Possible effects of space-time nonmetricity on neutrino oscillations

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Possible effects of space-time nonmetricity on neutrino oscillations

M. Adak*

Department of Physics, Pamukkale University, 20100 Denizli, Turkey T. Dereli†

Department of Physics, Koc¸ University, 34450 Sarıyer-I˙stanbul, Turkey

L. H. Ryder‡

Department of Physics, University of Kent, Canterbury, Kent CT2 7NF, United Kingdom

共Received 20 March 2003; published 1 June 2004兲

The contribution of gravitational neutrino oscillations to the solar neutrino problem is studied by construct-ing a Dirac Hamiltonian and calculatconstruct-ing the correspondconstruct-ing dynamical phase in the vicinity of the Sun in a non-Riemann background Kerr space-time with torsion and nonmetricity. We show that certain components of nonmetricity and the axial as well as nonaxial components of torsion may contribute to neutrino oscillations. We also note that the rotation of the Sun may cause a suppression of transitions among neutrinos. However, the observed solar neutrino deficit could not be explained by any of these effects because they are of the order of Planck scale.

DOI: 10.1103/PhysRevD.69.123002 PACS number共s兲: 95.30.Sf, 13.15.⫹g, 14.60.Pq

I. INTRODUCTION

Neutrinos have always attracted a lot of attention in high energy physics关1兴. A major problem of interest at present is the solar neutrino problem. The Sun is a strong source of electron neutrinos␯ebecause of the thermonuclear reactions

taking place in its core. According to the standard solar model, the number of␯eto be emitted from the Sun can be

predicted. At the same time, the flux of electron neutrinos coming from the Sun can be measured on Earth. The mea-sured amount of ␯e is approximately one-third of the

pre-dicted amount. Essentially, this is the so-called solar

neu-trino problem. One well-known solution to this problem is

provided by the assumption of neutrino oscillations 关1,2兴. Briefly stated, the neutrino oscillations imply that the elec-tron neutrinos coming out of the Sun may be converted to other neutrino species, muon ␯_␮ and tau ␯_␶, during their journey towards the Earth, assuming neutrinos to have a mass, whereas the standard electroweak model asserts zero mass for them. It should also be noted that all of the above arguments have been cast in Minkowski space-time. How-ever, we know that we live in a curved space-time—perhaps even in a curved space-time with torsion and nonmetricity. Therefore, in more recent years, physicists have turned their attention to specifically gravitational contributions to neu-trino oscillations—see 关3–7兴 and references therein. We re-cently investigated the effects of space-time torsion on neu-trino oscillations 关8兴—see also 关9,10兴. The essence of this work is to calculate the dynamical phase of neutrinos by finding the form of the Hamiltonian H from the Dirac equa-tion in a non-Riemannian space-time. The phase then follows from the formula

iប⳵␺

⳵t ⫽H␺⇒␺共t兲⫽exp

冉

⫺ i

ប

冕

Hdt

冊

␺共0兲, 共1兲 where ␺ is a Dirac 4-spinor and H is a 4⫻4 matrix. The Hamiltonian H will depend, for example, on momentum pជ, and this is expressed not as a differential operator but simply as a vector.1 In this paper we investigate within the same approach the possible effects of space-time nonmetricity on neutrino oscillations.

II. SPACE-TIME GEOMETRY

Space-time is denoted by the triple兵M ,g,ⵜ其where M is a four-dimensional differentiable manifold, equipped with a Lorentzian metric g which is a共0,2兲-type covariant, symmet-ric, nondegenerate tensor, and ⵜ is a connection which de-fines parallel transport of vectors共or more generally tensors兲. We shall give a coordinate system set up at a point p苸M by coordinate functions 共or independent variables兲兵x␣( p)其, ␣ ⫽0ˆ,1ˆ,2ˆ,3ˆ. This coordinate system forms a set of natural 共or

coordinate兲 reference frame at p as 兵(⳵/⳵x␣)( p)其, with shorthand notation ⳵_␣⬅⳵/⳵x␣. This natural reference frame is a basis vector set for the tangent space at p, denoted by

Tp( M ). Similarly, differentials兵dx␣( p)其 of coordinate

func-tions 兵x␣( p)其 at p form a natural共or coordinate兲 reference

coframe in the cotangent space at p, denoted by T_p*( M ). Interior product of the basis vectors with the basis covectors is defined by the Kroenecker symbol:

dx␣

冉

⳵

⳵x␤

冊

⬅ı⳵␤

dx␣⫽␦_␤␣. 共2兲 *Electronic address: madak@pamukkale.edu.tr

†_{Electronic address: tdereli@ku.edu.tr} ‡_{Electronic address: l.h.ryder@ukc.ac.uk}

1_{The exponential of exp}_{关⫺(i/ប)兰Hdt兴 is defined by its power} se-ries expansion.

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In general, any set of linearly independent vectors in tangent space, Tp( M ), can be taken as basis vectors and these

vec-tors can be orthonormalized by, for example, the Gram-Schmidt process. We denote a set like this by 兵Xa其, a

⫽0,1,2,3, and call it an orthonormal reference frame. In this case the metric defined on M satisfies the relation

g共X_a,X_b兲⫽␩_ab, 共3兲 where ␩ab is known as the Minkowski metric which is a

matrix whose diagonal terms are ⫺1,1,1,1 and off-diagonal terms are zero. The basis set dual to the orthonormal refer-ence frame is denoted by 兵ea其, a⫽0,1,2,3, and called the

orthonormal reference coframe. 兵X_a其, and its dual兵ea_其

sat-isfies the following set of equalities that is another manifes-tation of Eq.共2兲:

ea共Xb兲⬅ıX_b共ea兲⫽␦b a

. 共4兲

Here we adhere to the following conventions: indices de-noted by Greek letters ␣, ␤, . . .⫽0ˆ,1ˆ,2ˆ,3ˆ and ␮, ␯, . . . ⫽1ˆ,2ˆ,3ˆ are holonomic or coordinate indices, and a, b, . . . ⫽0,1,2,3 and i, j, . . . ⫽1,2,3 are anholonomic or frame in-dices. In terms of the local coordinate frame ⳵_␣( p), the or-thonormal frame X_a( p) can be expanded via the so-called vierbein共or tetrad兲 h␣a( p) as

Xa共p兲⫽h␣a共p兲⳵␣共p兲. 共5兲

In order for X_ato serve as an anholonomic basis, the h␣_a( p) are required to be nondegenerate—i.e., det h␣a( p)⫽0. In

T_p*( M ) an orthonormal coframe ea( p) can be expanded in terms of the local coordinate coframe dx␣( p) as

eb共p兲⫽hb_␤共p兲dx␤共p兲. 共6兲

The inverse vierbein hb

␤( p) has to be nondegenerate as well.

Moreover, the duality of the frame and the coframe requires for the vierbein and its inverse to satisfy

ı_X

ae b_⫽h␣

a共p兲hb␣共p兲⫽␦a

b_. _共7兲

We set the space-time orientation by the choice⑀0123⫽1. The nonmetricity 1-forms, torsion 2-forms, and curvature 2-forms are defined by the Cartan structure equations

2Qab⫽⫺D␩abª⌳ab⫹⌳ba, 共8兲

Ta⫽Deaªdea⫹⌳ab⵩eb, 共9兲

Rab⫽D⌳abªd⌳ab⫹⌳ac⵩⌳cb.

共10兲

d, D, ıa, and * denote the exterior derivative, the covariant

exterior derivative, the interior derivative, and the Hodge star operator, respectively. The linear connection 1-forms can be decomposed in a unique way according to 关11兴

⌳a

b⫽␻ab⫹Kab⫹qab⫹Qab, 共11兲

where␻ab are the Levi-Civita` connection 1-forms,

␻a

b⵩eb⫽⫺dea, 共12兲

Kab are the contortion 1-forms,

Kab⵩eb⫽Ta, 共13兲

and qab are the antisymmetric tensor 1-forms,

qab⫽⫺共ıaQbc兲ec⫹共ıbQac兲ec. 共14兲

It is cumbersome to take into account all components of nonmetricity and torsion in gravitational models. Therefore we will be content with dealing only with certain irreducible parts of them to gain physical insight. The irreducible de-compositions of torsion and nonmetricity invariant under the Lorentz group are summarily given below. For details one may consult Ref.关12兴. The nonmetricity 1-forms Q_abcan be split into their trace-free Q¯ab and trace parts as

Q_ab⫽Q¯_ab⫹1

4␩abQ, 共15兲

where the Weyl 1-form Q⫽Qaa and␩abQ¯ab⫽0. Let us

de-fine ⌳bªıaQ¯ab, ⌳ª⌳aea, ⌰bª*共Q¯ab⵩ea兲, ⌰ªeb⵩⌰b, ⍀aª⌰a⫺ 1 3ıa⌰, 共16兲

so as to use them in the decomposition of Qab as

Qab⫽Qab (1)_⫹Q ab (2)_⫹Q ab (3)_⫹Q ab (4) , 共17兲 where Q_ab(2)⫽1 3*共ea⵩⍀b⫹eb⵩⍀a兲, 共18兲 Q_ab(3)⫽2 9

冉

⌳aeb⫹⌳bea⫺ 1 2␩ab⌳

冊

, 共19兲 Q_ab(4)⫽1 4␩abQ, 共20兲 Q_ab(1)⫽Qab⫺Qab (2)_⫺Q ab (3)_⫺Q ab (4)_. _共21兲 We have ıaQ_ab(1)⫽ıaQ_ab(2)⫽0, ␩abQ_ab(1)⫽␩abQ_ab(2)⫽␩abQ_ab(3) ⫽0, and ea_⵩Q ab

(1)_{⫽0. In a similar way the irreducible} de-composition of Ta’s invariant under the Lorentz group is given in terms of

␣⫽ıaTa, ␴⫽ea⵩Ta, 共22兲

so that

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where Ta(2)⫽1 3e a_⵩_␣_, _共24兲 Ta(3)⫽1 3ı a_␴_, _共25兲

ta_ªTa(1)_⫽Ta_⫺Ta(2)_⫺Ta(3)_. _共26兲

Here ıata⫽ıaTa(3)⫽0, ea⵩ta⫽ea⵩Ta(2)⫽0. To give the

contortion components in terms of the irreducible compo-nents of torsion, we first write

2K_ab⫽ı_aT_b⫺ı_bT_a⫺共ı_aı_bT_c兲ec 共27兲

from Eq. 共13兲 and then substituting Eq. 共23兲 into above we find 2Kab⫽ıatb⫺ıbta⫺共ıaıbtc兲ec⫹ 2 3共ea⵩ıb␣⫺eb⵩ıa␣兲 ⫹2 3共ıaıb␴兲⫺ 1 3共ıaıbıc␴兲e c_. _共28兲

In components Kab⫽Kc,abec, ta⫽12tbc,aebc, ␣⫽Faea, ␴

⫽(1/3!)␴abceabcthis becomes

Kc,ab⫽

1

2共tac,b⫺tbc,a⫹tab,c兲⫹ 1

3共Fb␩ac⫺Fa␩bc兲⫺ 1 6␴abc.

共29兲 III. HAMILTONIAN OF A DIRAC PARTICLE

IN ARBITRARY SPACE-TIMES

The Dirac equation in a non-Riemannian space-time with torsion and nonmetricity is written as 关13–15兴

*␥⵩D␺⫹M*1␺⫽0 共30兲

in terms of the Clifford algebra Cᐉ_3,1-valued 1-forms ␥ ⫽␥a_e

a and M⫽mc/ប. We use the Dirac matrices

␥0_⫽i

冉

I 0 0 ⫺I

冊

,␥

i_⫽i

冉

0 ␴ i

⫺␴i ₀

冊

,

where ␴i are the Pauli matrices. ␺ is a 4-component complex-valued Dirac spinor whose covariant exterior de-rivative is given explicitly by

D␺⫽d␺⫹1 2⌳ [ab]_␴ ab␺⫹ 1 4Q␺, 共31兲 where ␴ab⫽ 1 4关␥a,␥b兴共32兲 are the spin generators of the Lorentz group. We write it out explicitly as D␺⫽⳵t␺dx0ˆ⫹⳵␮␺dx␮⫹ 1 2e c_⍀ c,ab␴ab␺⫹ 1 4e c_Q c␺ ⫽h0ˆ cec⳵t␺⫹h␮cec⳵␮␺⫹ 1 2e c_⍀ c,ab␴ab␺ ⫹1 4e c_Q c␺, 共33兲

where⌳[ab]ª⍀ab⫽⍀c,abecis the antisymmetric part of the

full connection 1-form and Q⫽Qaea, and using *␥

⫽␥a*ea and the identity*ea⵩eb⫽⫺␩ab*1 we calculate

*␥⵩D␺⫽

冉

⫺h0ˆ c␥c⳵t␺⫺h␮c␥c⳵␮␺ ⫺1 2⍀c,ab␥ c_␴ab_␺⫺1 4Qc␥ c_␺

冊

_*_1. _共34兲

Putting this into Eq. 共30兲 we obtain

h0ˆc␥c⳵t␺⫽⫺h␮c␥c⳵␮␺⫹M␺⫺

1 2⍀c,ab␥

c_␴ab_␺

⫺1₄Qc␥c␺. 共35兲

We multiply this from the left by

iប共h0â␥a兲⫺1⫽ ⫺iប b2 共h 0ˆ a␥a兲, 共36兲 where b2ª共h0ˆ0兲2⫹h0îh0î. 共37兲

When we compare the result with the Schro¨dinger equation

iប⳵␺

⳵t ⫽H␺, 共38兲

we deduce the Dirac Hamiltonian matrix 关7,8,15–20兴

H⫽ c b2 h0âh␮b␥a␥biប⳵␮⫺ imc2 b2 h0â␥a ⫹iបc 2b2 h0ˆd⍀c,ab␥d␥c␴ab⫹ iបc 4b2 h0âQb␥a␥b. 共39兲

The right-hand side of Eq. 共39兲 need not be a Hermitian matrix in general; e.g., if h0ˆ_i⫽0, then the mass term contains an anti-Hermitian part such as

H⫽H0⫹iH1, 共40兲 where H₀⫹⫽H0 and H1⫹⫽H1. However, the decomposition 共40兲 is frame dependent. That is, we can always find a local Lorentz frame in which Hamiltonian is fully Hermitian

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关17,18兴. First we can get rid of the anti-Hermitian part of the mass term by diagonalizing the matrix h␣avia a frame

trans-formation

⳵␣共x兲→⳵␤共x兲L⫺1␤␣共x兲,

g_␣␤共x兲→g_␥_␦共x兲L⫺1␥_␣L⫺1␦_␤. 共41兲 Thus2

h␣a共x兲 → f 共x兲␦a␣, 共42兲

where x stands for x␣ and f (x) is composed of h␣a(x).

Un-der this change Eq. 共39兲 goes over to

H→H⫽c f1共x兲␥0␥iiប⳵iˆ⫺imc2f2共x兲␥0

⫹iបc f3共x兲⍀c,ab␥0␥c␴ab⫹iបc f4共x兲Qb␥0␥b,

共43兲 where fi(x) are composed of h␣a(x). Putting in the

defini-tion

⍀c,ab⫽⑀abcdSd 共44兲

and using the identity

␥a_␴bc_⫽1 2␩ ab_␥c_⫺1 2␩ ac_␥b_⫺1 2⑀ abcd_␥ d␥5, 共45兲 where␥5⫽␥0␥1␥2␥3, the Hamiltonian matrix becomes

H⫽c f1共x兲␥0␥iiប⳵iˆ⫺imc2f2共x兲␥0⫹iបcNa共x兲␥0␥a

⫹iបc f5共x兲Sa␥0␥a␥5, 共46兲

where we introduced

N_aª f₃共x兲⍀b,_ba⫹ f₄共x兲Q_a. If we now define the canonical momenta

piª⫺iប

冉

⳵iˆ⫹

N_i共x兲

f1共x兲

冊

共47兲 and assume

pi⫹⫽pi, 共48兲

Eq. 共46兲 takes the form

H⫽ f1共x兲cpi␥0␥i⫹imc2f2共x兲␥0⫹iបc f5共x兲Sa␥0␥a␥5

⫺iបcN0共x兲. 共49兲

In order eliminate the last term in Eq. 共49兲 one may further perform a locally unitary transformation

␺共x兲 → U⫹_共x兲_{␺共x兲, H → U}⫹_{共x兲HU共x兲} _共50兲 and obtain H→H⫽ f1共x兲cpiU⫹共x兲␥0␥iU共x兲 ⫹imc2_f 2共x兲U⫹共x兲␥0U共x兲 ⫹iបc f5共x兲SaU⫹共x兲␥0␥a␥5U共x兲 ⫺iបc关 f1共x兲U⫹共x兲␥0␥i⳵iˆU共x兲⫹N0共x兲兴.

共51兲 Under the solvable matrix equation

U⫹共x兲␥0␥i⳵iˆU共x兲⫽⫺

N₀共x兲 f1共x兲

, 共52兲

we give the final form of our Hermitian Hamiltonian matrix 共up to a sign兲 by the expression

H⫽ f1共x兲cpi␥0␥i⫹imc2f2共x兲␥0⫹iបc f5共x兲Sa␥0␥a␥5. 共53兲 IV. NEUTRINO OSCILLATIONS IN THE KERR

BACKGROUND

Here we construct the Hamiltonian matrix of a Dirac par-ticle共i.e., a massive neutrino兲 of mass m in the background space-time geometry of a heavy, slowly rotating body of mass M such as the Sun. Its exterior gravitational field will be described by weak constant, uniform torsion and non-metricity fields, together with the Kerr metric关21兴:

⫹(1,3) where SO⫹(1,3) is special orthochronous Lorentz

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dea⫹␻ab⵩eb⫽0 ⇔ ␻ab⫽⫺ 1 2ıadeb⫹ 1 2ıbdea ⫹1₂共ıaıbdec兲ec, 共56兲

calculate the Levi-Civita` connection 1-forms

␻ 1 0 _⫽M G关r 2_⫺共a/c兲2_cos2_␪_兴 ␳4_c dt ⫹关共MG/c 2_⫺r兲␳2_⫺2MGr2_/c2_{兴a sin}2_␪ ␳4_c d␸, ␻ 3 2 _⫽2 M Gra cos␪ ␳4_c2 dt ⫹ ⌬

冉

a c

冊

2 sin2␪⫺关r2⫹共a/c兲2兴2 ␳4 cos␪d␸, ␻ 2

⫺1 dr丢dr ⫹rd␸丢rd␸⫺4 a c M G c2r2 cdt丢rd␸. 共58兲 We also write the orthonormal coframe approximately up to

O(a/rc) as e0⫽ f cdt⫺a f c d␸, e 1_⫽1 fdr, e2⫽0, e3⫽⫺a rdt⫹rd␸, 共59兲 where ⌬ ␳2⬅ f 2_⯝1⫺2 M G c2r . 共60兲

The inverses of these relations to the same order of approxi-mation are cdt⫽1 fe 0_⫹ a rce 3_, _dr_{⫽ f e}1_, _d_␪_⫽0, d␸⫽ a f r2ce 0_⫹1 re 3_, _共61兲 which give h0ˆ0⫽ 1 f, h 0ˆ 3⫽ a cr, h 1ˆ 1⫽ f , h3ˆ0⫽ a f cr2 , h3ˆ3⫽ 1 r, 共62兲 with all other components neglected. To this order of ap-proximation Eq.共57兲 gives

␻01⯝ f

⬘

e0⫹ a cr2 e3, ␻03⯝ a cr2 e1, ␻31⯝ a cr2 e0⫹ f re 3_, 共63兲 with the remaining ones neglected. Then the Hamiltonian matrix 共39兲 reads H⯝ f2c pr␥0␥1⫹ f cp␸␥0␥3⫹i f mc2␥0⫺ i 2បc f f

⬘

␥0␥1 ⫹3 2iបc f S a_␥ 0␥5␥a⫺iបc f N a_␥ 0␥a⫺ a f3 r pr␥3␥1 ⫹iaប f 2_f

_⬘

2r ␥3␥1⫺ iamc f2 r ␥3⫹ iaប f 2r2 ␥0␥2␥5 ⫹3iaប f 2 2r S a_␥ 3␥a␥5⫹ iaប f2 r N a_␥ 3␥a, 共64兲 where prª⫺iប

冉

⳵ ⳵r⫹ 1 r

冊

, 共65兲 p_␪ª0, 共66兲 p_␸ª⫺iប r ⳵ ⳵␸, 共67兲

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Naª 1 2

冉

t b a,b⫹Fa⫹ 5 4Qa⫺⌳a

冊

. 共68兲 Note that the contributions of axial components of torsion are given by Sa while certain components of nonmetricity and the nonaxial components of torsion occur only in Naand

冉

dr, 共77兲 ⌽↓_⫽ 1 បc

冕

冊

. 共79兲

In this case the relevant phase differences depending on non-metricity via Na _{and rotation via a come from the opposite}

spin polarization states

⌬⌽⫽⌽2↓⫺⌽1↑⫽ ⌬m2_c3 2បE ⌬r⫺ 2 បc

冕

冑

⌬rdr, 共80兲 ⌬⌽⫽⌽2↑⫺⌽1↓⫽ ⌬m2_c3 2បE ⌬r⫹ 2 បc

冕

冑

⌬rdr. 共81兲 We point out that ⌬r⫽Re ⌬r⫹i Im ⌬r implies

冑

⌬r⫽␣

⫹i␤ and hence the rotation of the Sun would suppress the transitions among the neutrinos via the phase difference equations 共80兲,共81兲 in opposite spin polarizations.

V. CONCLUSION

We have here extended our recent study of gravitationally induced neutrino oscillations 关8兴 by including the effects of rotation of the Sun and space-time nonmetricity and as well

as components of torsion other than the axial ones. The ro-tation of the Sun implies a damping of neutrino oscillations. However, this result is frame dependent as we explained in Sec. III in general. We have shown that there are contribu-tions coming from nonaxial components of space-time tor-sion and definite components of space-time nonmetricity de-pending on the polarizations of the spin states of the mass eigenstates. If we set the rotation parameter a⫽0, then Eq. 共79兲 gives

冑

⌬r⫽ 3 2បc

冋

共S0⫹S1兲 2_⫹

冉

_S 2⫺ 2 3f N3

冊

2 ⫹

冉

S3⫹ 2 3f N2

冊

2

册

1/2 , 共82兲 which means that there is no suppression among the neutri-nos and only N2 and N3 components of Na contribute to the oscillations. If we further set Na_{⫽0, we reach agreement}

with our previous results in 关8兴. It should be clear that the above scheme only works if the neutrino masses are different from each other and hence, in general, different from zero. This means there are right-handed neutrinos as well as left-handed ones which, however, must interact with matter very weakly as they have not yet been observed. Finally, we note that all possible contributions discussed here so far would be of the order of the Planck scale, and hence do not suffice to account for the observed solar neutrino deficit.

ACKNOWLEDGMENT

One of the authors 共M.A.兲 acknowledges partial support through the research project BAP2002-FEF007 by Pamuk-kale University, Denizli.

关1兴 W.C. Haxton and B.R. Holstein, Am. J. Phys. 68, 15 共2000兲. 关2兴 S.M. Bilenky and S.T. Petcov, Rev. Mod. Phys. 59, 671

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