Time-to-space conversion in quantum field theory of flavor mixing

Annals of Physics 315 (2005) 488–504 www.elsevier.com/locate/aop

Time-to-space conversion in quantum field theory of flavor mixing Chueng-Ryong Ji*, Yuriy Mishchenko Department of Physics, North Carolina State University, Raleigh, NC 27695-8202, USA Received 20 May 2004; accepted 16 September 2004 Available online 15 December 2004

Abstract We address the time-to-space conversion in quantum field theory of mixing. In the general theory of quantum field mixing (with an arbitrary number of mixed fields with either boson or fermion statistics) the mixing relations for flavor states are derived directly from the definition of mixing for quantum fields and the unitary inequivalence of the Fock space of energy- and flavor-eigenstates is found. The time dynamics of the interacting fields can be explicitly solved and the flavor time oscillation formulas can be derived in a general form. In this work, we analyze the conversion of these results to space-oscillations with a generalized method of wave-packets. Emphasizing the antiparticle content, we work entirely within the canonical formalism of creation and annihilation operators that allows us to include the effect due to the nontrivial flavor vacuum.
2004 Elsevier Inc. All rights reserved. PACS: 11.10.-z; 12.15.Ff; 14.60.Pq Keywords: Flavor mixing; Space oscillation

1. Introduction The mixing of quantum fields plays an important role in the phenomenology
0 and B0 B
0 bosons provides the of high-energy physics [1–3]. Mixing of both K 0 K *

Corresponding author. Fax: +1 919 5152471. E-mail address: [email protected] (C.-R. Ji).

0003-4916/$ - see front matter
2004 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2004.09.004

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evidence of CP violation in weak interaction [4] and gg 0 boson mixing in the SU(3) flavor group gives a unique opportunity to investigate the nontrivial QCD vacuum and fill the gap between QCD and the constituent quark model. In the fermion sector, neutrino mixing and oscillations provide striking evidences of neutrino masses and are the most likely solution of the famous solar neutrino puzzle [5–7]. In addition, the standard model incorporates the mixing of fermion fields through the Kobayashi–Maskawa (CKM) mixing of three quark flavors, a generalization of the original Cabibbo mixing matrix between d and s quarks [8–11]. Recently, it was found that the flavor mixing in quantum field theory introduces nontrivial relationships between the interacting and noninteracting (free) fields, which leads to unitary inequivalence between the Fock space of the interacting fields and that of the free fields [12–16]. The investigation of two-field unitary mixing in the fermion sector by Blasone and Vitiello [14,17–19] demonstrated a rich structure of the interacting fields vacuum as SU(2) coherent state and altered the oscillation formula to include the antiparticle degrees of freedom. Subsequent analysis of the boson case revealed a similar but richer structure of the vacuum of the interacting fields [13,20,21]. Attempts to look at the mixing of more than two flavors have also been carried out and a general framework for such a theory has been suggested [22]. Surprisingly, although a wide research effort has been undertaken in this direction, an important aspect of the formulation is still left uncultivated. Namely, the oscillation formulas were obtained for time oscillations, while the experiments measure only space oscillations [23]. In quantum mechanics, the means of conversion between time and space oscillations is provided by the approach of wave-packets in which the correct oscillation length is recovered and also the concept of correlation length is introduced. As we now know, in quantum field theory, the possibility of antiparticle admixtures changes the simple quantum mechanical dynamics of flavors [14]. It, therefore, becomes an interesting question how these effects translate into space-dimensions and what the result of such translation may be. In this work, we answer these questions by developing a field-theoretical treatment of time-to-space conversion for the flavor mixing. We must note that recently the wave-packet approach has been extended to the field theory using Green
s functions formalism [24]. However, such treatment is essentially similar to quantum mechanics and does not allow to simply consider the effect of antiparticle admixture in flavor states. We, on the other hand, intend to carry out our work entirely in the canonical formalism of creation/annihilation operators where such effects can be considered straightforwardly. In particular, the higher-frequency contribution to time-oscillations formulas can be retained and its effect in space-dimensions can be studied. In the next section (Section 2), we review the main points of the general quantum field theory of mixing. In Section 3, we present the canonical framework for the flavor oscillations in space-time and generalize the wave-packet method to work in the quantum field theory. In Section 4, we discuss the space-oscillations in the system of two scalar mesons as a demonstration of our method and present some relevant numerical results. Conclusion follows in Section 5.

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2. Flavor oscillations in quantum field theory Quantum mixing, first experimentally observed in weak interactions, occurs when the interaction and the propagation states of a particle are drastically different. As a result, a particle produced in some decay as a state with a decay signature A may propagate with time into a state with very different decay signature B. One usually treats this phenomenon in terms of interaction (or flavor A, B, . . .) and propagation (or energy a, b, . . .) eigenstates so that the flavor state is represented as a superposition of the energy-eigenstates. In quantum mechanics mixing of flavors may be described with Hamiltonian1 X X 1X X H¼ xl;k ayl;k al;k þ mlm ðayl;k am;k þ aym;k al;k Þ; ð1Þ 2 k l¼A;B;... k l;m¼A;B;... where ayA;B;... ðaA;B;... Þ are creation (annihilation) operators for quantum flavor-states. Note that this can be straightforwardly diagonalized by introducing creation/annihilation operators for energy eigenstates X ai;k ¼ U yil al;k ; ð2Þ l¼A;B;...

where U yil is an appropriate unitary mixing matrix, so that X X xi;k ayi;k ai;k þ const: H¼ k

ð3Þ

i¼a;b;...

Then, the time dynamics of the original flavor states can be solved straightforwardly X X jl; k; ti ¼ U li ji; k; ti ¼ U li e
ixi;k t ji; k; 0i ð4Þ i

i

and the flavor oscillation formulas can be derived
2


X
2 y
ixi;k t
jhm; kjl; k; tij ¼
U li U im e
:

i

ð5Þ

These are the famous Pontecorvo mixing formulas, e.g., for two flavors mixed with angle h they read x
x  a b 2 t ; jhAjA; tij ¼ 1
sin2 ð2hÞsin2 2 x
x  ð6Þ a b jhBjA; tij2 ¼ sin2 ð2hÞsin2 t : 2 In quantum field theory, analogously, mixing may be described by interaction Hamiltonian density 1 X HI ð/ðxÞÞ ¼ M lm /yl ðxÞ/m ðxÞ þ h:c: ð7Þ 2 l;m¼A;B;... 1 We use the Latin indexes i, j, k, . . . and small Latin letters a, b, . . . to label the mass-eigenstates and the Greek indexes l, m, q, . . . and capital Latin letters A, B, . . .to label the flavor eigenstates.

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491

Classically, this still can be diagonalized by a linear transformation from flavor-fields /l to free fields ui X /l ! ui ¼ U yil /l ð8Þ l¼A;B;...

with ui defined as X Z dk   pffiffiffiffiffiffiffiffiffi uikr aikr ðtÞeikx þ vikr byikr ðtÞe
ikx : ui ¼ 2xik r

ð9Þ

Here aikr ðtÞ ¼ e
ixik t aikr and bikr ðtÞ ¼ e
ixik t bikr (with the standard equal time commutation/anticommutation relationships) and uikr and vikr are the free particle and antiparticle amplitudes, respectively, where r is the helicity quantum number     k k  s uikr ¼ ruikr ;  s vikr ¼ rvikr ð10Þ jkj jkj and s is the spin operator. However, in quantum field theory the classical transformation given by Eq. (8) does not immediately imply a specific form for the mixing relations between flavor and energy eigenstates. In fact, it had been argued that, using regular perturbation approach in mixing of two fermions with spin 1/2, one arrives at a normalization problem demonstrated in [17]. Let us briefly present the argument of [17] here. For two fermion mixing the perturbation theory is simple and can be summed
j0i straightforwardly, e.g., for time-ordered two-point function S AA ¼ h0jT ½wA w A S AA ¼ S A ð1 þ m2AB S B S A þ m4AB S B S A S B S A þ   Þ ¼ S A ð1
m2AB S B S A Þ
1 þ ma þ mb ¼ cos2 ðhÞ 2 þ sin2 ðhÞ 2 ; 2 k
ma þ id k
m2b þ id

ð11Þ


1

where the ‘‘bare’’ propagators are S A;B ¼ ð
mA;B þ idÞ . Similarly, the transition amplitude for a fermion A to go, e.g., into the same particle after time t is given by 2

ixk;a t > P AA ðk; tÞ ¼ iury S AA ðk; tÞc0 urk;a ¼ cos2 ðhÞ þ sin2 ðhÞjU k j eiðxk;a
xk;b Þt ; k;a e

ð12Þ

where S > AA ðtÞ is forward (t > 0) propagation function and ua,b are the bi-spinors used to expand the mass-eigenstate fields so that 1 X ry s 2 2 jU k j ¼ ju u j : 2 r;s k;b k;a Upon computing |Uk|2 one can explicitly observe that |Uk|2 < 1 and thus PAA(t fi + 0) „ 1 [17]. Similar normalization issue can be found in boson case as well [13]. These results indicate that special care needs to be taken in quantum field theory to properly define quantum flavor states.

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In the field theory, any operators with the same conserved quantum numbers can mix. This means that in general X   alkr ¼ ali ðkÞaikr þ bli ðkÞbyi
k
r ; i¼a;b;...

byl
k
r

X 

¼

 ali ðkÞbyi
k
r þ gbli ðkÞaikr ;

ð13Þ

i¼a;b;...

where al(ai) stands for flavor-eigenstate (mass-eigenstate) particle annihilation operator and bl(bi) stands for flavor-eigenstate (mass-eigenstate) antiparticle annihilation operator. Factor g = (
1)2S with S being the spin of the mixed fields (g is +1 for bosons and
1 for fermions) is required to guarantee [alkr,bl
k
r]± = 0. Respectively, we have introduced two mixing matrices (ali and bli) to describe the particle–particle mixing and the particle-antiparticle cross-mixing2. For Eq. (13) to preserve commutation/anticommutation relationships it must hold that ( 2 2 2 jali j þ jbli j ¼ jU li j fermions; ð14Þ 2 2 2 jali j
jbli j ¼ jU li j bosons; so that

cosðhli Þ fermions;



coshðhli Þ bosons; sinðhli Þ fermions;

ali ¼ U li bli ¼ U li

ð15Þ

sinhðhli Þ bosons:

Eq. (13) also furnishes a representation of the classical mixing transformation (8) in the linear space of quantum fields. Then, it follows that the following relation holds hli
hli0 ¼ hi0 i

ð16Þ

independent of l. Furthermore, the flavor vacuum state |Xæ of the Fock space of flavor eigenstates shall be defined as a state annihilated by all flavor particle/antiparticle annihilation operators. Its explicit structure can be obtained by solving these conditions [22]: ! N X 1 y y jXi ¼ exp Z ij ai b
j j0i; ð17Þ Z i;j¼1 where Zij is an (i,j) element of the matrix Z =
a
1 Æ b. The normalization constant y y Z is fixed by ÆX|Xæ = 1; Z ¼ det1=2 ð1 þ Z^ Z^ Þ for fermions and Z ¼ det
1=2 ð1
Z^ Z^ Þ for bosons. 2

Here, we suppress momentum notation implying that all quantities are taken at given momentum k and helicity r or (
k,
r) as will be indicated by the sign in front of the flavor/mass-eigenstate index (i.e., ai will stand for aikr and b
i for bi
k
r).

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493

The time dynamics of the quantum theory is described by its nonequal time commutation/anticommutation relationships, which are found to be [22] X



  F lm ðtÞ ¼ ½al ðtÞ; aym  ¼ alk amk0 ak e
ixk t ; ayk0  þ blk bmk0 by
k eixk t ; b
k0  k;k 0

¼

X

 alk amk e
ixk t
gblk bmk eixk t ;

k

½b
l ðtÞ; by
m  ¼ F ml ðtÞ;  X



  alk bmk0 b
k e
ixk t ; by
k0  þ gblk amk0 ayk eixk t ; ak0  Glm ðtÞ ¼ ½b
l ðtÞ; am  ¼ k;k 0

¼

X

 alk bmk e
ixk t
blk amk eixk t :

k

ð18Þ We also note that for t fi + 0 Eq. (18) reduces to Flm(0) = dlm and Glm(0) = 0 and also that Flm(t)* = Fml(
t), Glm(t)* =
Gml(t). Using these, the evolution of the flavor
qmq Þ and charge Qqmq ¼ N qmq
N
qmq m particle number (Nqmq), antiparticle number ðN expectation values in a beam with a given 3-momentum k can be found as N qmq ¼ h0jaq aym ðtÞam ðtÞayq j0i ¼ jF mq ðtÞj2 þ Z m ðtÞ;
qmq ¼ h0jaq by ðtÞb
m ðtÞay j0i ¼ gjGmq ðtÞj2 þ Z m ð
tÞ; N
m q 2

ð19Þ

2

Qqmq ¼ jF mq ðtÞj
gjGmq ðtÞj : Now, since in quantum field theory, in general, two matrices are required to describe flavor mixing, one needs to establish a connection between the classical mixing matrix U and the quantum mixing matrices a and b. Blasone and Vitiello
s quantum field theory of mixing provides such connection based on the observation that an explicit quantum transformation K(U,t) in the linear space of quantum fields can be constructed out of fields u and their canonical momenta p that provides a representation of the classical relation (8) X y /A ðtÞ ¼ U Ai ui ðtÞ ¼ KðU ; tÞ ua ðtÞKðU ; tÞ; etc: ð20Þ i

In the simplest case of two-scalar fields, such transformation is explicitly Z Kðh; tÞ ¼ exp ih dxðpya ðx; tÞub ðx; tÞ
pyb ðx; tÞua ðx; tÞ þ h:c:Þ : It can be checked by a straightforward computation that this indeed generates the mixing transformation for 2-flavors: /A ¼ cosðhÞua þ sinðhÞub ; /B ¼
sinðhÞua þ cosðhÞub and diagonalizes the quantum Hamiltonian. In the associate Fock-space, K(U,t) acts similarly jAi ¼ KðU ; 0Þy jai; etc:;

ð21Þ

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C.-R. Ji, Y. Mishchenko / Annals of Physics 315 (2005) 488–504

thus the ladder operators are transformed in the same way y

aA ðtÞ ¼ KðU ; tÞ aa ðtÞKðU ; tÞ; etc:

ð22Þ

These definitions allow one to find explicit form of a and b. Two-flavor mixing and three-flavor mixing (where explicit finding of K(U,t) is a much more difficult task) for both spin-1/2 fermions and scalar bosons have been analyzed along this line in the literature [14,18,19,21,25]. General relations, using somewhat different approach, have been also found [22].

3. Oscillations of flavor in space-time Let us now consider a particle that was created initially in quantum state |iæ and that propagates in space and time. The number of particles of sort q to be detected at the space-time position x = (t,x) is given by the expectation value of the number operator N q ðxÞ ¼ hijayq ðxÞaq ðxÞjii;

ð23Þ

where ayq ðxÞðaq ðxÞÞ is creation (annihilation) operator for the particle of flavor q at space-time position x. These can be defined via creation (annihilation) operators with 3-momentum k as X 1 pffiffiffiffiffiffiffiffiffiffi e
ikx ayq;k ðtÞ: ayq ðt; xÞ ¼ ð24Þ 2xqk k Substituting the definition given by Eq. (24) into Eq. (23), one obtains N q ðxÞ ¼

X k;k0

0

eiðk
kÞx y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hijaq;k ðtÞaq;k0 ðtÞjii: 2 xqk xqk0

ð25Þ

We thus find that the number of particles expected at the space-time position x can be found using Eq. (25) once hijayq;k ðtÞaq;k0 ðtÞjii is known for all k and k 0 . In the case of free fields, i.e. aq;k ðtÞ ¼ aq;k ð0Þ e
ixq;k t ; it reduces trivially to the square of conventional transition amplitude. To specify more precisely the initial state |iæ, we note that the original state of a particle, created in some interaction, is a superposition of different definite-flavor states. Thus, for mixing problem with N flavors we shall consider the initial state |iæ in the form jii ¼

N X

ðgq;k ayq;k þ hq;k byq;
k ÞjXi;

ð26Þ

q¼1;k

where again, ayq;k and byq;
k are the particle and antiparticle creation operators for flavor q and |Xæ is the flavor vacuum annihilated by all aq,k and bq,k. Eq. (26) represents a single particle created initially in a state such that the probability to observe it as a

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495

flavor q-particle (antiparticle) is simply |gq,k|2 (|hq,
k|2). Then, 2N functions gq,k, hq,k are the form-factors for the initial state |iæ3. For notation convenience we shall adopt following convention: we will let a be both positive and negative (a =
N, . . . ,N, excluding a = 0) with negative a enumerating antiparticles and positive a enumerating particles, respectively. In this notation jii ¼

N X

fa;k aya;k jXi;

ð27Þ

a¼
N ;k

where we have introduced aya;k :¼ aya;k for a > 0 and aya;k :¼ by
a;
k for a < 0. Analogously, fa,k: = ga,k for a > 0 and fa,k: = h
a,k for a < 0. Also, from now on, in summations over a we imply a „ 0. Furthermore the creation operator may be introduced for form-factors F:{fa,k, a =
N . . .
1, 1, . . . ,N} as ay ðF Þ ¼

N X

fa;k aya;k ;

ð28Þ

a¼
N ;k

so that concisely |iæ = a(F)|Xæ. From Eq. (18) it is straightforward to obtain the nonequal time commutation/anticommutation relationships for a(F) and a(G) ½aðF Þ; ay ðGÞ;equal

time

N X

¼

 fa;k ga;k ;

a¼
N ;k

½at ðF Þ; ay ðGÞ ¼

N X

 fa;k Fab;k ðtÞgb;k ;

ð29Þ

a;b¼
N ;k

½at ðF Þ; aðGÞ ¼

N X

 fa;k Gab;k ðtÞgb;k ;

a;b¼
N ;k

where Fab;k ðtÞ, Gab;k ðtÞ in our notation are defined as 8 ! > < G
a;b ðtÞ; a < 0; b > 0; 0 gGT ð
tÞ Gab ðtÞ ¼ ½aa;k ðtÞ; ab;k0  ¼ gG
b;a ð
tÞ; a > 0; b < 0; ¼ ; > GðtÞ 0 : 0; otherwise: 8 >   < F a;b ðtÞ; a > 0; b > 0; F ðtÞ 0 y Fab ðtÞ ¼ ½aa;k ðtÞ; ab;k0  ¼ F
b;
a ðtÞ; a < 0; b < 0; ¼ : > 0 F T ðtÞ : 0; otherwise: ð30Þ 3 In Eq. (26), we assumed that particles and antiparticles are distinguishable. Although this is feasible in the case of, e.g., neutrinos, for many cases of meson mixing the field operators are self-adjoint and thus particles may not be distinguished from antiparticles. However, Eq. (26) can still be used by redefining b ” a and gq,k ” hq,
k. Given this remark, we will continue with general formulation keeping in mind that the mixing of the self-adjoint fields can be obtained with straightforward adjustments from our final results.

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C.-R. Ji, Y. Mishchenko / Annals of Physics 315 (2005) 488–504

Our primary interest is the process in which a ‘‘flavor’’ particle is created in some initial state |iæ with form-factors G = {ga,k} and is detected at time t as a state |f æ with form-factors F = {fa,k}. Then, the expected number of particles to be detected is given by

 ð31Þ hijN F ðtÞjii ¼ XjaðGÞayt ðF Þat ðF Þay ðGÞjX : After some algebra we find

 2 2 hijN F ðtÞjii ¼ j½at ðF Þ; ay ðGÞ j þ gj½aðGÞ; at ðF Þ j þ ayt ðF Þat ðF Þ
2
2




X
X


  
fa;k Fab;k ðtÞgb;k
þ g
fa;k Gab;k ðtÞgb;k
¼



a;b;k
a;b;k X  þ fa;k Za;k ðtÞfa;k ;

ð32Þ

a;k

where the last term Zbg ¼ densation

PN

 a¼
N ;k fa;k Za;k ðtÞfa;k

Za;k ðtÞ ¼ hXjaya;k ðtÞaa;k ðtÞjXi:

is related to the flavor vacuum conð33Þ

Generally, this term is not zero for t „ 0 and for many choices of F it may be infinite. On the other hand, one may notice that this contribution is independent from the initial state |iæ. Moreover, for the case of point-like detector (f(k)  e
ikx) this contribution is also x-independent and thus may be interpreted as constant background due to the vacuum condensation picked up by the detector. Oscillations of flavor charge can be obtained from Eq. (32). For a single particle with flavor b and form-factor f(k), the flavor charge oscillations are given by Qb ðtÞ ¼ N ½b ðtÞ
N ½
b ðtÞ;

ð34Þ

where [b] = {fa,k = f(k)da,b} and [
b] = {fa,k = f(
k)d
a,b}. It is straightforward to derive from Eqs. (32) and (34) the field-theoretical formula for oscillations in space, in whichpcase ffiffiffiffiffiffiffiffiffiffithe detector shall be characterized by the form-factor ½b ¼ ffa;kp¼ffiffiffiffiffiffiffiffiffiffi ðe
ikx Þ=ð 2xbk Þda;b g for a single particle of sort b or ½
b ¼ ffa;k ¼ ðeikx Þ=ð 2xbk Þd
a;b g for a single antiparticle of the same sort.
2

2



X eikx
X eikx



Fba;k ðtÞga;k
þg
Gba;k ðtÞga;k
þZbg ; hijN b ðt;xÞjii¼
1=2 1=2


a;k ð2xbk Þ
a;k ð2xbk Þ
2

2



X e
ikx
X e
ikx



F ðtÞg þ g G ðtÞg hijN
b ðt;xÞjii¼


þZbg ;

ba;k
ba;k a;k a;k


a;k ð2xbk Þ1=2
a;k ð2xbk Þ1=2 hijQb ðt;xÞjii ¼hijN b ðt;xÞjii
hijN
b ðt;xÞjii: ð35Þ As in the quantum-mechanical wave-packet method, our final result depends on the kind of initial state assumed for the flavor particle. For example, one may consider the initial state as a state with definite momentum k and definite flavor b. Then, dropping the constant background contribution Zbg ,

C.-R. Ji, Y. Mishchenko / Annals of Physics 315 (2005) 488–504
1

2


1

497 2

hbkjN a ðt; xÞjbki ¼ ð2xak xbk Þ jFab;k ðtÞj þ gð2xak xbk Þ jGab;k ðtÞj :

ð36Þ

This has no dependence on x and thus one can not observe any space oscillations. One also may consider a particle of sort b created at position x 0 and observed at position x at time t as a particle of sort a. For that case, similarly dropping constant background, we write
2
2




X eiðx
x0 Þk
X eiðxþx0 Þk



hbx0 jN a ðt; xÞjbx0 i ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fab;k ðtÞ
þ g
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gab;k ðtÞ



k 2 xak xbk
k 2 xak xbk ¼ j Fab ðjx0
xj; tÞj2 þ gj Gab ðjx0 þ xj; tÞj2 ; ð37Þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 where  FðjzjÞ is the Fourier transform of ð2 xak xbk Þ FðjkjÞ and  G is defined similarly. In practice, we are interested in the case when a flavor particle is produced in a small (but finite) region of space with (nonzero) momentum k. This can be represented by a well-peaked initial state |i æ with the form-factor g(k) such that a single particle of sort b appears as a wave-packet of momentum k0 with a small spread r. Taking explicit form of F and G and leaving detector point-like, we obtain
2


X 2 2 2 2

hbgjN a ðt; xÞjbgi ¼
ðaab;c e
iwc t e
r ðvc t
xÞ =2 þ bab;c eiwc t e
r ð
vc t
xÞ =2 Þ


c
2


X 2 2 2 2

þ g
ðcab;c e
iwc t e
r ðvc t
xÞ =2 þ d ab;c eiwc t e
r ð
vc t
xÞ =2 Þ
;

c vc ¼

dwc ðkÞ j : dk k0

ð38Þ

In the above derivation we used the following identity which can be proved by using stationary phase approximation for function g(k) sharply peaked around k = k0 and slow-varying functions f(k) and S(k): Z ~ 0 ÞÞ2 =2Þ; dk gðkÞf ðkÞeiSðkÞ eikx  ð2pr2 Þ3=2 gðk0 Þf ðk0 ÞeiðSðk0 Þþk0 xÞ expð
r2 ðx þ rSðk r2 ¼


gðk0 Þ : g00 ðk0 Þ

ð39Þ

The explicit form of F and G is taken as (for specific values of aab;c(k), bab;c(k), cab;c(k), dab;c(k) see, for example [22]) N   X pffiffiffiffiffiffiffiffiffiffi
1 ð 2xak Þ Fab;k ðtÞ ¼ aab;c ðkÞe
iwck t þ bab;c ðkÞeiwck t ; c¼1 N   X pffiffiffiffiffiffiffiffiffiffi
1 ð 2xak Þ Gab;k ðtÞ ¼ cab;c ðkÞe
iwck t þ d ab;c ðkÞeiwck t

ð40Þ

c¼1

and all amplitudes in Eq. (38) are taken at k = k0. We observe that Eq. (38) represents a single wave-packet propagating through the space and Æbg|Na(t,x)|bgæ

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C.-R. Ji, Y. Mishchenko / Annals of Physics 315 (2005) 488–504

reaches maximum when the ‘‘center’’ of the wave-packet passes over the observation point x hvit  x

or

h
vit  x:

ð41Þ

To observe space oscillations in conventional sense, we should take in Eq. (35) an average over time which would correspond to a continuous observation Z W bg ðaxÞ lim dthbg jN a ðt;xÞjbg i T !1 T 0

2

2 1 Z


X eikx
X eikx



pffiffiffiffiffiffiffiffiffiffi Fab;k ðtÞgb;k
þg
pffiffiffiffiffiffiffiffiffiffi Gab;k ðtÞgb;k
A:  lim dt@
T !1 T


b;k 2xak
b;k 2xak ð42Þ Using Eq. (40), we may rewrite this as XZZ dk dk0 dðwck
wc0 k0 Þ½ðaab;c ðkÞaab;c0 ðk0 Þ þ bab;c ðkÞbab;c0 ðk0 ÞÞ c;c0 0

þ gðcab;c ðkÞcab;c0 ðk0 Þ þ d ab;c ðkÞd ab;c0 ðk0 ÞÞgb;k gb;k0 eiðk
k Þx : When the mass difference m2c
m2c0 is small, the functional dðwck
wc0 k0 Þ½ðaab;c ðkÞaab;c0 ðk0 Þ þ bab;c ðkÞbab;c0 ðk0 ÞÞ  ðcab;c ðkÞcab;c0 ðk0 Þ 0

þ d ab;c ðkÞd ab;c0 ðk0 ÞÞgb;k gb;k0 eiðk
k Þx

ð43Þ

reaches maximum at k  k 0  k0. Then we can use the stationary phase approximation again to find Xh W ðxÞ  ðaab;c ðkÞaab;c0 ðkÞ þ bab;c ðkÞbab;c0 ðkÞÞ c;c0

(  ) 2 0 Dk cc þgðcab;c ðkÞcab;c0 ðkÞ þ d ab;c ðkÞd ab;c0 ðkÞÞ exp
r2 x2 þ iDkcc0 x ; 2k 0 i

ð44Þ where k  k0 ; Dkcc0 ¼ oscillation length as L

2k : Dm212

Dm2 0

cc 2jk0 j2

k0 ¼

m2c
m20 2jk0 j

c 2

k0 . For the mixing of two flavors we recover the ð45Þ

We should point out that, although field-theoretical effects do reveal themselves in the shape of the space oscillations, here no second major oscillation mode is found. This is different from the case of the flavor oscillations in time where the additional high-frequency terms were very prominent. While this result is certainly in part attributed to the approximation we made, one may also notice that in Eq. (42) Z dt eiðxck xc0 k0 Þt 6¼ 0

C.-R. Ji, Y. Mishchenko / Annals of Physics 315 (2005) 488–504

499

only when xck ± xc 0 k 0 = 0 and for that the frequencies must come in with the opposite signs. Thus, no high-frequency terms may survive the integration over time. We will examine the physical meaning of this observation in the next section. However, the field-theoretical effect does reveal itself in the shape of the space oscillations. 4. Space-oscillations for two flavors In the case of two spin-zero fields the mixing can be described completely by a single real angle which relates the flavor fields (u) to the free fields (/) by      cosðhÞ sinðhÞ /1 u1 ¼ : ð46Þ
sinðhÞ cosðhÞ u2 /2 The time evolution in this system has been studied in the quantum field theoretical framework [20,21]. It was found that the time evolution of the flavor fields can be described by nonequal time commutation relations [20,22]: F 11;k ðtÞ ¼ cos2 ðhÞe
ix1k t þ c2þ sin2 ðhÞe
ix2k t
c2
sin2 ðhÞeix2k t ; F 12;k ðtÞ ¼ F 21;k ðtÞ ¼ cþ sinðhÞ cosðhÞðe
ix2k t
e
ix1k t Þ; 2

F 22;k ðtÞ ¼ cos ðhÞe


ix2k t

þ

c2þ sin2 ðhÞe
ix1k t




ð47Þ

c2
sin2 ðhÞeix1k t

and G11;k ðtÞ ¼ cþ c
sin2 ðhÞðe
ix2k t
eix2k t Þ; G12;k ðtÞ ¼
G21;k ð
tÞ ¼ c
sinðhÞ cosðhÞðe
ix1k t
eix2k t Þ; 2

G22;k ðtÞ ¼ cþ c
sin ðhÞðe


ix1k t


e

ix1k t

ð48Þ

Þ:

Then, the formula for the flavor charge oscillations was found to be Q1 ¼ 1 þ sin2 ðhÞðc2
sin2 ðXtÞ
c2þ sin2 ðxtÞÞ; Q2 ¼ sin2 ðhÞðc2þ sin2 ðxtÞ
c2
sin2 ðXtÞÞ;

ð49Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where x = (x1k
x2k)/2 and X = (x1k + x2k)/2, c ¼ ð x1k =x2k  x2k =x1k Þ=2. In this section, we intend to consider space-propagation of a single particle born as the flavor a = 1 with a sharp gaussian form-factor centered at 3-momentum k0. We make use of Eq. (38). In Eq. (38), we notice that for given a and b either Fab (e.g., a > 0 fi b > 0) or Gab (e.g., a > 0 fi b > 0) is equal to zero (i.e., never both of them together nonzero). Then, implying that aab and bab are nonzero only for the particle–particle sector and cab and dab are nonzero only for the particle–antiparticle sector (i.e. c11 pffiffiffiffiffiffiffiffiffiffiffi actually is c1 fi
1), we shall set up modulo a normalization factor 1= 2xak0 : a11;1 ¼ cos2 ðhÞ; a11;2 ¼ c2þ sin2 ðhÞ; b11;1 ¼ 0; b11;2 ¼
c2
sin2 ðhÞ; a12;1 ¼ a12;2 ¼
cþ sinðhÞcosðhÞ; b12;1 ¼ b12;2 ¼ 0; c11;1 ¼ 0; c11;2 ¼ cþ c
sin2 ðhÞ; d 11;1 ¼ 0; d 11;2 ¼
cþ c
sin2 ðhÞ; c12;1 ¼ c
sinðhÞcosðhÞ; c12;2 ¼ 0; d 12;1 ¼ 0; d 12;2 ¼
c
sinðhÞ cosðhÞ:

ð50Þ

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C.-R. Ji, Y. Mishchenko / Annals of Physics 315 (2005) 488–504

We choose m1 = 540 MeV, m2 = 930 MeV, and h = p/4 close to the parameters of g– g 0 system. As can be seen in Figs. 1 and 2, a typical wave-packet propagates oscillating into the other flavor. After some time, one flavor is almost completely converted into the other after which the reverse process takes place. With time the original Gaussian wave-packet deforms so that the two separate gaussians eventually emerge. This corresponds to the two mass-eigenstates completely separated in space: no flavor oscillations occur after that point. If r is large (almost point-like source) and k0 „ 0, the two mass-eigenstates separate almost immediately and propagate independently producing no flavor oscillations at all (Fig. 4). The additional effect

0

-20

1

1

1

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

-10

20

10

30

-40

-20

20

40

-40

-20

20

4

1 1

0

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

-20

20

40

-40

-20

20

-40

40

-20

20

40

Fig. 1. Example of propagation of gaussian wave-packet for particle of flavor a = 1 at k0  0.35 GeV. The distance scale is GeV
1 and time flows left-to-right and up-to-down.

0

0

-20

1

1

1

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

-10

-20

10

20

30

-40

-20

20

40

-40

-20

20

1

1

1

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

20

40

-40

-20

20

40

-40

-20

20

4

40

Fig. 2. Example of propagation of gaussian wave-packet for particle of flavor a = 2 at k0  0.35 GeV. The distance scale is GeV
1 and time flows left-to-right and up-to-down.

C.-R. Ji, Y. Mishchenko / Annals of Physics 315 (2005) 488–504

501

due to the nontrivial flavor vacuum is noticeable as sporadic distortions of the wavepacket shape at the moments when one of the flavors is almost completely extinct due to the flavor conversion, as can be seen in Fig. 3. Also, remarkable is the trace of the antiparticle wave-packet propagating in the opposite direction to that of the main wave-packet (Fig. 4). This coherent beam of ‘‘recoil’’ antiparticles is due to the terms of the form exp[(
vt
x)2] in Eq. (38); it is correlated with the positive wing at all times via the mechanism similar to the EPR-effect. Clearly, the contribution from the high-frequency term, prominent in time-evolution, translates in space as interference between these parts of the wave-packet, propagating to the right and to the left, respectively, and thus dies very quickly.

0

0.1

0.1

0.1

0.08

0.08

0.08

0.06

0.06

0.06

0.04

0.04

0.04

0.02

0.02

0.02

-20

20

40

-40

-20

20

-0.02

0

40

-40

-20

-0.02

20 -0.02

0.1

0.1

0.1

0.08

0.08

0.08

0.06

0.06

0.06

0.04

0.04

0.04

0.02

0.02

0.02

-20

20

40

-40

-20

20

-0.02

40

-40

-0.02

-20

20 -0.02

Fig. 3. Example of field-theoretical fluctuations of the flavor charge distribution at the ‘‘cross-over’’ time when particle of flavor a = 1 has been almost completely converted in flavor a = 2. The plotted quantity is flavor charge, thus the negative parts correspond to presence of flavor antiparticles in the wave-packet. The distance scale is GeV
1 and time flows left-to-right and up-to-down.

1 0.25 0.8

0.2 0.15

0.6

0.1 0.4

0.05

0.2

-4

-2

2

4

-0.05 -4

-2

2

4

-0.1

Fig. 4. Two snapshots of distribution of flavor charge Q vs. distance for originally well-localized wavepacket give an example of the coherence loss by a point-like flavor source. Also, clearly seen is EPRcorrelated antiparticle wave-packet traveling in the opposite direction. As above, plotted is the flavor charge and the negative piece is antiparticle wave-packet. The distance scale is GeV
1.

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C.-R. Ji, Y. Mishchenko / Annals of Physics 315 (2005) 488–504

Fig. 5. Space oscillations for flavor charge (A) and antiparticle (B) population for flavor a = 1 (k  0.35 GeV). The distance scale is GeV
1.

Fig. 6. Space oscillations of flavor charge (A) and antiparticle (B) population for flavor a = 2 (k  0.35 GeV). The distance scale is GeV
1.

The space oscillations of flavor in conventional sense can be observed through the change of the maximum amplitude of the wave-packet as the particle flies through the space. To observe this quantity explicitly we numerically traced the position of the maximum of the wave-packet (38). We found that the maximum propagates at approximately constant speed consistent with 2k 0 v ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : k 20 þ m21 þ k 20 þ m22

ð51Þ

With that in mind, we reproduced the plot of the maximum amplitude of the wavepacket vs. distance. In Figs. 5 and 6 we observe that when the momentum of the particle is sufficiently low, the form of the space-oscillations is significantly distorted from the quantum-mechanical prescription: at certain points the flavor charge even becomes negative meaning there is a good chance to detect flavor antiparticle at these points. The field-theoretical corrections decrease as the energy increases and die out with the distance. 5. Conclusion We studied the problem of time-to-space conversion in general quantum field theory of flavor mixing. We considered this problem with the most general wave-packet

C.-R. Ji, Y. Mishchenko / Annals of Physics 315 (2005) 488–504

503

approach utilizing the canonical formalism of creation/annihilation operators. This allowed us to account for the nontrivial flavor vacuum effect which is otherwise lost. We derived the space-oscillation formulas for a flavor particle initially created as a sharp wave-packet. Differently from the time-oscillations, in which the flavor vacuum effect introduces a prominent high-frequency terms, we found only one major mode of space-oscillations. The higher-frequency time-oscillation terms translate in space as an interference between the parts of the wave-packet propagating in the opposite directions and thus diminish rather rapidly. Nevertheless, both quantitative and qualitative differences from the quantum mechanical prediction are present in the shape and in the antiparticle content of the space oscillations in the quantum field theory, especially, in the vicinity of the flavor-particle creation site. We further applied our general formalism to a specific case of two-flavor mixing, in which we considered in great details the evolution of the flavor-particle wavepackets. We observed the space oscillations of flavor through the dependence of the wave-packet maximum on the position of the maximum. We found that the propagation of the flavor particle in the above setting is consistent with the group velocity v  k/E. We observed differences between quantum-mechanical and fieldtheoretical results for flavor oscillations and found that the flavor charge may become negative at certain points in space. Also, a correlated beam of antiparticles, propagating in the direction opposite to that of the main wave-packet, was illuminated in our simulation. While for most known physical mixed systems the field-theoretical corrections to space-oscillations are small, the theory has many interesting indirect experimental consequences. For example, we expect to be able to observe traces of flavor antiparticles in the beam of flavor-particles. Also the presence of anti-directed antiparticle recoil beam may imply, via total charge conservation, a substantial excess/deficit of flavor particles at the detection site. This, of course ‘‘fictitious,’’ charge conservation violation is caused by the fact that the antiparticle beam carries a fraction of charge away from the detector. Some interesting implications for cosmology and cosmological constant has been also suggested [26].

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