In this section we study the allowed form of couplings for the electromagnetic .... Here we list the numerical values for the NCR by DMS for future reference ...
-.__-__.. i. .
C H I N E S E 30URNAL OF PIIYSICS
VOL. 31. NO. 1
Electromagnetic Properties of the NeutrinoT
K. L. Ng D e p a r t m e n t of Physics, Noti’onal
Taiwan liniversity, Taipei 106, Taitcan, R.0.c’. (Received September 5, 1992)
In this paper, I will discuss the electromagnetic properties of the neutrino by imposing the hermiticity condition, CPT and CP invariance on the electromagnetic current matrix element. These invariances imply certain constraints on neutriuo ’s elect.romagnetic form factors. Then, I discuss the gauge invariance and divergence problem in defining the neutrino charge radius (NCR) within the Standard Model (SM). Mext, I present the calculation of NCR in the IV = 1 Minimal Supersymmetric Standard Model (MSSM), and compare the supersymmetric results with the SSI value.
I. INTRODUCTION The neutrino was introduced by W. Pauli in 1930 in order to explain the continuous electron spectrum accompanying nuclear beta decay. The experimental evidence for the existence of a neutrino was first reported by Reines and Coivan in 1953. Even though it is over 50 years after the discovery, we only know that the neutrino is a neutral, spin l/2 and weakly interacting particle. We don’t know if the neutrino has a mass or not. If neutrino has mass, then the question of whether neutrino is a Dirac or Majorana type particle arises naturally. This is because the neutrino may be its own anti-particle (Majorana particle). The difference between a Dirac and Majorana neutrino is clearly exhibited in the neutral current interaction process [l], observation of neutrinoless double beta decay, and their electromagnetic properties [2,3]. For example, a spin l/2 Majorana neutrino can only have the anapole moment form factor, if CPT invariance holds. This result was generalized to an arbitrary half integral spin [a], and arbitrary spin [5] !Jajorana fermion. Since a major part of this talk will be published [6] thus I will keep the discussion to be minimal. In section 2, I review the electromagnetic properties of the Dirac and t Refereed version of the invited paper presented at the First Workshop on Particle Physics Phenomenofogy, May 22-34, 1992, Kenting National Park, Pingtung, Taiwan, R.O.C. 157
@ 1993 T H E P H Y S I C A L S O C I E T Y OF THE REPUBLIC OF CHINA
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ELECTROMAGNETIC PROPERTIES OF THE NEUTRINO
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Majorana neutrinos. In section 3, I discuss the conceptual difficulties in defining neutrino charge radius (NCR) in the Standard Model (SM) and review various attempts to fix the gauge dependence problem for the definition. In section 4, I present the calcuation of NCR in the Minimal Supersymmetric Standard Model (MSSM) and investigate the dependence of the NCR on the supersymmetric parameters and conclude in section 5. II. ELECTROMAGNETIC FORM FACTORS OF THE NEUTRINO In this section we study the allowed form of couplings for the electromagnetic current, J:" , matrix element between two neutrino states. In my discussion I will closely follow the notations used in Ref. [3].
.
II- 1. General analysis Consider the neutrino decay process vi f of + y, where Y; and of are two Dirac neutrinos with masses mi and rnf (m; > mr) and y is the photon. The transition amplitude for this process is given by
where IV; > and < “~1 are the initial and final neutrino states respectively, and (I,)f; is the dressed vertex function that characterize the above neutrino decay process. Lorentz invariance implies the dressed vertex function in general can have ten types of coupling: five vector types and five pseudo-vector types of coupling. The five possible vector types of coupling have the following forms: qa, ye, P,, o,pqp, and a,pPfl, where q = pf - pi, P = pf + P; a n d ocrP = ;[Y~,Y~]. The pseudo-vector types of coupling are obtained by the addition of a 75 factor. Using the Dirac equation, (yPpp - m)u = 0, one obtains identities which relate the various types of coupling (the Gordon decomposition relation), hence, reduce the number of independent couplings to six. Therefore, the electromagnetic current matrix element between two Dirac neutrino states is given by < Vf(Pfjl
JFIv;(Pi) >
where V and A are the vector and pseudo-vector type form factors respectively. Conservation of the electromagnetic current, q*Jz” = 0, will further reduce the number of independent types of coupling, thus the most general electromagnetic current matrix element between two neutrino states is given by
L.’
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K. L. NG
159
= u(P/)[(% - %d?/d/~2)(~ t A2Y5) (2.3) t~upqP(1;3tA375)]/,2L(1)i)
V2 and Vs are called the charge moment form factor and magnetic dipole moment form factor respectively, A2 and A3 are called the anopole moment form factor and electric dipole moment form factor respectively. where
11-2. Electromagnetic form factors of a Dirac neutrino
i-_ qaJa It follows from the hermiticity of the electromagnetic tensor operator, J, where 7 = (-l,l, 1, l), that
’ =< vi(pt)lJTIVj(pj) >
(2.4)
thus implies -%(ra)ij
P-5)
(V27V37A27A3)ji = (V~,VS,A~,-AS)~~
(2-Q
YO(Ta)~iYCl =
hence .
For the off-diagonal case, ~j # vi, hermiticity does not imply any restriction on the form factors. For the diagonal case, hermiticity implies that all the form factors are real except the electric moment form factor A3. Under the CPT transformation, Jim ‘P T -Jim. This implies CPT
CPT = -
.
(2.7)
In terms of the Dirac spinor, the left-handed side is given by CPT
CPT = ~CPT(-Pi)(rcr)ji”C PT(-pj)
cw
and UCPT( -P) = ‘Y oV~‘( C~~(~))*
(2.9)
where uCpT(p) is the CPT conjugate of the spinor u(p), VT is the time-reversal matrix, t denotes the transpose operation, and r, is the dressed vertex function describes the process Vi -+ tij + 7, where fi denotes the anti-neutrino state. Using Eq. (2.8) and the properties of the gamma matrices under C and VT in Eq. (2.7), we obtain
CVT(r,)jiV~‘C -’ = (I?a)ij which implies
_-.-_
IL
-_,
.
.
.
(2.10)
ELECTROhl.4GNETIC
160
PROPERTIES OF THE NEUTRINO
(%, T/3: &, _&),, = (-l/2. -I,>, -AZ, -_4&
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(2.11)
as a result of CPT invariance. Under the CP transformation, Jz” % q,.Jz”. CP < ~_f(~_f)lJFl~;(~;) >CP = 70
(2.12)
and the left-handed side is given by CP
CP = ‘1 Lcp(--p:)(l;l,)jiZ1cp(--p;)
(2.13)
where p: = -GP, = (Qo, -p), r’ d enotes the dressed vertex function with q replaced by q’ and (2.14)
WPh') = YoC~%q.
Inserting Eq. (2.13) and Eq. (2.14) into Eq. (2.12), one obtains YOc(T/,);ic-‘YO
=
(2.15)
-77a(raJji
If CP invariance holds. we obtain
(Vz, I&/,, A2,_A3)j; = (-1’2, -1’3, -AZ, As)ji
(2.16)
It follows from Eq. (2.11) and Eq. (2.16) that .4 szi = 0. That means in a CPT invariant theory, a Dirac neutrino cannot have the electric dipole moment form factor A3 if C P invariance holds. 11-3. Electromagnetic form factors of a Majorana neutrino Under the CPT transformation, a Majorana neutrino, v”, transforms as [7] (2.U)
W7+M(p,s) > = &&M(p, -s) >
where 71;~~ is a phase factor that depends on the spin of the particle, and q&T = -_rlc&. Assuming CPT inva.riance for the electroma.gnetic CPT
CPT = -
(2.18)
For a Majorana neutrino. the left-hand side of Eq. (2.18) can be written as CPT
CPT = ~PT(Pj)(ra)jiUPTh)
(2.19)
where u~T(I)) = -JQV~~U*(~). This implies I,F’(r;)j;&-
= -(r&j
Using lhc properties of the gamma matrices under \‘l. in Eq. (2.20), we obtain
(2.20)
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K. L. NG
(V2, V3, AZ, -43)fi =
(-V., -V,, A27 -hIif
161 (2.21)
For the same neutrino species, i = f, CPT invariance implies A2 # 0, that is a Majorana neutrino has the anapole moment form factor only. Under CP transformation, a Majorana neutrino transforms as CPIYyP,s) > = 17;plv"(-P,4 >
(2.22)
where r$& is the CP parity of the Majorana neutrino and q& = fi. Assuming CP invariance, we have
The left-hand side of Eq. (2.23) can be written as C P < V~(P_f)/J~mI~Y(Pi) >CP = UP(P;)(r~)fi~P(P:)
(2.24)
where up(p’) = you(p). Using Eq. (2.23) and Eq. (2.24), one obtains .
r?+70(rL)fi70 = 77cr(Fa)fi
(2.25)
where qcp = iq. Eq. (2.25) implies
The amplitude of the Majorana neutrino decay process, VM --f vy + 7, depends on t h e relative CP parity of the initial and final neutrino states. For instance, if $qf = 1, a Majorana neutrino has pseudo-vector types of coupling (electric dipole type transition), A 2 and As, while for $qf = -1, a Majorana neutrino has vector types of coupling (magnetic dipole type transition) V2 and Va only. III. NEUTRINO CHARGE RADIUS OF THE NEUTRINO In this section, I discuss the subtlety of defining the NCR in SM and review various attempts to fix the problem. The matrix element of the electromagnetic current J,“, between two neutrino states is given by (2.3). It is tempting to define the mean square charge radius of the neutrino as
Even though neutrino remains carrying zero net charge under arbitrary quantum fluctuations, it can have a nonzero charge radius due to the fluctuation of the charge distribution in the surroundings. This definition is motivated via the elastic electron scattering
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ELECTROMAGNETIC PROPERTIES OF THE KEUTRINO
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off a static charge distribution. Bardeen, Gastmans. and Lautrup [S] calculated < r2 > in SM and found that it is ultraviolet divergent and concluded it can not be a physical quantity. Several authors have tried to remedy this problem by increasing the number of Feynman diagrams involved in the definition of V2(q2). Lucia, Rosado, and Zepeda [9] studied neutrino-lepton neutral current scattering, where the amplitude is given by M =
terms
ie2
(3.2)
w h e r e Lz = 2Lf-ypu;, Lz = tij[--$r,(l- ys) + s2yP] u,, and LE = VfyP(l - 7s)~;. F. and kz are finite and gauge-invariant functions. The authors proposed a generalization of charge radius to the “electroweak radius”. defined as < .
r2
>EW = -6
&12
(3.3)
92 =o
which is then a finite and gauge-invariant quantity. Subsequently, Degrassi, Marciano, and Sirlin (DMS) [lo] reconsidered the radiative corrections to the neutral current v-lepton and u-hadron scatterings. DMS concluded that the minimal set of diagrams that will give a gauge-invariant and finite result of the NCR are given in Figs. 1 and 2. The 22 box diagram is gauge-invariant by itself and does not contribute to the NCR if masses of the external fermions are ignored. The sum of the contributions of the minimal set of diagrams mentioned above can be written as SM em
---.-!?_ < flJ,“,li
=
2M$
> LLA(“)(q2)
(3.4)
where Aty)(q2) is the induced electromagnetic form factor of interest. This led DMS to define the effective electromagnetic form factor of the neutrino, a finite, gauge invariant function of the momentum transfer that can be associated with low energy neutrino scattering processes. The corresponding effective charge radius of the neutrino is defined as
D,j,fS = -- A(“)(O).
(3.5)
M;,
Here we list the numerical values for the NCR by DMS for future reference purposes. For m2 = 120GeV and mH = lOOGeV, one obtains
DhfS =
6.6 x 10-34cm2 38 x 10-34cm2 55 x 10-34cm2
for u,, for vP,, for v,.
Recent experimenta.1 bounds from u,e and v,e elas,tic seattering on NCR is [ll]
(3.6)
-..
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FIG. 1. Electromagnetic proper vertex, yZ mixing and related counterterms in v-lepton scattering. f
”
Ii
IL
W
Z
W IL
Y
”
(b)
f
Y
Ii
Ii Z
Z IL
1r ”
1
Cd)
FIG. 2. Vertex correction to Z mediated amplitude and box diagrams in v-lepton scattering
.
.
164
ELECTROMAGNETICPROPERTIESOFTHENEUTRINO -281x 10-34~m2 2 < r2 >", < 51 x 10-34cm2
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31
(3.7)
and most recently, the LAMPF obtained the first bound on < r2 >“, -274 x 10-34cm2 < < r2 >“, < 488 x 10-34cm2
(3.8)
with 90% CL. IV. NEUTRINO CHARGE RADIUS IN MINIMAL SUPERSYMMETRIC STANDARD MODEL
.
In this section, I study the effect of supersymmetry on the neutrino charge radius. As in the case of SM, the supersymmetric corrections to the NCR arise from vacuum polarizations (two-point functions), vertex corrections (three-point functions), box diagrams (four-point functions), and counterterm diagrams. The gauge-independence and finitness of the SUSY corrections is shown in [6]. Of all the contributions to the form factor A”(q2), we expect the oblique two-point functions are the dominant one. A general expression for the electromagnetic form factor AcV)(q2) in terms of the two, three, and four point functions has been obtained by DMS [lo]. It is given by
A(“)(q2)
=
icA+jq2)
+ (il)ZRe[Az;y) _ Aw~2Mi?d] M’
t
;F;r(q2) + 2B(q2)
(4.1)
where (c, s) = ( COST,, sin8,) and Bw is the Weinberg angle. The general form of the selfenergy of the gauge bosons IIF, (V = yy, 72, IYW, 22) is given by IIF = Av(q2)g’“” t Bv(q2)qpq”
(4.2)
By using the Dirac equation, it is straightforward to show that only the transverse part Av(q2) is relevant in our problem. Fy,z are the form fa,ctors of the yvv and Zff vertices, and B arises from the box diagrams. IV-l. Numerical Results The interested reader can find a detailed study of the dependence of the NCR for each of the three different sectors: Riggs sector, scalar matter sector, and the gaugino sector within the MSSM in Ref. [6]. In this section I will summarize the various contributions from each of the three sectors and give a brief discussion on the MSSM results. Our numerical results show that vacuum polarizations give the dominant contributions to the NCR, while the TVV vertex, Zee vertes and the box diagrams give sn~all contributions only.
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