Neutrino matter potentials induced by Earth

0.16 MB·English

by  J. Linder

Neutrino matter potentials induced by Earth
J. Linder∗
Department of Theoretical Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
(Dated: Received February 2, 2008)
An instructive method of deriving the matter potentials felt by neutrinos propagating through
matter on Earth is presented. This paper thoroughly guides the reader through the calculations
involvingtheeffectiveweakHamiltonianforleptonandquarkscattering. Thematterpotentialsare
well-knownresultssincethelate70’s, butadetailedandpedagogicalcalculation ofthesequantities
is hard to find. We derive potentials due to charged and neutral current scattering on electrons,
neutronsandprotons. Intendedreadershipisforundergraduates/graduatesinthefieldsofrelativis-
tic quantum mechanics and quantum field theory. In addition to the derivation of the potentials
for neutrinos, we explicitely study the origin of the reversed sign for potentials in the case of
antineutrino-scattering.
6
0 PACSnumbers: 13.15.+g,25.30.Pt
0
2

n
a
J I. INTRODUCTION

5
1 In his famous paper from 1978 [1], Wolfenstein showed that neutrino oscillations [2] are altered in the presence of
matter; a phenomenon which came to be known as the MSW-effect after Mikheyev, Smirnov, and Wolfenstein. This
4
occurs as a result of neutrinos experiencing potentials due to charged and neutral current scattering on electrons,
v
neutrons,andprotonsastheypropagatethroughmatteronEarth. Theexplicitexpressionsforthesematterpotentials
4
canbefoundinmuchliterature,andabriefsketchforhowthesequantitiesarederivedisgivenin[3],whichdealswith
6
2 thespecialcaseofchargedcurrentelectron-neutrinoscatteringonelectrons. Followingtheoutlinesofthissketch,this
4 paper provides a thorough and pedagogical walk-through of calculating all relevant matter potentials that neutrinos
0 experience when propagating through matter in Earth. The calculations include many techniques that are relevant
5 when setting out to find expectation values in the second quantized formalism, and should hence be of interest
0
for students in the process of being equipped to deal with scattering reactions in the framework of quantum field
/
h theory. The physics underlying the calculation is also explained to make the derivation more comprehensible, with
p particular emphasis on how the procedure of finding the potentials can be considerably simplified by observing that
- they are independent of the axial coupling constant, in addition to an analysis of why the potential sign is reversed
p
e when considering antineutrinos. Especially the latter argumentis of particular value for the intended audience, since
h practically all literature simply states that the sign is reversed instead of explaining why it is reversed.
:
v
i
X II. NEUTRINO EIGENSTATES AND EFFECTIVE HAMILTONIAN
r
a
The vacuum Hamiltonian describing a propagating neutrino flavor state is as previously mentioned altered in the
presenceofmatter. This is readilyseenby consideringthe energycontributionto the Hamiltonianfromscatteringon
nuclear components such as electrons and neutrons. First of all, the flavor eigenstates produced in weak interactions
arecoupledto the masseigenstatesthrough[ν ν ν ]T =U[ν ν ν ]T, whereU is the unitarymixing matrixusually
e µ τ 1 2 3
parametrized as in Ref. [2],
1 0 0 c 0 s e−iδCP c s 0
13 13 12 12
U =U U U = 0 c s 0 1 0 s c 0 . (1)
23 13 12 23 23 12 12
  − 
0 s c s eiδCP 0 c 0 0 1
− 23 23 − 13 13
   
Here, s = sinθ and c = cosθ and θ are the mixing angles between the neutrino states. The δ is the
ij ij ij ij ij CP
CP-violating phase with allowed values δ [0,2π]. For δ 0,π we have CP invariance, while the violation
CP CP
∈ ∈ { }
is at its largest when δ π/2,3π/2 . For our purposes, δ is set to zero. This is motivated by the fact that
CP CP
∈ { }
∗Electronicaddress: [email protected]
2
CP-violation in the standard model requires a non-zero value of θ , and since experiments have placed stringent
13
limits onthe value of θ , the effective CP-violationis quite small(see e.g. Ref. [3]). Note that we haveintentionally
13
left out diagonal Majorana phases that are of no consequence for oscillation experiments.
The evolution of the mass eigenstates ΨM(x) in vacuum is described by
d
i ΨM(x)=HMΨM(x), (2)
dt 0
where H =diag(E ,E ,E ). E ,i=1,2,3, are the energy eigenvalues for mass eigenstates. To obtain the Hamilto-
0 1 2 3 i
nian in flavor space, we perform H = UHMU−1, where U is the mixing matrix from Eq. (1). When the neutrinos
0 0
propagate through matter, we must add the contributions from scattering on particles present. Accordingly, one
obtains H =H +H , written out as
0 I
H =Hn+Hp +He +He , (3)
I Z Z Z W
where the interaction Hamiltonians are given as Hi = diag(Vi,Vi,Vi), i = n,p,e, and He = diag(Ve ,0,0).
Z Z Z Z W W
The superscript refers to the scattering component while the subscript indicates which gauge boson mediates the
reaction, i.e. neutral or charged current. For example, Ve represents the effective matter potential due to NC
Z
(neutral-current) scattering on electrons. Note that we have only included matter potentials arising from reac-
tionswithelectrons,protons,andneutrons,sincethetheconcentrationsofµandτ particlesisvirtuallyzeroonEarth.
In deriving the matter potentials, we are going to need the effective weak CC (charged-current) and NC interaction
Hamiltonians
G
eff = FJ Jµ†, (4)
HW √2 Wµ W
4G
eff = FJµJ . (5)
HZ √2 Z Zµ
The Fermi-constant is G =√2g2/8m2 , while the currents are defined as
F W
Jµ = lγµ(1 γ )ν +d γµ(1 γ )u ,
W − 5 l θ − 5
l
X(cid:2) (cid:3)
1
Jµ = ψ γµ I3(1 γ ) 2Q sin2θ ψ , (6)
Z 2 i i − 5 − i W i
Xi h i
where i =(l,ν ,u,d), while I3 is the accompanying particle isospin and Q is the particle charge in units of e. Now,
l i i
d =dcosθ +ssinθ . The Cabbibo angle θ has been experimentally determined to cosθ 0.98, so we shall
θ C C C C
≈
exclude the s quark part from now on and set d =d. This notation has been adapted from Refs. [4, 5].
θ
The derivation of neutrino matter potentials is certainly not new, and has been studied in e.g. Refs. [1, 6, 7, 8].
This paper contributes to the study of matter potentials by presenting an instructive method to derive the matter
potentials that applies in the same way to leptons as nucleons.
III. DERIVATION OF Ve
W
First, we seek the quantity
Ve = ν (p ,s )e(p ,s )H ν (p ,s )e(p ,s ) , (7)
W h e 1 1 2 2 | W| e 1 1 2 2 i
where H is the charged-current contribution due to scattering on electrons, omitting the superscript eff for clarity.
W
Since we are dealing with elastic scattering, it is fair to assume that the neutrinos and electrons conserve their
momentumasshowninEq. (7). Thelow-energyeffective HamiltoniandensityrelevantforCCν escatteringisfound
e
in Eq. (4), namely
G
(x)= F e(x)γβ(1 γ )ν (x) ν (x)γ (1 γ )e(x) . (8)
W 5 e e β 5
H √2 − −
(cid:2) (cid:3)(cid:2) (cid:3)
3
Nave insertion of this density into
H = (x)dx, (9)
W W
H
ZV
isnotcorrect. Thepresenceofelectronsinamediumleadstotwoimportantmodifications. Firstofall,the statistical
energy distribution of the electrons in the medium is accounted for by integration over the Fermi function f(E ,T)
e
which is normalized to f(E ,T)dp = 1. Secondly, since we do not know the polarization of the electrons, an
e e
averaging over spins 1/2 is needed. In total, this corresponds to transforming Eq. (8) to
R s
P G 1
(x)= f(E ,T) F e(x)γβ(1 γ )ν (x)
W e 5 e
H √2 × 2 −
Z Xs2 (cid:2) (cid:3)
ν (x)γ (1 γ )e(x) dp . (10)
e β 5 2
× −
Since only electrons with (p,s)=(p ,s ) w(cid:2)ill contribute to Eq.(cid:3)(7), we obtain
2 2
e(x)γβ(1 γ )ν (x) ν (x)γ (1 γ )e(x) ν (p ,s )e(p ,s )
5 e e β 5 e 1 1 2 2
− − | i
1
(cid:2) = (cid:3)(cid:2) a† (p )a (p )u(cid:3)(p )γβ(1 γ )ν (x) ν (x)γ (1 γ )u (p ) ν (p ,s )e(p ,s ) . (11)
2VE (p ) s2 2 s2 2 s2 2 − 5 e e β − 5 s2 2 | e 1 1 2 2 i
2 2
(cid:2) (cid:3)(cid:2) (cid:3)
Making the identification of the number operator Num (p ) = a† (p )a (p ), insertion of Eq. (11) into Eq. (7)
s2 2 s2 2 s2 2
produces
G Num (p )
Ve = ν (p ,s )e(p ,s ) F f(E ,T) s2 2 u (p )γβ(1 γ )ν (x)
W h e 1 1 2 2 |4√2V ×Z Z e Xs2 Ee(p2) (cid:2) s2 2 − 5 e (cid:3)
ν (x)γ (1 γ )u (p ) dx dp ν (p ,s )e(p ,s ) . (12)
× e β − 5 s2 2 2| e 1 1 2 2 i
It should be clear th(cid:2)at the ν (x) and ν (x) s(cid:3)ymbols refer to the second quantized fields
e e
1
ψ(x)= a (p)u (p)e−ipx+b†(p)v (p)eipx
s2Vωp s s s s
Xs,p h i
1
ψ(x)= b (p)v (p)e−ipx+a†(p)u (p)eipx , (13)
s2Vωp s s s s
Xs,p h i
where ψ is a lepton or a quark. On the other hand, the state vectors ν (p ,s ) , i=1,2, are simply representations
e i i
| i
forneutrinoswith4-momentump andspins . Inordertocontinuewithouttoomanycomplications,assumethatthe
i i
materialis isotropic and holds anequal number of electronswith spin up as spin down, i.e. a non-magnetic material.
This leads to
G f(E ,T)N (p )
Ve = ν (p ,s )e(p ,s ) F e e 2 u (p )γβ(1 γ )ν (x)
W h e 1 1 2 2 |4√2 ×Z Z Ee(p2) Xs2 (cid:2) s2 2 − 5 e (cid:3)
ν (x)γ (1 γ )u (p ) dx dp ν (p ,s )e(p ,s )
× e β − 5 s2 2 2| e 1 1 2 2 i
G f(E ,T)N (p )
= ν ((cid:2)p ,s )e(p ,s ) F (cid:3) e e 2 ν (x)γ (1 γ )ν (x)
h e 1 1 2 2 |4√2 ×Z Z Ee(p2) e β − 5 e
u (p )γβ(1 γ ))u (p ) dx dp ν (p ,s )e(p ,s ) . (14)
× s2 2 − 5 s2 2 2| e 1 1 2 2 i
Xs2 (cid:2) (cid:3)
Here, we have used the Fierz identity to re-arrange the ν and e spinors in such a fashion that makes it possible to
e
extract the neutrino-spinor part from the summation over s . The remaining sum is evaluated by
2
u (p )γβ(1 γ )u (p ) =Tr (/p +m )γβ(1 γ )
s2 2 − 5 s2 2 { 2 e − 5 }
Xs2 (cid:2) (cid:3)
=p Tr γαγβ(1 γ ) =4pβ, (15)
2α { − 5 } 2
4
with efficient use of standard trace relations. Eq. (15) in synthesis with (14) leads to
G f(E ,T)N (p )
Ve = ν (p ,s )e(p ,s ) F e e 2
W h e 1 1 2 2 |√2 ×Z Z Ee(p2)
ν (x)γ (1 γ )ν (x)pβdx dp ν (p ,s )e(p ,s )
× e β − 5 e 2 2| e 1 1 2 2 i
G N
= ν (p ,s )e(p ,s ) F e ν (x)γ (1 γ )ν (x)dxν (p ,s )e(p ,s ) , (16)
e 1 1 2 2 e 0 5 e e 1 1 2 2
h | √2 × − | i
Z
where we have exploited the isotropy p f(E ,T) dp = 0 and the expression for the total electron density
2 e 2
f(E ,T)N (p ) dp =N . Only integration over x remains, such that we obtain
e e 2 2 e
R
R G N
Ve = ν (p ,s )e(p ,s ) F e ν (x)γ (1 γ )ν (x)dxν (p ,s )e(p ,s )
W h e 1 1 2 2 | √2 × e 0 − 5 e | e 1 1 2 2 i
Z
G N 1
= ν (p ,s )e(p ,s ) F e Tr (/p +m )γ0(1 γ5)
h e 1 1 2 2 | √2 × 2VEνe Z { νe νe − }
Nums1(p1)
a† (p )a (p )dxν (p ,s )e(p ,s )
× s1 1 s1 1 | e 1 1 2 2 i
= ν (zp ,s )}e|(p ,s{) GFNe 1 4E dxν (p ,s )e(p ,s ) . (17)
h e 1 1 2 2 | √2 × 2VEνe Z νe | e 1 1 2 2 i
Assuming normalized state vectors ν (p ,s )e(p ,s ) , Eq. (17) reduces to
e 1 1 2 2
| i
G N 2
Ve = F e dx ν (p ,s )e(p ,s ) ν (p ,s )e(p ,s ) =√2G N . (18)
W √2 × V h e 1 1 2 2 || e 1 1 2 2 i F e
Z
IV. DERIVATION OF Vn
Z
We now set out to find Vn due to ν n, α = e,µ,τ scattering. This reaction is mediated by the Z0 boson, so we
Z α
must use the effective Hamiltonian density Eq. (5). Now, the neutron consists of one u and two d. The u part of the
Hamiltonian is
G 8
F u(x)γµ(1 γ sin2θ )u(x) ν (x)γ (1 γ )ν (x) , (19)
5 W α µ 5 α
2√2 − − 3 −
(cid:2) (cid:3)(cid:2) (cid:3)
while the d part is
G 4
F d(x)γµ(1 γ sin2θ )d(x) ν (x)γ (1 γ )ν (x) . (20)
5 W α µ 5 α
− 2√2 − − 3 −
(cid:2) (cid:3)(cid:2) (cid:3)
These contributions are to be added in the ratio 1:2 to obtain the relevant Hamiltonian for ν n scattering. In total,
e
this gives
G 8 4
F ψ γµ (1 γ sin2θ ) 2(1 γ sin2θ ) ψ [ν (x)γ (1 γ )ν (x)
2√2 n − 5− 3 W − − 5− 3 W n α µ − 5 α
h G(cid:2) (cid:3) i i
= F ψ γµ(1 γ )ψ ν (x)γ (1 γ )ν (x) . (21)
−2√2 n − 5 n α µ − 5 α
(cid:2) (cid:3)(cid:2) (cid:3)
We are left with the effective Hamiltonian
G 1
(x)= F f(E ,T) ψ (x)γµ(1 γ )ψ (x) ν (x)γ (1 γ )ν (x) dp . (22)
HZ −2√2 n × 2 n − 5 n α µ − 5 α n
Z s
X(cid:2) (cid:3)(cid:2) (cid:3)
Here, we have introduced the statistical Fermi distribution for neutrons f(E ,T) and summation over the neutron
n
spins due to the assumption of unpolarized medium, just as for the electrons. We see that Eq. (22) is of the same
form as Eq. (10) if we use the earlier mentioned Fierz identity, such that the rest of the analysis is equivalent to the
derivation of Ve . Following the same procedure as above, we find
W
5
G N
Vn = ν (p ,s )n(p ,s ) F n ν (x)γ (1 γ )ν (x)dxν (p ,s )n(p ,s )
Z −h α 1 1 2 2 | 2√2 × α 0 − 5 α | α 1 1 2 2 i
Z
G N 1
= ν (p ,s )n(p ,s ) F n Tr (/p +m )γ0(1 γ5)
−h α 1 1 2 2 | 2√2 × 2VEνα Z { να να − }
Nums1(p1)
a† (p )a (p )dxν (p ,s )n(p ,s )
× s1 1 s1 1 | α 1 1 2 2 i
= νz (p ,s}|)n(p ,{s ) GFNn 1 4E dxν (p ,s )n(p ,s ) . (23)
−h α 1 1 2 2 | 2√2 × 2VEνα Z να | α 1 1 2 2 i
With normalized state vectors ν (p ,s )n(p ,s ) , Eq. (23) leads to
α 1 1 2 2
| i
G N 2 G N
Vn = F n dx ν (p ,s )n(p ,s ) ν (p ,s )n(p ,s ) = F n. (24)
Z − 2√2 × V h α 1 1 2 2 || α 1 1 2 2 i − √2
Z
V. DERIVATION OF Ve
Z
Now, the expression for Ve is found by including the relevant terms from Eq. (5), which for NC ν e scattering reads
Z e
G
F ν (x)γµ(1 γ )ν (x) e(x)γµ(1 γ 4sin2θ )e(x) . (25)
e 5 e 5 W
−2√2 − − −
Taking into accountthe Fermi dis(cid:2)tribution ofelectrons a(cid:3)(cid:2)nd the averagingofspins, the ex(cid:3)pressionfor Ve(x) takes the
Z
form
G 1
(x)= F f(E ,T) ν (x)γµ(1 γ )ν (x) e(x)γ (1 γ 4sin2θ )e(x) dp . (26)
Z e e 5 e µ 5 W e
H −2√2 × 2 − − −
Z s
X(cid:2) (cid:3)(cid:2) (cid:3)
As before, the rest of the analysis is equivalent to the derivation of Ve . In analogy to Eq. (17), it is found that
W
G N
Ve = ν (p ,s )e(p ,s ) F e (1 4sin2θ )ν (x)γ (1 γ )ν (x)dxν (p ,s )e(p ,s )
Z −h e 1 1 2 2 | 2√2 × − W α 0 − 5 α | e 1 1 2 2 i
Z
G N 1
= ν (p ,s )e(p ,s ) F e (1 4sin2θ )Tr (/p +m )γ0(1 γ5)
−h α 1 1 2 2 | 2√2 × 2VEνe Z − W { νe νe − }
Nums1(p1)
a† (p )a (p )dxν (p ,s )e(p ,s )
× s1 1 s1 1 | e 1 1 2 2 i
= νz (p ,s}|)e(p ,{s ) GFNe 1 (1 4sin2θ )4E dxν (p ,s )e(p ,s ) . (27)
−h e 1 1 2 2 | 2√2 × 2VEνe Z − W νe | e 1 1 2 2 i
With normalized vectors ν (p ,s )n(p ,s ) , Eq. (27) gives
e 1 1 2 2
| i
G N 2
Ve = F e (1 4sin2θ ) dx ν (p ,s )e(p ,s ) ν (p ,s )e(p ,s )
Z − 2√2 × V − W h e 1 1 2 2 || e 1 1 2 2 i
Z
G (1 4sin2θ )N
F W e
= − . (28)
− √2
VI. DERIVATION OF Vp
Z
Inorderto deriveVp we attackthe probleminthe same wayas forVn. A protonconsistsoftwou andoned quarks,
Z Z
sofirstwefind considerthe individualcontributionsfromeachquark. FromEqs. (19)and(20)the totalcontribution
to the proton is seen to be
G 8 4
F ψ γµ 2(1 γ sin2θ ) (1 γ sin2θ ) ψ ν (x)γ (1 γ )ν (x)
2√2 p − 5− 3 W − − 5− 3 W p α µ − 5 α
hG (cid:2) (cid:3) ih i
= F ψ γµ(1 γ 4sin2θ )ψ ν (x)γ (1 γ )ν (x) , (29)
2√2 p − 5− W p α µ − 5 α
(cid:2) (cid:3)(cid:2) (cid:3)
6
when added in the ratio 2:1, and including the extra factor of 2 due to the double quark contribution from Eq. (5).
This gives the effective Hamiltonian for ν p, α=e,µ,τ scattering, namely
α
G 1
p(x)= F f(E ,T) ψ (x)γµ(1 γ 4sin2θ )ψ (x) ν (x)γ (1 γ )ν (x) dp . (30)
HZ 2√2 p × 2 p − 5− W p α µ − 5 α n
Z s
X(cid:2) (cid:3)(cid:2) (cid:3)
The further analysis is then just as for Ve , and leads to
W
G N
Vp = ν (p ,s )p(p ,s ) F p (1 4sin2θ )ν (x)γ (1 γ )ν (x)dxν (p ,s )p(p ,s )
Z −h α 1 1 2 2 | 2√2 × − W α 0 − 5 α | α 1 1 2 2 i
Z
G N 1
= ν (p ,s )p(p ,s ) F p (1 4sin2θ )Tr (/p +m )γ0(1 γ5)
h α 1 1 2 2 | 2√2 × 2VEνα Z − W { να να − }
Nums1(p1)
a† (p )a (p )dxν (p ,s )p(p ,s )
× s1 1 s1 1 | α 1 1 2 2 i
= ν z(p ,s })|p(p ,s{) GFNp 1 (1 4sin2θ )4E dxν (p ,s )p(p ,s ) . (31)
h α 1 1 2 2 | 2√2 × 2VEνα Z − W να | α 1 1 2 2 i
As before, we require the state vectors ν (p ,s )p(p ,s ) to be normalized. Eq. (31) then gives
α 1 1 2 2
| i
G N 2
Vp = F p (1 4sin2θ ) dx ν (p ,s )p(p ,s ) ν (p ,s )p(p ,s )
Z 2√2 × V − W h α 1 1 2 2 || α 1 1 2 2 i
Z
G N (1 4sin2θ )
F p W
= − . (32)
√2
VII. JUSTIFICATION OF TREATING NEUTRONS AND PROTONS AS POINT-PARTICLES
At first sight, it might seem startling that the neutron and proton are treated as point particles with respect to the
neutrino scattering, instead of taking the well-known nucleon form factors into account. But it turns out that these
are actually irrelevant as far as the matter potentials are concerned. Notice how all NC potentials are directly pro-
portionalto the correspondingvector coupling constantg of the scattering particle, i.e. N ge =2sin2θ 1/2,
N gn = 1/2, N gp = 1/2 2sin2θ . The nucleVon vector coupling constants areeob∝taiVned by quitWe s−imply
n ∝ V − p ∝ V − W
adding the quark equivalents for the u and d in the correct ratio; 2:1 for p and 1:2 for n.
An attempt to apply the same logic to the axial coupling constant g would not be successful. The axial current
A
anomaly (see e.g. Ref. [9]) modifies the value of this quantity, thus yielding for instance gp 1.27/2 instead of
V ≃
1/2 (this particular number has been determined by experiments, see e.g. Ref. [10]). One compensates for this
fact by operating with form-factors in scattering reactions on nucleons. For neutrino forward scattering (essentially
the MSW-effect), one needs to include form factors that are non-vanishing in the limit of zero momentum transfer.
Because of the mentioned axial current anomaly, the form factor for the axial current will give results different from
a combination of quark currents in the respective ratios for neutrons and protons. But the potentials were generally
derivedto be independent of g , regardlessofscattering component. As a consequence,we neednot worryabout the
A
axial form-factors. Left with only the vector part, the nucleon can simply be treated as the superposition of three
quarksin this scenario. Of course,suchanuncomplicated picture is certainly the exception ratherthan the rule with
respect to nucleon scattering.
VIII. HOW TO TREAT A VARYING DENSITY
Eqs. (24), (28), (32) are equally valid even if the particle density N fluctuates in space. One would compensate for
i
such behavior by evaluating the matter potential at each separate point in an artifical lattice-space, or alternatively
approximating the particle density with a contineous function of the space-coordinate x. The two cases constant vs.
non-constant particle density differ crucially in terms of how they affect the probability for a neutrino oscillation to
have occured (see e.g. Ref. [2] for an introduction to the topic of neutrino oscillations). Dependent on which of the
two scenarios that is applicable, the oscillation probability takes on a certain characteristic spatial-dependence. This
is also dependent on the nature of the particle density fluctuations, i.e. adiabiatic, unit-step profile, et.c. However,
oscillations are not the topic of this paper, and we shall settle with the derivation of Eqs. (24), (28), (32) since these
are heuristically valid even in the case of a varying particle density.
7
IX. POTENTIAL SIGN FOR ANTINEUTRINOS
The matter potentials derived so far has been for neutrinos. One cannot generalize these results to be valid for
antineutrinos, and here is why. Consider the induced matter potential for ν due to NC scattering on electrons. This
e
potential is given as (see Sec. V)
Ve = ν (p ,s )e(p ,s )He ν (p ,s )e(p ,s ) , (33)
Z h e 1 1 2 2 | Z| e 1 1 2 2 i
while the induced matter potential for ν for the same type of reaction reads
e
Ve = ν (p ,s )e(p ,s )He ν (p ,s )e(p ,s ) . (34)
Z h e 1 1 2 2 | Z| e 1 1 2 2 i
In the general case, the operator part of Eq. (34) takes the form
0b (p) b (k)b†(k′) b†(p′)0 , (35)
h | ν ν ν ν | i
hXk,k′ i
where the sum over b (k)b†(k′) comes from the second quantized fields ν (x)ν (x). Using the commutation relation
ν ν e e
[b (k),b†(k′)]=δ(k′ k), Eq. (35) becomes
ν ν −
0b (p) b†(k′)b (k) b†(p′)0 . (36)
−h | ν ν ν ν | i
hkX,k′ i
The only non-vanishing contribution from this term is obtained by taking k = p′,k′ = p. In our case of elastic
scattering, we impose the limit p p′. A similar argument for Eq. (33) produces the operator sequence
→
0a (p) a†(k′)a (k) a†(p′)0 . (37)
h | ν ν ν ν | i
hkX,k′ i
Here, a (p) and a†(p) (b (p) and b†(p)) are interpreted as annihilation and creation operators for neutrinos
ν ν ν ν
(antineutrinos). Thematterpotentialsdiffer insigndue tothe relativeminussignbetweenEqs. (36)and(37). Thus,
Ve = Ve.
Z − Z
X. SUMMARY
TherelevantmatterpotentialsforneutrinospropagatingthroughEarthhavebeenderivedusingamethodthatapplies
inthe samewaytobothleptonsandnucleons. We haveexplicitelyshownwhythe matterpotentialsignisreversedin
the caseofantineutrinos. Specifically,itisworthtonotethatthe effectivepotentialinanelectricallyneutralmedium
with respect to neutrino oscillations is given by He in Eq. (3). From Eqs. (28) and (32) it is easily seen that
W
Ve+Vp =0. (38)
Z Z
Ne=Np
(cid:12)
(cid:12)
Thus, there is no effective matter potential felt by neutri(cid:12)nos due to scattering on protons and electrons mediated by
Z0 in an electrically neutral medium. Also, since He in Eq. (3) is diagonal, it gives an overall phase shift to all
Z
neutrino flavors, which is of no significance in the oscillation scenario. Our results are summarized in Tab. I.
TABLE I: Neutrinomatter potentials induced byEarth.
Type of reaction Matter potential
VZn ∓GFNn/√2
VZp ±GF(1−4sin2θW)Np/√2
VZe ∓GF(1−4sin2θW)Ne/√2
VWe ±√2GFNe
8
Generalization to matter potentials such as Vµ are obtained simply by substituting N N . The upper sign refer
Z e → µ
to neutrinos, while the lower gives the matter potential for antineutrinos.
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