow region Dz << z1 Instability develops outside the production region averaging exponential grow asymptotics z0 Dz z1 Wide region Instability develops inside the production region Dz > z1 asymptotics non-linear exponential grow z0 z1 Faster due to increase of m Bi-polar osc. with changing parameters ~

  32. Linear stability analysis

  33. Density matrix in linear approximation - 1 - S* - S 1 1 S* S - 1 2r = I + 2r = I + From evolution equations for r and r z(t) 0 d dt i S(ze,t) = (- w + Ve) S(ze,t) + m [S(z’e,t) - S(z’e,t)] pn(z’e) dz’e z(t) 0 d dt i S(ze,t) = (w + Ve) S(ze,t) + m [S(z’e,t) - S(z’e,t)] pn(z’e) dz’e

  34. Solution Search for solution for the individual modes t te S(ze,t) S(ze,t) Q(ze) Q(ze) = exp -iW(t’) dt’ determined by the initial conditions Solution integrated (averaged) over the production region t te z 0 1 fn(z) Sint(z) = dzepn(ze) Q(ze) exp -iW(t’) dt’ Similar for Sint(z) with Q(ze)  Q(ze) Equation for integrated S: ~ ~ - w + Ve - m - Wm - m w + Ve + m - W Sint Sint = 0 ~ ~

  35. Exponentially growing modes Non-trivial solution: Det M (W) = 0 ~ W(z) = Ve +/- w[2m(z) + w] Exponentially growing collective mode if Im (W) > 0 ~ w < 0 2m(z) > |w| Conditions: inverted ordering  onset of collective oscillations |w| 2m z0 = Dz Initial condition • sin2q • Ve Q(ze) ~ Q(ze) ~ 2r12(ze) ~ sin2qm ~

  36. Solution z 0 t te sin2qm fn(z) Sint(z) = dzepn(ze) exp -iW(t’) dt’ Integration over dt’ - explicitly Dependences on z and ze factorize in the integrand sb sin2qm Dzfn(z) |Sint(z)| = |G(zmax)|exp[-iI(z)/sb] zmax 0 |G(zmax)| = dze exp[-i (Veze + I(ze))/sb] expressions for I(ze) are different for W and N regions z, wide region Dz, narrow region zmax = |G(zmax)| ~ 1/Ve Approximation:

  37. Narrow region fn = 1 Dz < z1 sb sin2qm DzVe |Sint(z)| = exp[|IN(z)|/sb] 2|w|z0 3sb |IN(z)|/sb = (Dz/z0 - 1)3/2 + (3Dz/2z0) (Dz/z0 - 1)1/2 (z/Dz – 1) Dependence on z For very narrow production region Dz <<1/(2 2mw ) sb sin2qm DzVe |Sint(z)| = exp sb-1 |w|(2m - |w|)(z -Dz)

  38. Wide region fn = z/Dz Dz > z1 - instability completely developed in production region sb sin2qm z Ve |Sint(z)| = exp[|Iw(z)|/sb] 2|w|z0 3sb |Iw(z)|/sb = (z/z0 - 1)3/2 faster increase, since new neutrinos joint system

  39. Survival probability and 1-2 correlation Very large production region Dz >> z1 zoomed Evolution of individual modes Produced at different ze Modes produced in the linear regime zw < 50 strongly correlate

  40. Non-linear part z1- Dz More neutrinos joint system In the first minimum, zmin |w| 2m(zmin) ree(zmin) = ~ Frequency of the bi-polar oscillations increases with z ~ |w|m(z) 2log q wbp= The average of bi-polar oscillations decreases converging to the asymptotic value <ree>as = 0.5 ( 1 +|w|/2m) The depth of oscillations decreases: the modes emitted at z > z1 have higher frequencies than collective ensemble created before. Due to collective effects their depth is modulated, they are partially adjusted to (correlate with) the rest of system, and therefore in spite of random production points, do not average strongly.

  41. Delay of strong conversion From the condition |Sint(z1 )| = 1 (using results for the wide region) Dz1/3 2|w|m z1 ~ [1.5sb (ln D-1 - lnDz/z1*)]2/3 where z1* is given by the formula above but without second log sb sin2qm DzVe is the initial condition suppression factor D = For surface emission: z1s~ sb (2|w|m)-1/2ln (sb sin 2qm )-1 For realistic parameters z1 = 5 m  100 – 200 m z1 /z1s ~ Dz1/3 (|w|m)1/6

  42. Conclusions

  43. Effective width of neutrino production region in SN is much bigger than the oscillation length D ~ 1/ reffVe ~ 10-10 - 10-8 Integration over the production region leads to suppression of conversion effects determined by D. Approximation of surface emission is not correct. Effect is best seen in terms of correlation r12 - the off-diagonal element of density matrix in the basis of eigenstates, which determines the onset of collective oscillations. Initial r12 gets additional suppression factor D. Effect has been explored for the simplest system of two intersecting beams. Analytic solution of the linearized equations is in very good agreement with results of numerical computations.

  44. For this simplified example Averaging does not eliminate development of instabilities (exponential grow of transition) but leads to delay of the transition. (The latter can be artifact of symmetries in the example.) Delay is given by the log of suppression factor |ln D| For wide production region the integration modifies exponential grow: exp(Az3/2), instead of exp(Az) for the narrow region. It also modifies oscillations in non-linear regime as well as asymptotics. Delay depends on model of emission (distribution of sources) Qualitative results are generic: additional suppression due to integration must be taken into account which can lead to delay of strong collective transformation or even to their elimination

  45. Backup slides

  46. Parametric resonance Dependence of the depth of parametric oscillations on wk/wp Ab= 1.1001, Vn0= 100 wp Vn0= 100wp Parametric resonance condition: - cos2qwp + Ve + Vn0 = AbDm Frequency of oscillations of the probe neutrino determined by the averaged density Frequency of modulations of the potentials Width of the resonance G / wk= sin2qx (1 + Ab) proportional to x

  47. Finite width of production region 1. Simple averaging: Sources of neutrinos are distributed in the region of size r, after emission neutrino propagates without inelastic collisions l l-r dnn(l’) dl < xcosfm(l) > = dl’ cos (AbDml’) Dm ~ Ve Suppression factor for the depth of modulations: 1 Ab r Ve ~ 10-8 - 10-6 for r ~ (0.1 – 1) km, r ~ (1011 - 1012) g/cm3 For suppressed modulations, parametric conversion scale becomes 104 km and not 1/ ~ 10-2 km

  48. Integration over production region 2. More sophisticated approach: Introduce realistic density, temperature, distributions, Cross sections of neutrino production and absorption sabs -r/r0 For V = V0 e the suppression factor 2 r0 rabs 1 r0 V0 rabs = 1/sabs n0 - gives similar suppression 3. The most consistent way – to use quantum kinetic equations which simultaneously take into account oscillation effects and inelastic interactions

  49. Parametric enhancement V. Ermilova,V. Tsarev, V. Chechin, Krat. Soob. Fiz. # 5, 26, (1986) Strong transition if there is a harmonic modulation of density profile n (x) = <n> + n1coswdx Parametric resonance condition: k wd = 2Dm(<n>) , k = 1, 2 … where frequency of oscillations for the average density Dm2 2 E Dm(<n>) = [(cos2q – 2VE/Dm2)2 + sin22q ]½ P With modulations Without modulations In astrophysical objects? x

  50. Analytic solution The evolution equation with this Hamiltonian  oscillation solution. The solution in the flavor basis (making transformations back) Pet = g-2 (½f’)2 + (g/d)2(d – AbDm)2 + (gf’/d)(d – AbDm)cosfm sin2½gt depth of oscillations has fast modulated usual oscillatory factor Probability averaged over fast modulations Pet = g-2 [(½f’)2 + (g/d)2(d – AbDm)2 ] sin2½gt d = AbDm Parametric resonance condition: In resonance g = ½f’ determined by neutrino density Pet = sin2½gt = sin2(f’t/4)

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