Mixing of fermions and spectral representation of propagator

0.23 MB·English

by  A. E. Kaloshin

Mixing of fermions and spectral representation of propagator
A.E. Kaloshin1,∗ and V.P. Lomov2,†
1
Physical Department, Irkutsk State University,
K. Marx str. 1, 664003, Irkutsk, Russia
2
Laboratory 1.2, Institute for System Dynamics and Control Theory,
RAS, Lermontov str. 134, 664043, Irkutsk, Russia
We develop the spectral representation of propagator for n mixing fermion fields
in the case of P-parity violation. The approach based on the eigenvalue problem
for inverse matrix propagator makes possible to build the system of orthogonal pro-
jectors and to represent the matrix propagator as a sum of poles with positive and
6
negative energies. The procedure of multiplicative renormalization in terms of spec-
1
0 tral representation is investigated and the renormalization matrices are obtained in
2
a closed form without the use of perturbation theory. Since in theory with P-parity
r
a violation the standard spin projectors do not commute with the dressed propagator,
M
they should be modified. The developed approach allows us to build the modified

3 (dressed) spin projectors for a single fermion and for a system of fermions.
1

] PACS numbers: 12.12.Ff, 11.10.Gh
h
p Keywords: fermion mixing; matrix propagator;renormalization; spin projectors
-
p
e
h
[

I. INTRODUCTION
3
v
7 The problem of neutrino oscillations has been in the spotlight since last decades, both from
3
3 experimental and theoretical points of view. This phenomenon is generated by mixing in the
6
neutrinos system, when mass states differ from the flavor ones. Since quantum field theory is
0
. a proper theoretical framework for describing these effects, the essential efforts were devoted
1
0 to application of QFT methods for neutrinos mixing problem [1–10]. What we cited here
5
is only a small part of relevant publications (see also the references cited therein), which is
1
: directly related to problem of neutrino oscillations in the QFT. The mixing effects also play an
v
i essential role in the quarks system, where radiative corrections lead to modification of the bare
X
Cabbibo-Kobayashi-Maskawa (CKM) matrix and to necessity to renormalize this matrix (see,
r
a
e.g. Refs. [11–14]). Note that in the mixing problem there exist some delicate theoretical issues
related with dependence on renormalization scheme, possible gauge dependence and properties
of renormalized mixing matrix [15–17].
In studying of mixing and oscillation phenomena in the QFT the matrix propagator plays
the central role. In recent series of papers [18–20] the properties of dressed matrix propagator
in the presence of P-parity violation were investigated in detail. The dressed propagator was
represented inaclosed algebraicform, which satisfies themainphysical requirements andallows
to build the renormalized propagator. The pole scheme of renormalization was investigated
∗Electronic address: [email protected]
†Electronic address: [email protected]
2
and wave-function renormalization (WFR) matrices were obtained in a closed analytical form
without recourse to perturbation theory.
In the present paper we develop a convenient algebraic construction for consideration of
fermion matrix propagator and mixing effects in the QFT frameworks. The main feature of
suggested construction is that propagator is represented as a sum of single poles with positive
and negative energies. Note, that it is made in a covariant manner 1/(W m ) and this is
i
±
a general property of considered eigenvalue problem, see e.g. (7) for free fermion propagator.
The obtained very simple expression for WFR matrices (126) confirms the old opinion that just
W is the natural variable in fermion case.
Another important feature of the suggested approach is related with spin properties of the
dressed propagator. In theory with γ5 the usual spin projectors do not commute with dressed
propagator and should be somehow modified. Standard procedure of Dyson summation (in
particular, in Refs. [18–20]) does not touch the spin projectors, having in mind their existence.
For the developed here approach the generalized spin projectors (94), (96) are the necessary
elements of construction, used to prove the completeness condition.
Technically, the suggested construction is based on so called spectral representation of an
operator (see, e.g. textbook [21]). In this representation the self-adjoint operator Aˆ takes the
form (in quantum-mechanical notations):
Aˆ = λ i i = λ Π ,
i i i
| ih |
i i
X X
where λ are eigenvalues of the operator, i are eigenvectors
i
| i
Aˆ i = λ i
i
| i | i
and Π = i i are corresponding orthogonal projectors (eigenprojectors). In the case of non-
i
| ih |
self-adjoint operator the similar decomposition also exists but to construct it, one needs solu-
tions of both left and right eigenvalue problems.
If we have n fermion fields with the same quantum numbers, they begin to mix at loop level
even in the case of diagonal mass matrix. In the QFT the main object of studying is the dressed
matrix propagator G(p). To build the spectral representation of G(p), first of all one needs to
solve the eigenvalue problem for inverse propagator S(p) 1
SΠ = λ Π . (1)
i i i
If we have the complete system of orthogonal eigenprojectors2
2n
Π Π = δ Π , Π = 1, (2)
i k ik k i
i=1
X
then we obtain the spectral representation of inverse propagator S(p)
2n
S(p) = λ Π . (3)
i i
i=1
X
1 Here S (and Π also) has two sets of indices S , where α,β = 1,...,4 are the Dirac γ-matrix indices and
αβ;ij
i,j =1,...,naregenerationindices. Notethat,fromthebeginningwearelookingforeigenprojectorsinstead
of eigenvectors (following to Ref. [22]), to avoid cumbersome intermediate expressions.
2 The completeness condition and closely related with it spin projectors are discussed in Section IV.
3
The matrix propagator G(p) is obtained by reversing of (3)
2n
1
G(p) = Π . (4)
i
λ
i
i=1
X
If the projectors satisfy the orthogonality property, then the same Π are solutions of two
i
eigenvalue problems: left (1) and right one
Π S = λ Π . (5)
i i i
As will be shown later, the representation (3) looks very simple and evident in the case of
P-parity conservation and the main technical problems are related with appearance of γ5 in
vertex and dressed propagator. In Ref. [22] we constructed the representation (3) for a single
fermion (n = 1) in the case of parity violation and investigated the renormalization procedure.
In the present paper we build the spectral representation for the case of n mixing fermion fields
and study the main properties of this representation.
The paper is organized as follows. In Sec. II we consider the eigenvalue problem for inverse
matrix propagator in theory with P-parity violation and build the corresponding set of orthog-
onal projectors, which are solutions of both left and right eigenvalue problems. In Sec. III the
case of CP-conserving theory is considered, that is reflected in symmetry of matrix coefficients
and leads to essential simplification of the spectral representation. Sec. IV is devoted to the
completeness condition for the obtained eigenprojectors, which is equivalent to the existence of
the generalized (dressed) spin projectors with the necessary properties. We indicate the explicit
form of generalized (in theory with γ5) spin projectors, which are closely related with the ob-
tained eigenprojectors. In Sec. V we formulate the multiplicative renormalization requirements
for matrix propagator in terms of the obtained spectral representation. It gives very simple
conditions for the renormalization constants and allows to write down the answer in a closed
form.
II. EIGENVALUE PROBLEM FOR INVERSE MATRIX PROPAGATOR
A. Preliminary
In the following it’s convenient to use the off-shell γ-matrix projectors3
1 pˆ
Λ±(p) = 1 , (6)
2 ± W
(cid:16) (cid:17)
where W = p2 is in general a complex variable and for positive p2 it is the center-of-mass
energy. In the study that follows we do not impose any restrictions on the sign of p2. For free
p
inverse propagator S = pˆ m these projectors are solutions of eigenvalue problem and free
0
−
propagator is represented as
1 1 1
G(p) = = Λ+ + Λ−, (7)
pˆ m W m W m
− − − −
3 Many people used these off-shell projectors for different purposes, the first known for us case is related with
the problem of fermion Regge poles, see papers of V.N. Gribov and co-authors [23, 24].
4
so we obtain a covariant separation of poles with positive and negative energies.
In the case of parity conservation the eigenprojectors Π are just Λ±, multiplied by flavor
i
matrix, see(14)below. Inthetheorywithγ5 theγ-matrixprojectorsΛ± appearatintermediate
stage of the Π building but they are useful to simplify the algebra.
i
In the case of parity violation we introduce the following set of matrices
= Λ+, = Λ−, = Λ+γ5, = Λ−γ5 (8)
1 2 3 4
P P P P
and use them as a basis to expand the self-energy and propagator. The inverse matrix propa-
gator may be written as
4
S(p) = G−1(p) = S (W), (9)
M M
P
M=1
X
where the matrix coefficients S have the obvious symmetry properties:
M
S (W) = S ( W), S (W) = S ( W) (10)
2 1 4 3
− −
and are calculated as4
1 1
S = Sp( S), S = Sp( S),
1 1 2 2
2 P 2 P
(11)
1 1
S = Sp( S), S = Sp( S).
3 4 4 3
2 P 2 P
If the parity is conserved, the self-energy
•
Σ(p) A(p2)+pˆB(p2) =
≡ (12)
= (A(W2)+WB(W2))+ (A(W2) WB(W2))
1 2
P P −
contains only two terms in the decomposition (9). In this case the eigenvalue problem
(1) is reduced to eigenvalue problem for n n matrices S
1,2
×
S π (A(W2)+WB(W2))π = λπ ,
1 1 1 1
≡ (13)
S π (A(W2) WB(W2))π = λπ
2 2 2 2
≡ −
and eigenprojectors Π take the factorized form
i
Π = Λ+π(i), i = 1,...,n,
i 1
(14)
Π = Λ−π(i), i = n+1,...,2n
i 2
for positive and negative energy poles respectively.
If P-parity is violated, the spectral representation (3) for inverse propagator becomes less
•
evident. For single fermion (n = 1 in the above) it was built and investigated in Ref.
[22]. The eigenvalues λ (W) are defined by the characteristic equation
1,2
λ2 λ(S +S )+(S S S S ) = 0, (15)
1 2 1 2 3 4
− −
4 Here spur is taken over γ-matrix indices.
5
where the numbers S (W) are coefficients in the decomposition (9). The eigenprojectors
i
in general case are
1
Π = (S λ ) +(S λ ) S S ,
1 2 1 1 1 1 2 3 3 4 4
λ λ − P − P − P − P
2 −1 1(cid:16) (cid:17) (16)
Π = (S λ ) +(S λ ) S S .
2 2 2 1 1 2 2 3 3 4 4
λ λ − P − P − P − P
1 2
− (cid:16) (cid:17)
Finally, note that if to use the γ-matrix basis for inverse propagator
S = a+nˆb+γ5c+nˆγ5d = a+nˆ(b+nˆγ5c+γ5d), (17)
then the eigenprojectors (16) may be rewritten in the very simple form
1 b+nˆγ5c+γ5d
Π = 1 nˆ , (18)
1,2
2 ± · √b2 +c2 d2
(cid:18) − (cid:19)
where nµ = pµ/W is the unit vector.
B. Left eigenvalue problem
Let us consider the mixing problem for n fermion fields in the theory with parity viola-
tion. The inverse propagator is defined by decomposition (9) with arbitrary matrix coefficients
S (W). Following Ref. [22], we solve the eigenvalue problem
M
SΠ = λΠ (19)
in matrix form, i.e. from the beginning we are looking for eigenprojectors Π instead of eigen-
vectors. The sought-for eigenprojectors may also be written as decomposition (9)
4
Π = A , (20)
M M
P
M=1
X
with matrix n n coefficients A (W). Due to simple multiplicative properties of the basis
M
×
(8), it’s easy to reduce the eigenvalue problem (19) to the following set of linear equations for
unknown matrices A
M
(S λ)A +S A = 0,
1 1 3 4
−
(S λ)A +S A = 0,
2 2 4 3
− (21)
(S λ)A +S A = 0,
1 3 3 2
−
(S λ)A +S A = 0.
2 4 4 1
−
In fact we have two separated subsystems for unknown A , A and A , A , so it’s convenient
1 4 2 3
to express A , A by
3 4
A = S−1(S λ)A , A = S−1(S λ)A (22)
3 − 4 2 − 2 4 − 3 1 − 1
and to obtain the homogeneous equations for n n matrices A , A
1 2
×
OˆA [(S λ)S−1(S λ) S ]A = 0,
1 ≡ 2 − 3 1 − − 4 1 (23)
Oˆ′A [(S λ)S−1(S λ) S ]A = 0.
2 ≡ 1 − 4 2 − − 3 2
6
Here we introduced the short notations Oˆ, Oˆ′ for appeared λ-dependent operators. One can
see that matrices Oˆ, Oˆ′ are connected by similarity relationship
Oˆ′ = (S λ)S−1 Oˆ (S λ)−1S = S (S λ)−1 Oˆ S−1(S λ), (24)
1 − 4 · · 1 − 3 3 2 − · · 4 2 −
so equations (23) give the same characteristic equation for λ
det[(S λ)S−1(S λ) S ] = 0. (25)
2 − 3 1 − − 4
In the absence of degeneration this equation gives 2n different eigenvalues λ (W).
i
Thus, the matrix solution of left eigenvalue problem (19) may be written as
Πi = Ai + Ai S−1(S λ )Ai S−1(S λ )Ai, (26)
P1 1 P2 2 −P3 4 2 − i 2 −P4 3 1 − i 1
where matrices Ai, Ai are solutions of equations
1 2
Oˆ Ai Oˆ(λ = λ )Ai = 0,
i 1 ≡ i 1 (27)
Oˆ′Ai Oˆ′(λ = λ )Ai = 0
i 2 ≡ i 2
and eigenvalues λ (W) are defined by equation (25).
i
C. Right eigenvalue problem
It was noted in the above that orthogonal projectors should satisfy both left and right
eigenvalue problems. So as the next step consider the right eigenvalue problem for inverse
propagator
Π S = λΠ . (28)
R R
We can look for the right eigenprojectors Π in the same form (20) with matrix coefficients
R
B . Similar calculations give the matrix solution of the right problem
M
Πi = Bi + Bi BiS (S λ )−1 BiS (S λ )−1, (29)
R P1 1 P2 2 −P3 1 3 2 − i −P4 2 4 1 − i
where Bi, Bi are solutions of the right homogeneous equations
1 2
BiOˆ′ = 0, BiOˆ = 0 (30)
1 i 2 i
and eigenvalues λ (W) are defined by the same equation (25).
i
D. Left and right problems together
Let us require the “matrix” Π to be solution of both left and right eigenvalue problems. It
means that expressions (26), (29) should coincide with each other.
First of all Bi = Ai, Bi = Ai, as is seen from , terms. Coefficients at , give two
1 1 2 2 P1 P2 P3 P4
relations between A and A
1 2
Ai = S−1(S λ ) Ai S (S λ )−1,
2 3 1 − i · 1 · 3 2 − i (31)
Ai = (S λ )−1S Ai (S λ )S−1.
2 2 − i 4 · 1 · 1 − i 4
7
Now the matrices A , A should satisfy both left and right homogeneous equations
1 2
Oˆ Ai = 0, AiOˆ′ = 0,
i 1 1 i
(32)
Oˆ′Ai = 0, AiOˆ = 0,
i 2 2 i
where the matrices Oˆ , Oˆ′ are defined by (23).
i i
Note that homogeneous equations for A lead to the following equalities
1
S−1(S λ ) Ai = (S λ )−1S Ai,
3 1 − i · 1 2 − i 4 · 1 (33)
Ai (S λ )S−1 = Ai S (S λ )−1,
1 · 1 − i 4 1 · 3 2 − i
so one can see that two relations (31) actually coincide. Moreover, one can convince yourself
that equations for Ai (32) are consequence of relation (31) and equations for Ai. Therefore, it
2 1
is sufficient to require the left and right homogeneous equations for Ai (first line in (32)) and
1
connection between Ai and Ai (one of (31)).
2 1
At last, note that the matrix Ai has zeroth determinant and may be represented in the split
1
form
Ai = ψ (ψ˜)T, (34)
1 i i
˜
where vectors ψ , ψ (columns) are solutions of homogeneous equations
i i
Oˆ ψ = 0, (ψ˜)TOˆ′ = 0 or (Oˆ′)Tψ˜ = 0 . (35)
i i i i i i
(cid:16) (cid:17)
Then solution of both left and right eigenvalue problems may be represented as
Π = ψ (ψ˜)T + S−1(S λ )ψ (ψ˜)T(S λ )S−1
i P1 i i P2 3 1 − i i i 1 − i 4 −
ψ (ψ˜)T(S λ )S−1 S−1(S λ )ψ (ψ˜)T. (36)
−P3 i i 1 − i 4 −P4 3 1 − i i i
˜
For short notations, it is convenient to introduce the vectors φ , φ as
i i
φ = S−1(S λ )ψ , (φ˜)T = (ψ˜)T(S λ )S−1. (37)
i 3 1 − i i i i 1 − i 4
Inthesetermsthe“matrix”Π , whichisasolutionofbothleftandrighteigenvalueproblems,
i
takes very elegant form
Π = ψ (ψ˜)T + φ (φ˜)T ψ (φ˜)T φ (ψ˜)T. (38)
i 1 i i 2 i i 3 i i 4 i i
P · P · −P · −P ·
˜
Recall, that the auxiliary vectors φ , φ also satisfy the following homogeneous equations
i i
(consequence of definition)
Oˆ′φ = 0, (φ˜)TOˆ = 0. (39)
i i i i
E. Eigenprojectors
So we have Π (38) — solutions of both left and right eigenvalue problems. Let us require
i
these “matrices” (with two sets of indices) Π to be orthogonal projectors
i
Π Π = δ Π . (40)
i k ik k
8
It gives four equations if to use the decomposition (9)
ψ (ψ˜)Tψ +(φ˜)Tφ δ (ψ˜ )T = 0,
i i k i k ik k
−
h i
φ (ψ˜)Tψ +(φ˜)Tφ δ (ψ˜ )T = 0,
i i k i k ik k
−
(41)
h i
ψ (ψ˜)Tψ +(φ˜)Tφ δ (φ˜ )T = 0,
i i k i k ik k
−
h i
φ (ψ˜)Tψ +(φ˜)Tφ δ (φ˜ )T = 0,
i i k i k ik k
−
h i
which are equivalent to the orthonormality condition for vectors involved in (38)
(ψ˜)Tψ +(φ˜)Tφ = δ . (42)
i k i k ik
If i = k the condition (42) is consequence of equation on ψ and (ψ˜)T. To see it, let us
k i
• 6
˜
rewrite (42) in terms of the vectors ψ and φ :
i i
(φ˜)T (S λ )S−1 +S−1(S λ ) ψ = δ . (43)
i 2 − i 3 3 1 − k k ik
h i
˜
Now let us write down the homogeneous equations for ψ and φ
k i
0 = Oˆ ψ = S−1λ2 λ (S S−1 +S−1S )+S S−1S S ψ ,
k k 3 k − k 2 3 3 1 2 3 1 − 4 k
(44)
h i
0 = (φ˜)TOˆ = (φ˜)T S−1λ2 λ (S S−1 +S−1S )+S S−1S S .
i i i 3 i − i 2 3 3 1 2 3 1 − 4
h i
Multiplying first of these equations by (φ˜)T from the left, second one by ψ from the
i k
right, and subtracting one equation from another, we obtain
(λ λ ) (φ˜)T (S λ )S−1 +S−1(S λ ) ψ = 0 (45)
k − i · i 2 − i 3 3 1 − k k
h i
and at λ = λ it gives the condition (42).
i k
6
At i = k equation (42) defines the normalization (with weight) of the vector ψ in respect
i
•
˜
to ψ .
i
III. CASE OF CP CONSERVATION
In the case of CP conservation, the matrix n n coefficients of the self-energy contribution
×
4
Σ(p) = Σ (W) = A(p2)+pˆB(p2)+γ5C(p2)+pˆγ5D(p2) (46)
M M
P
M=1
X
have the following symmetry properties (see, e.g. Ref. [25])
AT = A, BT = B, DT = D, CT = C, (47)
−
which are equivalent to
(Σ )T = Σ , (Σ )T = Σ . (48)
1,2 1,2 3 4
−
9
Since the inverse propagator S(p) has the same symmetry properties (48), it connects matrices
Oˆ and Oˆ′ (23)
Oˆ′ = (Oˆ)T. (49)
−
Eigenprojectors have the form (38) but now two equations (35) coincide
Oˆ ψ = 0, Oˆ ψ˜ = 0. (50)
i i i i
˜
Then, in the absence of degeneration, we have ψ = c ψ and the coefficient c may be
i i i i
absorbed by redefinition of vector. From the limiting case of parity conservation (see Sec. IIIA)
it follows that c should have different signs for solution with positive and negative energies.
i
˜
So, the most convenient choice is ψ = ε ψ , where ε = 1 is the sign of energy.
i i i i
±
So, the eigenprojectors (38) in the case of CP conservation take the form
Π = ε ψ (ψ )T φ (φ )T + ψ (φ )T φ (ψ )T (51)
i i 1 i i 2 i i 3 i i 4 i i
P · −P · P · −P ·
(cid:0) (cid:1)
and the vector φ is related to ψ by
i i
φ = S−1(S λ )ψ , or (φ )T = (ψ )T(S λ )S−1. (52)
i 3 1 − i i i − i 1 − i 4
In the case of CP conservation, we need to solve the homogeneous equation for vector ψ for
i
every λ
i
Oˆ ψ = (S λ )S−1(S λ ) S ψ = 0, i = 1,...,2n (53)
i i 2 − i 3 1 − i − 4 i
and to calculate φ accord(cid:2)ing to (52). Note that φ sa(cid:3)tisfies the homogeneous equation (conse-
i i
quence of (53), (52))
OˆTφ = (S λ )S−1(S λ ) S φ = 0. (54)
i i − 1 − i 4 2 − i − 3 i
(cid:2) (cid:3)
The orthonormality condition Π Π = δ Π leads to simple property of vectors
i k ik k
ε (ψ )Tψ (φ )Tφ = δ . (55)
i i k i k ik
−
(cid:0) (cid:1)
As it was shown before, this is not a new requirement: at i = k it follows from homogeneous
6
equation and at i = k it defines normalization of vectors ψ .
i
But to keep the solutions with positive and negative energies onequal footing (see Sec. IIIA)
one should proceed in a different way.
a) For the positive energy solution (i = 1,...,n) we solve the equation for vector ψ (53)
i
and after it calculate φ according to (52).
i
b) For the negative energy solution (i = n+1,...,2n) we find vector φ from the equation
i
(54). Then we can calculate the vector ψ from relation5 (52)
i
ψ = S−1(S λ )φ . (56)
i 4 2 − i i
5 In fact one can avoid the solution of equation (54) due to W W replacement — see, e.g. a particular
→ −
case (68).
10
A. Case of parity conservation
Let us consider a particular case of the spectral representation of propagator, when parity
is conserved6. It allows to clarify some details of general construction.
In this case the eigenprojectors Π
i
SΠ = λ Π , Π S = λ Π (57)
i i i i i i
take the factorized form, see (14). Here n n matrices π satisfy the homogeneous equations
i
×
(13)
S π = λπ ,
1 1 1
(58)
S π = λπ
2 2 2
and also the right equations (see (5))
π S = λπ ,
1 1 1
(59)
π S = λπ .
2 2 2
It’s known that the eigenvalues of left and right problems coincide and since the matrices
S (W), S (W) are symmetric ones, the solutions (vectors) of both left and right eigenvalue
1 2
probems also coincide. So the matrices π may be represented in a split form.
i
Projectors which correspond to positive energy poles are given by
•
Π = Λ+(p)ψ ψT, i = 1,...,n. (60)
i i i
Vectors ψ satisfy the eigenvalue equation
i
S ψ = λ ψ , i = 1,...,n, (61)
1 i i i
where λ are solutions of characteristic equation
i
det(S (W) λI ) = 0. (62)
1 n
−
Projectors onto the negative energy poles are
•
Π = Λ−(p)φ φT, i = n+1,...,2n. (63)
i i i
Equation for vectors φ is
i
S φ = λ φ , i = n+1,...,2n. (64)
2 i i i
Corresponding characteristic equation is
det(S (W) λI ) = 0. (65)
2 n
−
6 We suppose that the mixing fermionfields Ψ , Ψ havethe same parity(quarks or leptons). If they have the
1 2
oppositeparities(baryonfieldsineffectivetheories),theself-energycontainsγ5 incaseofparityconservation
and the dressed matrix propagatorhas absolutely different form, see Refs. [26, 27].