Majorana in dense maer and strong magnec fields* Maxim Dvornikov (1) Instute of Physics, University of São Paulo, Brazil (2) Instute of Terrestrial Magnesm, Ionosphere and Radiowave Propagaon, Troitsk, Russia

*This talk is based on M.Dvornikov, Found. Phys. 42, 1469 (2012) arXiv:1106.3303 [hep-th]; M.Dvornikov, Nucl. Phys B 855, 760 (2012), arXiv:1108.5043 [hep-ph]; M.Dvornikov and D.M.Gitman, arXiv:1211.5367 [hep-th]. Plan of the talk • Introducon • Problems in the descripon of a massive Majorana (Weyl) field in vacuum • Brief review of the previous approaches to the studies of massive Weyl fields • Hamilton formalisms for Majorana parcles • New way of quanzaon in vacuum • Comparison with the standard descripon of Weyl • of massive Weyl fields in background maer • Classical and quantum field theory descripons of Weyl in external fields • Self-interacng Weyl spinors • Discussion Introducon

• Aer the experimental confirmaon (SNO, KamLAND, BOREXINO, etc.) of the existence of oscillaons, we believe that neutrinos are massive parcles. • The mass of the neutrinos is small. From the probes of the large-scale structure and of the cosmic microwave background anisotropies we get that ∑mν < 1 eV (Wong, 2011). • The smallness of the neutrino mass is explained in a most natural way by suggeson that neutrino mass eigenstates are Majorana parcles (Schechter & Valle, 1980). • Despite great experimental efforts (CUORICINO, NEMO, COBRA, etc.) to reveal the nature of neutrinos, i.e. to say whether neutrinos are Dirac or Majorana, this issue is sll open. Majorana and Weyl spinors

• Majorana condion in an c i,12 ∗ extended sense ψγψκψκ== = • We can choose a specific form of γ-matrices to make a Majorana real. • One can express a Majorana ⎛⎞iση∗ ⎛⎞ξ ψψ==2 or spinor in terms of the two ηξ⎜⎟ ⎜⎟i ∗ component Weyl spinors. ⎝⎠η ⎝⎠− σξ2 • The representaon via Weyl spinors is preferable since neutrinos are le-handed parcles. •  ∗ The wave equaons for a Weyl η − (σ ∇)η + mσ 2η = 0 spinors can be formally derived  ξ + (σ ∇)ξ − mσ ξ ∗ = 0 from the Dirac equaon. 2 Variaon methods to describe Weyl fields dynamics • A Lagrangian for a Dirac field is valid for both first- and L =ψγ(iµ∂−m ) ψ second-quanzed spinors µ • A Lagrangian for a Weyl field i(† µ ) can be obtained by a direct L =ησ∂µ η substuon of a Majorana iiT†∗ spinor (expressed in terms of −mmηση22+ ηση Weyl spinors) in the Dirac 22 Lagrangian i T • Mass term in the Weyl Lm =− mηση2 2 Lagrangian vanishes if η is a i c-numbers spinor +=mηση† ∗ 0 2 2 Possible soluons of the mass term puzzle • Case (1957) suggested that, since η is a ½ field, it should be expressed via an-commung variables. This kind of suggeson is reasonable but not logical. Case (1957) also “quanzes” this field, i.e. he expresses it via creaon and annihilaon operators. However, this procedure is just a re-expression of already quanzed objects in terms new variables rather than a generic quanzaon. • Schechter & Valle (1981) claimed that a massive Weyl field does not have a proper descripon in terms of first quanzed c-number spinors. The only possible descripon of a classical Weyl field can be made using Grassmann variables (g-numbers). However it is in contradicon to the fact that a classical fermionic system should have equivalent c- & g-numbers descripons. • Ahluwalia et al. (2010) introduced an exoc filed, Eigenspinoren des LadungsKonjugationsOperators (ELKO), which allowed them construct a Lagrange formalism for a c-number Majorana spinor. Hamilton approach for the descripon of massive Weyl fields (c-numbers)   H = d3r π T (σ ∇)η −η† (σ ∇)π ∗ + m⎡η†σ π + π †σ η⎤ ∫ { ⎣ 2 2 ⎦} If the parcle mass m is real, the funconal H is also real, as it should be for a classical Hamiltonian.

δ H  ∗ δ H  ∗ ∗ η = = (σ ∇)η − mσ 2η , π = − = (σ ∇)π + mσ 2π δπ δη

If we introduce the new variable ξ = iσ2 π, we reproduce the wave equaon for a right-handed neutrino. The canonical equaons for η and π decouple. It means that one cannot reconstruct a Lagrangian. Quanzaon of c-number Weyl spinors 1d3p Em⎡⎤⎛⎞⎛⎞ m η()xawaweawawe=+ 1 −−∗iipx++ ∗ px ∫ 3/2 ⎢⎥⎜⎟⎜⎟−− ++ +− −+ 2(2)π ||pp⎣⎦⎝⎠⎝⎠EE++ || || p id3p Em⎡⎤⎛⎞⎛⎞ m ξ()xbwbwebwbwe=++ 1 −∗iipx +− ∗ px ∫ 3/2 ⎢⎥⎜⎟⎜⎟++ −− −+ +− 2(2)π ||pp⎣⎦⎝⎠⎝⎠EE++ || || p

1 ⎛⎞E HdE3****ppppppppp1()()()()()()()() abbaabba =+⎜⎟{{ −− + − −−− ++ + + 4||∫ ⎝⎠p 2 ⎛⎞m ⎡⎤ab()pp**** () ba () pp () ab () pp () ba () pp () ++⎜⎟⎣⎦−− − −−− ++ + + ⎝⎠E+ ||p

m −2iEt +i()()()()()()()()eab[ −−pp−+ b −− pp a −+ b +− pppp a ++ ab ++−] ||p +ea−2iEt ⎡⎤**()ppbbabaab (− )+ * (− pp ) * ()+ * (− pppp ) * ()+ ** () (− ) ⎣⎦−− − − + + ++ }} Two independent ways of quanzaon (2) a (p) = b (p) ±  a (p) b (p) (1) ± = ± 1 acc* ±=±( −+) ⎡a (k),a* (p)⎤ = δ 3(k − p) 2 ⎣ σ σ ⎦+ ⎡c (k),c* (p)⎤ = δ 3(k − p) ⎣ σ σ ⎦+ ⎡⎤⎛⎞aa**⎛⎞aa ⎛⎞ ⎛⎞ ⎡⎤⎛⎞aa**⎛⎞aa ⎛⎞ ⎛⎞ 3 −−++Pppd3 −−++ HE=d,p ⎢⎥⎜⎟**×+ ⎜⎟×=⎢⎥⎜⎟**×⎜⎟+ ⎜⎟× ⎜⎟ ∫ cc⎜⎟cc ⎜⎟ ∫ cccc ⎣⎦⎢⎥⎝⎠−⎝⎠− ⎝⎠+ ⎝⎠+ ⎣⎦⎢⎥⎝⎠−⎝⎠− ⎝⎠+ ⎝⎠+ Expressions (1) and (2) may be regarded as quantum Majorana conditions. Lagrange and Hamilton formalisms for g-number Weyl spinors

• We start from the †T†µ ii∗ L =i(ησ∂− ) ηmm ηση22+ ηση Lagrangian µ 22 Φ = π − iη* = 0, Φ = π * = 0, • 1 2 This system has two ∂ L ∂ L π = r , π * = r second class constraints ∂η * ∂η ii • The Hamiltonian H =i(ηη†T†σ∇ )+mm ησηηση− ∗ 2222 • The extended +ΦΦλλ+ Hamiltonian H=H11122

ηη(,),(,)xytt**= ηηη ,−ΦΦ , C , ηδ *3= ( xy− ), • The Dirac bracket { }D { } { i} ij{ j } −1 Cij={ΦΦ i, j }  η = η,H = (σ ∇)η + iσ mπ , • Wave equaon in the { }D 2  equivalent form π = π ,H = (σ T∇)π + iσ mη { }D 2 Quanzaon of massive Weyl fermions (g-numbers) The quanzaon of a fermionic field can be made by the replacement (Gitman & Tyun 1990) η → ηˆ, π → πˆ, ⎡ηˆ(t,x),πˆ(t,y)⎤ = i η(t,x),π (t,y) = iδ 3(x − y) ⎣ ⎦+ { }D The realizaon of the quanzed fields

1d3ppEm+ ||⎡⎤⎛⎞⎛⎞ m ηˆ()xawaweawawe=ˆˆˆˆ−−i†px++ †i px , ∫ 3/2 ⎢⎥⎜⎟⎜⎟−− ++ +− −+ 2(2)π 2EE⎣⎦⎝⎠⎝⎠++ ||pp E || †3 †† ⎡⎤aaˆˆ(),(kk′′′′′ )=δδ ( kkkk− ), aa ˆˆ (),( ′ )= 0, ⎡⎤ aa ˆˆ (),( kk ′ ′ ) = 0. ⎣⎦σσ++ σσ[ σσ]+ ⎣⎦ σσ The total energy of quanzed fields EEaaaa==dd33††rpˆˆ ++ ˆˆ divergentterms tot ∫∫H ( −− ++) Majorana neutrinos in external fields

• Now we generalize our treatment to arbitrary neutrino types. Thus wave funcons acquire an index a = 1,2… • We will be mainly interested in neutrino interacons with a background maer and an external electromagnec field. • Despite the neutrino charge is zero, a neutrino may parcipate in electromagnec interacon owing to the possible presence of anomalous magnec moments (Giun & Studenikin, 2009). • The interacon of acve neutrinos with maer is diagonal in the flavor basis. • We work with mass eigenstates since only in mass eigenstates basis we can say if neutrinos are Dirac of Majorana parcles (Schechter & Valle, 1980). • The effecve potenals should be transformed to the mass eigenstates basis. • The interacon with external fields is non-diagonal in the mass basis unless we discuss a specific case. Lagrangian for Majorana neutrinos in background maer and electromagnec fields ii L =i(ησ†T†*†µµ∂− ) ηmmg ηση + ηση− ηση aµµ a22 a a22 a a a a ab a b 1 −−⎡⎤µη†*†Tσ(i)BE ση+ () µησ (i) BE +σ η 2 ⎣⎦ab a22 b ab a b 1 −⎡⎤εη†*†Tσ(i)EB+++ ση () ε ησ (i) EBσ η 2 ⎣⎦ab a22 b ab a b

•The matrices of magnec moments (µab) and electric dipole moments (εab) are Hermian and ansymmetric. No diagonal interacon with electromagnec field is possible. µ •The matrix of the interacon with background maer (g ab) is Hermian. The components of this this matrix for different oscillaons channels were found by 0 MD & Studenikin (2002). The zero component, g ab, is proporonal to the effecve maer density. The spaal components, gab, are the linear combinaons of maer velocies and polarizaons •The complete system which involves terms nondiagonal in neutrino types can be analyzed only in frames of the perturbaon theory. That is why we shall keep only diagonal interacon with external fields. Propagators of massive Weyl fermions (g-numbers) in background maer

µµiimg 0,ii** mg 0 The evoluon equaons σησπµµµµµµ∂−22 += σησπση∂− + σπ = The causal (Gitman & Tyun 1990) ⎛⎞()iσT*gmµ i †Tˆˆ 4∂−t ∇−σσµ − 2 Sxcc()0()()0,(,),()(),− y= TΞΞ x y Ξ=πη KSx =− δ x K=⎜⎟ { } ⎜⎟i()imgσ µ ⎝⎠−∂−∇σσ2 t +µ The modified propagator

S = −S S S γ 0γ 5, S = diag(σ ,σ ), ⎡Γ µ (i∂ + ig Γ5 ) − mΓ5 ⎤S = δ 4 (x), Γ n = −iγ 5γ µ ,−iγ 5 , ⎡Γ k ,Γ n ⎤ = 2η kn c 2 c 2 2 2 0 ⎣ µ µ ⎦ c ( ) ⎣ ⎦+ The path integral representaon of a propagator

x ⎛ ∂ ⎞ ∞ out S = exp iΓ n l de dχ M(e)De Dx Dπ Dν Dψ exp i(S + S ) +ψ (1)ψ n (0) c ⎜ n ⎟ ∫ 0 ∫ 0 ∫ ∫ ∫ ∫ ∫ { cl GF n } ⎝ ∂θ ⎠ 0 e x ψ (0)+ψ (1)=θ 0 in θ=0 1 ⎡ z2 e ⎛ 2 ⎞ ⎤ S = − − M 2 + x d µ + iχ mψ 5 + ψ µ d − iψ ψ n dτ , cl ∫ ⎢ 2e 2 µ ⎜ 3 µ ⎟ n ⎥ 0 ⎣ ⎝ ⎠ ⎦ z µ = xµ + iχψ µ , M 2 = m2 + g 2 +16∂ g µψ 0ψ 1ψ 2ψ 3, d = 2iε gνψ αψ β µ µ µναβ Wave equaons for Weyl fermions (c-numbers) in a background maer and an external electromagnec field

The wave equaons for η and ξ can be derived directly from the Dirac equaon. µ We should recall that the vector term ~ g ab γµ ψb of the maer interacon is µ 5 absent for Majorana neutrinos and the axial vector term ~ g ab γµ γ ψb is twice the corresponding contribuon for Dirac neutrinos.

 *  * 0  ηa − (σ ∇)ηa + maσ 2ηa − µabσ (B − iE)σ 2ηb + i(gab +σ gab )ηb = 0    ξ + (σ ∇)ξ − m σ ξ * + µ σ (B + iE)σ ξ * − i(g0 −σ g )ξ = 0 a a a 2 a ab 2 b ab ab b

µ 0 The interacon with maer is characterized by a four vector g ab = (g ab, gab). We discuss the situaon when only acve neutrinos are present and CP µ invariance is conserved. In this case the matrix (g ab) is symmetric. Hamiltonian for Weyl spinors (c-numbers) in external fields

⎡   H = d3r π T (σ ∇)η −η† (σ ∇)π * + m ⎡η†σ π + π †σ η ⎤ ∫ ⎢∑{ a a a a a ⎣ a 2 a a 2 a ⎦} ⎣ a     + µ ⎡π Tσ (B − iE)σ η* +ηTσ σ (B + iE)⎤ − i⎡π T g0 +σ g η −η† g0 +σ g π * ⎤ ⎤ ∑{ ab ⎣ a 2 b a 2 ⎦ ⎣ a ( ab ab ) b a ( ab ab ) b ⎦}⎦⎥ ab

δ H     ( ) m * (B iE) * i g0 g , ηa = = σ ∇ ηa − aσ 2ηa + µabσ − σ 2ηb − ( ab +σ ab )ηb δπ a δ H    π = − = (σ ∗ ∇)π + m σ π * − µ σ σ (B + iE)π * + i g0 +σ *g π a δη a a 2 a ab 2 b ( ab ab ) b a Quanzaon of Weyl fields (c-numbers) in maer

We separate the total Hamiltonian into H0 and Hint parts. H0 contains terms diagonal in neutrino types whereas Hint is nondiagonal in neutrino types.

Let us discuss the case nonmoving and unpolarized maer, gab = 0.

3 ⎧⎫−++**− (0)d p ⎪⎪⎡⎤⎡⎤−−−iiiiEtaaaammaa++ Et ipr Et−− Et i pr ηaa(,)r taweaweeaweawee=3/2⎨⎬−− 0 aa+++− 0 a+ ∫ (2 )⎢⎥⎢⎥Eg+ |pp |( ) Eg− | | ( ) π ⎩⎭⎪⎪⎣⎦⎣⎦aaa+− aaa++ 3 ⎧⎫+−**−+ (0)d p ⎪⎪⎡⎤+−−iiEtaama − Et ipr⎡⎤−iiEtaama + Et −i pr ξaa(,)r tbwebwee=+3/2⎨⎬+ 0 a−+bweaa+−0 bwee− ∫ (2 )⎢⎥Eg− |p | ⎢⎥( ) Eg+ ||p ( ) π ⎩⎭⎪⎪⎣⎦aaa++ ⎣⎦aaa+−

2 Energy levels of a neutrino in a background maer: E ± = m2 + | p | g0 a a ( aa ) * a± (p) E ± | p | g0 4b± (p) | p | g0 , ⎡a± (k),⎡a± (p)⎤ ⎤ = δ δ 3(k − p) a ( a +  aa ) = a (  aa ) ⎢ a ⎣ a ⎦ ⎥ ab ⎣ ⎦+

** ** HEaaEaad,3p ⎡⎤−− − ++ + Ppp=+d3 ⎡⎤aa−− aa ++ =+∑ aa( ) a aa( ) a ∫ ∑ ⎢⎥( aa) ( aa) ∫ a ⎣⎦⎢⎥a ⎣⎦ Nondiagonal neutrino interacon We work in the forward scaering regime, i.e. only the terms, which conserve the number of parcles, should be le. δρ3*()()()(),(,)pk− p=aa p k A =± a iρ = ⎡ρ, H ⎤ AB B A ⎣ int ⎦− We discuss the case when E = 0 and two ultrarelavisc neutrinos, a = 1,2, with 0 µ12 = iµ, µ21 = - iµ, and k >> max(ma, g aa). 2 22 00 00 mm ± ma 0 † −Φi(+gt11 ) i(Φ− gt 22 ) −Φ− i( gt 11 ) i( Φ+ gt 22 ) 12− E =| k | +  g + ρρqm ==UU,diag, U eeee , , , Φ= a 2 | k | aa ( ) 2|k |

⎛ Φ + g0 g0 0 −iµ | B | sinϑ ⎞ ⎜ 11 12 kB ⎟ 0 0 ⎜ g21 −Φ + g22 iµ | B | sinϑkB 0 ⎟ iρqm = ⎡H qm ,ρqm ⎤ , H qm = ⎜ ⎟ ⎣ ⎦− 0 0 ⎜ 0 −iµ | B | sinϑkB Φ − g11 −g12 ⎟ ⎜ ⎟ ⎜ iµ | B | sinϑ 0 −g0 −Φ − g0 ⎟ ⎝ kB 21 22 ⎠ Here we reproduce the standard effecve Hamiltonian for the descripon of neutrino flavor and spin-flavor oscillaons in a background maer and a magnec field (Lim & Marciano, 1988; Mannheim 1988, Giun, et al., 1992). The self-interacon of neutrinos Neutrinos can interact between themselves by exchanging a Z-boson. This kind of interacon is significant when the neutrino density is high, e.g., in a supernova explosion. HGGdd,3Lrrµµ 3†† GG SL=∑∑ab cdψγψ aµµ b⋅→⋅ ψγψ c d ab cd ηση a b ηση c d ∫∫abcd abcd In the forward scaering approximaon for ultrarelavisc neutrinos we get the contribuon to the effecve quantum mechanical Hamiltonian:

⎛⎞ρρ−− −+ HHHqm()k == diag( −− , ++ ) ,ρ qm ⎜⎟ ⎝⎠ρρ+−++ d3p H =2 1− cosϑρρGG tr⎡⎤⎡⎤ (pp ) − GTT ( )+ G ρρ ( pp )− ( ) G , −−∫ (2π )3 ( kp ){ ⎣⎦⎣⎦ −− ++ −− ++ } d3p H =2 1− cosϑρρρρGGTT tr⎡⎤⎡⎤ (pp ) − G ( )+ G T ( pp )− T ( ) G T ++∫ (2π )3 ( kp ){ ⎣⎦⎣⎦ ++ −− ++ −− } We can reproduce the case of Dirac neutrinos (Sigl & Raffelt, 1993) if we set T H++ = 0 and H -- ≠ 0, as well as introduce the anneutrino density matrix ρ = ρ + + . Discussion: Weyl fields in vacuum

• We applied the Hamilton formalism to the descripon of classical (or c-number, or first-quanzed) massive Weyl fields. Previously it was claimed by Schechter & Valle (1981) that a massive Weyl spinor cannot be descbed in terms of a first- quanzed field. • We carried out a canonical quanzaon of a system. It was found that, owing to the degeneracy of the energy levels, two independent ways to quanze a Weyl field are possible. • We introduced quantum Majorana conditions, which connects “parcle”, η, and “anpatricle”, ξ, degrees of freedom. The classical Majorana condion, ψc ~ ψ, cannot be regarded as equality of “parcles” and “anparcles” since it is applied on a classical level before quanzaon. • The descripon of massive Weyl fermions in terms of c-number spinors is consistent with the standard g-number descripon. Discussion: Interacng Weyl fields

• We generalized our formalism to include the interacon with a background maer and an external electromagnec field. • We obtained a classical Hamiltonian which then was used to quanze the system in case when only the diagonal interacon with nonmoving and unpolarized maer is present. • The account of the nondiagonal interacon with external fields, if we work in the forward scaering approximaon, allowed us to reproduce the effecve Hamiltonian for the descripon of neutrino spin-flavor oscillaons in maer and magnec field. • We also studied the neutrino self-interacon. Again in the forward scaering limit, we derived the contribuon to the effecve Hamiltonian. It was shown that the self-interacon does not directly cause a spin-flip, but certainly it influences the process of spin-flavor oscillaons of neutrinos. Although there is no direct correspondence to Dirac neutrinos case, we showed how one can reconstruct an effecve Hamiltonian for Dirac neutrinos. • We derived the most general representaon of a propagator of a massive Weyl field interacng with a background maer. The characteriscs of the background maer (number density, velocity, and polarizaon) can depend on me and spaal coordinates. Acknowledgements

• I am thankful to A.E. Lobanov, J. Maalampi, G.G. Raffelt, and V.B. Semikoz for helpful discussions; • FAPESP (Brazil) for a grant; • The organizers of SILAFAE 2012 for the invitaon.