Neutrino Electromagnetic Form Factor and Oscillation Effects on Neutrino Interaction With Dense Matter

0.15 MB·English

by  C. K. Williams

FIS-UI-TH-04-01
Neutrino Electromagnetic Form Factor and Oscillation Effects on Neutrino
Interaction With Dense Matter
C. K. Williams, P. T. P. Hutauruk, A. Sulaksono, T. Mart
Departemen Fisika, FMIPA, Universitas Indonesia, Depok 16424, Indonesia
(Dated: February 2, 2008)
The mean free path of neutrino - free electron gas interaction has been calculated by takinginto
account the neutrino electromagnetic form factors and the possibility of neutrino oscillation. It is
shownthattheformfactoreffectbecomessignificantforaneutrinomagneticmomentµν ≥10−10µB
and for a neutrino radius R ≥ 10−6 MeV−1. The mean free path is found to be sensitive to the
νe−νµ and νe−νeR transition probabilities.
PACSnumbers: 13.15.+g,13.40.Gp,25.30.Pt,97.60.Jd,14.60.Pq
5
0
Neutrino interaction with dense matter plays an im- nosandneutrinooscillationsisinevitable. Asafirststep
0
portant role in astrophysics,e.g., in the formation of su- beforethat,inthisreportwecalculatethemeanfreepath
2
pernova and the cooling of young neutron stars [1, 2, 3, of neutrino-free electrons gas where those effects are in-
n
4, 5, 6, 7, 8]. Earlier calculation on neutrino interac- cluded. Here we assume that neutrinos are massless and
a
tions with electrons gas,dense and hot matter, based on the RPA correlationscanbe neglected. Furthermore, we
J
thestandardmodelhasbeenperformedbyHorowitzand use zero temperature approximation in this calculation.
7
Wehrberger [2, 3]. Some relativistic calculations of neu- In the standard model, where the momentum transfer
1
trino mean free path in hot and dense matter have been ismuchlessthanthe W mass,directZ0 andW± contri-

1 alsodoneinRefs. [4,5,6,7]. Recently,duetoademand butions to the matrix element can be written as an
M
v onamorerealisticneutrinomeanfreepathforsupernova effective four-point coupling [3, 22]
8 simulations, a mean free path calculation by taking into
4 account the weak magnetism of nucleons has been also = GF[U¯(k′)γµ(1+γ5)U(k)][U¯(p′)J U(p)], (1)
1 performed [8]. MW √2 µ
1
0 However, certain phenomena such as solar neutrinos, where GF is the coupling constant of weak interaction,
5 atmospheric neutrinos problems, and some astrophysics U(k)andU(p)areneutrinoandelectronspinors,respec-
0
and cosmology arguments need explanations beyond the tively, and the current Jµ is defined by
/
h standardmodelassumptionofneutrino’spropertiessuch
p as neutrino oscillation [9, 10], the helicity flipping of Jµ =γµ(CV +CAγ5). (2)
-
p neutrinos [11, 12, 13, 14] and neutrino electromagnetic ThevectorandaxialvectorcouplingsC andC canbe
V A
e form factors. We note that the upper bound of the writtenintermsofWeinbergangleθ (wheresin2θ
W W
h neutrino magnetic moment extracted from the Super- 0.223 [3, 4]) as C = 2sin2θ 1/2 and C = 1/≈2
v: Kamiokande solar data [15, 16] falls in the range of (the upper signisVforν , the loWwe±rsignis forνA and±ν ).
i (1.1 1.5) 10−10µB, where µB = e/2me stands for The electromagneticeproperties of Dirac neuµtrinos aτre
X − ×
the Bohr magneton. Other experimental limits [17, 18]
described in terms of four form factors, i.e., f ,g ,f
r give µ < 1.0 10−10µ , whereas signals from Super- 1ν 1ν 2ν
a nova1ν987A(SN×1987A)rBequirethatµ 1.0 10−12µ . and g2ν, which stand for the Dirac, anapole, magnetic,
ν B and electric form factors, respectively. The matrix ele-
≤ ×
Theseboundshavebeenderivedbyconsideringthehelic-
mentfortheneutrino-electroninteractionwhichcontains
ity flipping neutrino scattering in a supernova core [19].
electromagnetic form factors reads [22]
Inthe case ofrandommagnetic fields inside the sun, one
can obtain a direct constraint on the neutrino magnetic 4πα
moment of µν ≤ 1.0 ×10−12µB, similar to the bounds MEM = q2 [U¯(p′)γµU(p)](U¯(k′)"fmνγµ
obtained from the star cooling [20]. In addition, data
from muon neutrino- and anti neutrino-electron scatter- Pµ
ings [21, 22] and a close examination to the data over + g1νγµγ5 (f2ν +ig2νγ5) U(k) . (3)
− 2me# )
the years from Kamiokande II and Homestake according
to Moura˜o et al. [24], similarly give a neutrino average where f =f +(m /m )f , Pµ =kµ+kµ′, m and
squared radius R2 25 10−12 MeV−2 with R2= R2 mν 1ν ν e 2ν ν
+ R2 . The defin∼itions×of R2 and R2 will bhe eVxi- me areneutrinoandelectronmasses,respectively. Inthe
h Ai h Vi h Ai static limit, the reduced Dirac form factor f1ν and the
plained later.
neutrinoanapoleformfactorg arerelatedtothevector
1ν
Therefore, in connection with the demand on realistic andaxialvectorchargeradiihRV2iandhRA2ithrough[22]
neutrino meanfree path in dense and hot matter, an ex-
tension of the previous study [4, 5, 6, 7, 8] which takes 1 1
into account the electromagnetic form factors of neutri- f1ν(q2)= 6hRV2iq2 and g1ν(q2)= 6hRA2iq2. (4)
2
Inthelimitofq2 0,f andg definerespectivelythe volume V for scattering of neutrinos with the initial en-
2ν 2ν
neutrino magneti→c moment µm =f (0)µ and the (CP ergy E and final energy E′ on the electrons gas. It
ν 2ν B ν ν
violating)electricdipolemomentµe =g (0)µ [22,23]. consists of the contributions from weak (W) interaction,
ν 2ν B
Here we use µ2=µm 2+ µe 2. electromagnetic (EM) interaction, as well as their inter-
ν ν ν
Next, we can obtain the differential cross section per ference (INT) term, i.e.,
2
1 d3σ 1 E′ G 4πα 2 8G πα
= ν F LµνΠIm(W)+ LµνΠIm(EM)+ F LµνΠIm(INT) . (5)
V d2Ω′dEν′ !νe −16π2Eν " √2! ν µν (cid:18) q2 (cid:19) ν µν q2√2 ν µν #
For each contribution, the neutrino tensors are given by
Lµν(W) =8[2kµkν (kµqν +kνqµ)+gµν(k q) iǫαµβνk k′], (6)
ν − · − α β
Lµν(EM) = 4(f2 +g2 )[2kµkν (kµqν +kνqµ)+gµν(k q)] 8if g ǫαµβν(k k′)
ν mν 1ν − · − mν 1ν α β
f2 +g2
2ν 2ν(k q)[4kµkν 2(kµqν +qµkν)+qµqν], (7)
− m2 · −
e
Lµν(INT) =4(f +g )[2kµkν (kµqν +kνqµ)+gµν(k q) iǫαµβνk k′], (8)
ν mν 1ν − · − α β
whereas the polarizations read If we take into account the possibility of the ν ν
e µ
−
transition, the cross section can be written in the form
ΠIµmν(W) = CV2ΠIµmνV +2CVCAΠIµmν(V−A)+CA2ΠIµmνA, of [25, 26]
ΠIm(EM) = ΠImV,
µν µν
ΠIm(INT) = C ΠImV +C ΠIm(V−A). (9) d3σ d3σ d3σ
µν V µν A µν d2Ω′dE′ =Pee d2Ω′dE′! +(1−Pee) d2Ω′dE′! .
Due to the current conservation and translational in- νe νµ
variance, the vector polarization ΠImV consists of two (12)
µν
independent components which we choose to be in the Here (d3σ/d2Ω′dE′) is the crosssectionofthe ν e
frame of qµ (q , ~q ,0,0), i.e., νe e−
≡ 0 | | scattering. If CV and CA are replaced with CV 1
−
and C 1, respectively, then the cross section becomes
ΠITmV = ΠV22 =ΠV33 and (d3σ/dA2Ω−′dE′) ,i.e.,thecrosssectionoftheν escat-
ΠILmV = −(qµ2/|~q|2)ΠV00. tering. Pee isνtµhe νe’s flavor survival probabµil−ity as a
function of the neutrino energy.
The axial-vector and the mixed pieces are found to be
Due to the assumption of massless neutrino, the ν
e
helicity flip from left- to right-handed is only possible
ΠIm(V−A)(q)=iǫ q Π , (10)
µν αµ0ν α VA throughit’sdipolemoment. Thus,thecrosssectionafter
taking into account this possibility (ν νR transition)
and e − e
reads [17]
ΠImA(q)=ΠImV(q)+g Π . (11)
µν µν µν A d3σ d3σ d3σ
=(1 P ) +P .
The explicit forms of ΠV , ΠV , Π and Π are given d2Ω′dE′ − LL d2Ω′dE′! LL d2Ω′dE′!
22 00 VA A LR νe
in Ref. [3]. Thus the analytical form of Eq. (5) can be (13)
obtained from the contraction of every polarization and where (d3σ/d2Ω′dE′) is the ν e scattering via neu-
neutrino tensors couple (LµνΠ ). trino dipole moment LaRnd P iset−he probability of ν to
µν LL e
3
10 10 35 35
µν = 0 µB R = 0 MeV-1 Pee = 0.00 PLL = 0.00
8 µµµννν == = 11 1000--11-940 µµµBBB 8 R = RR5 ==× 111000---655 MMMeeeVVV---111 2350 PPPeeeeee === 000...257505 2350 PPPLLLLLL === 000...257505
6λ (10 m) 46 RP L=L 5= ×1 10-6 MeV-1 46 µPνLL = = 1 10-14 µB 6(10 m) 1250 PRµν e= e= 5= 1 ×01 -.101000 -µ6 BMeV-1 1250 PLL = 1.00
λ
2 2 10 10 R = 5 × 10-6 MeV-1
µν = 10-10 µB
5 5
0 0
50 60 70 80 90 100 50 60 70 80 90 100
0 0
kF (MeV) kF (MeV) 50 60 70 80 90 100 50 60 70 80 90 100
k (MeV) k (MeV)
F F
FIG.1: Totalmeanfreepathcomparedtothemeanfreepath
of weak interaction with various neutrinomagnetic moments FIG.3: Totalmeanfreepathofνe thatallowsforνe−νµand
µν and radii R as a function of Fermi momentum kF. In νe−νeR transitions with various PLL and Pee as a function
the left panel the neutrino charge radius is fixed, while the
of Fermi momentum kF. In the left panel we vary the νe’s
neutrinomagneticmomentisvaried. Intherightpanel,wefix
flavorsurvivalprobability,whileintherightpanelthehelicity
theneutrinomagneticmoment,butvarytheneutrinoradius.
flippingprobability of neutrino is varied.
3.0 3.0
Total, k = 100
) F Thetotalmeanfreepathisthecoherentsumoftheweak,
m Weak, k = 100
c 2.5 F 2.5 electromagnetic and the interference contributions.
-90/ MeV- 2.0 WToetaaEklν,, kk= FF 5 == M 55e00V 2.0 µν = 10-9µΒ Rpa2Tn≈ehle2or5ef×Fa1irg0e.−1a1l2wsoMe ueeVvsei−dR2en[=2ce1s,52t2h,1a20t4−]R6.2TMh≈eeVre1−f01o−rae3n2,dincvmtah−rey2leµofνrt
E’ (1ν 1.5 qR1 = = 5 5 × M 1e0V-6 MeV-1 1.5 1b0e−tw12eµen 0asatnhde1st0r−o9nµgBes.tIbnouth×nedroinghtthepnaneuelt,riwneomusaegµnνet=ic
d B
2Ω’ 1.0 µν = 10-10µΒ 1.0 momentwhileRisvariedbetween0and5×10−5MeV−1.
d It is evident from the left panel of Fig.1 that for
V
3σd / 0.5 0.5 fiµxνe=d 1R0,−t1h0eµBm.eaAnsfwreeecapnatsheeinfcrroemasFesigr.a2pitdhliys ionnclryemafetnetr
isduetothesignificantdifferencebetweentotalandweak
0.0 0.0 cross sections starting from µ = 10−10µ . The sum-
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 ν B
mation of the longitudinal and transversal terms of the
q (MeV) q (MeV)
0 0
electromagnetic contribution is responsible for this. The
right panel shows that for fixed µ , the total mean free
FIG. 2: Total cross section compared to the cross section of ν
weakinteractionasafunctionofenergytransferq0wheremo- pathandthemeanfreepathofweakinteractionshowsig-
mentumtransferq1isfixed. Here,twodifferentneutrinomag- nificant variance for R 10−6 MeV−1. This is also due
≥
netic moments µν and Fermi momenta with a same neutrino to the fact that the summation of the longitudinal and
chargeradiusareused. Intheleftpanelweuseµν =10−10µB, transversaltermsoftheelectromagneticpartofthecross
while in theright panel µν =10−9µB. section increases rapidly starting at R=10−6 MeV−1.
Figure3 shows the effects of neutrino oscillations on
theneutrinomeanfreepath. Inthiscasewedonotcalcu-
be still left handed.
late the transition probabilities. Instead, we only study
Finally we can compute the mean free path from Eqs.
the variation of neutrino mean free path with respect
(5), (12), and (13), by using
to the transition probabilities of a left handed massless
1 2Eν−q0 2Eν ~q 1 d3σ neutrino electron, νe, oscillatesto a left handedmassless
= d~q dq | | 2π .
λ(Eν) Zq0 | |Z0 0Eν′Eν V d2Ω′dEν′ enleeucttrrionno, νmRu.on, νµ, or flips to a right handed neutrino
(14) e
Bycomparingthe possibility ofν ν transition(left
e µ
In this calculation we use a neutrino energy of 5 MeV. panel of Fig.3) and ν νR transitio−n (right panel), we
e− e
Figure1 shows the total mean free path compared to can clearly see that these effects lengthen the neutrino
the meanfreepathofweakinteractionwithvariousneu- meanfree path,wherethe ratedepends ontheir survival
trino effective moments µ , and neutrino chargeradii R. probabilities. ForsmallerP (largeflippingpossibility),
ν LL
4
thepathincrementbecomesmoresignificant. Thiseffect form factors and neutrino oscillations. It is found that
can be traced back to the value of (d3σ/d2Ω′dE′) in the electromagnetic form factor has a significant role if
Eq.(13) which is smaller than that of (d3σ/d2Ω′dELR′) . µ 10−10µ and R 10−6 MeV−1. We note that
νe ν ≥ B ≥
Ontheotherhand,forsmallP thepossibilityofν ν these values are larger than their largest upper bounds.
ee e µ
−
oscillation does not change the neutrino mean free path It would be interesting to see whether or not such phe-
dramatically. This fact arises because the difference be- nomenon would also appear if contributions from the
tween (d3σ/d2Ω′dE′) and (d3σ/d2Ω′dE′) in Eq.(12) neutrino-nucleon scatterings were taken into account.
νµ νe
isnotaslargeasinthecaseofEq.(13). Thereforediffer- Future calculation should address this question. The
ent from the mean free path with flavor changing possi- mean free path is also found to be sensitive to the neu-
bility,themeanfreepathwithhelicityflippingpossibility trino oscillations and depends on the transition proba-
dependsstronglyonthevalueofµ . Forexamplewehave bilities of ν ν and ν νR. This result clearly in-
ν e − µ e − e
also found that with decreasing P the mean free path dicates that realistic mean free path calculations in the
LL
grows more rapidly when we use µ = 10−12µ rather future should be performed with appropriate values of
ν B
than µ =10−10µ . the ν ν and ν νR transition probabilities.
ν B e− µ e− e
In conclusion, we have studied the sensitivity of the TM and AS acknowledge the support from the QUE
neutrino mean free path to the neutrino electromagnetic project.
[1] Sanjay Reddy, M. Prakash, J. M. Lattimer, J. A. Pons, [14] W.Grimus,P.Stockinger,Phys.Rev.D57,1762(1998).
Phys.Rev.C 59, 2888 (1999). [15] D. W. Liu et al.,Phys.Rev. Lett 93, 021802 (2004).
[2] C.J.Horowitz andK.Wehrberger,Phys.Rev.Lett. 66, [16] J. F. Beacom and P. Vogel, Phys. Rev. Lett 83, 5222
272 (1991); (1999).
[3] C. J. Horowitz and K. Wehrberger, Nucl. Phys. A531, [17] W. Grimus et al.,Nucl.Phys. B 648, 376 (2003).
665 (1991); Phys.Lett. B 226, 236 (1991). [18] Z. Daraktchieva et al.,Phys. Lett B 564, 190 (2003).
[4] R. Niembro, P. Bernados, M. Lo´pez-Quele, S. Marcos, [19] N. Nunokawa, R. Toma´s and J. W. F. Valle, Astropart.
Phys.Rev.C 64, 055802 (2001). Phys. 11, 317 (1999); and references therein.
[5] L.MornasandA.Parez,Eur.Phys.J.A13,383(2002). [20] O. G. Miranda, T. I. Rashba, A. I. Rez and J. W. F.
[6] L. B. Leinson, Nucl. Phys. A 707, 543 (2002). Valle, Phys.Rev.Lett 93, 051304 (2004).
[7] G.FabbriandF.Matera, Phys.Rev.C54,2031 (1996). [21] R. C. Allen et al.,Phys. Rev.D 43, 1 (1991).
[8] C. J. Horowitz and M. A. Pe´rez-Garc´ia, Phys. Rev. C [22] B. K. Kerimov, M. Ya Safin and H. Nazih, Izv. Ross.
68, 025803 (2003). Akad.Nauk.SSSR.Fiz. 52, 136 (1998).
[9] T. K. Kuo and J. Pantaleone, Rev. Mod. Phys. 61, 937 [23] Enrico Nardi, AIPConf. Proc. 670, 118 (2003).
(1992); and references therein. [24] A. M. Moura˜o, J. Pulido, and J. P. Ralston, Phys. Lett.
[10] J. Pulido, Phys. Rept. 211, 167 (1992); and references B 285, 364 (1992).
therein. [25] H. P. Simanjuntak and A. Sulaksono, Mod. Phys. Lett.
[11] A.Ayala, J. C.D’Olivoand M. Tores, Phys.Rev.D 59, 9A, 2179 (1994); A. Sulaksono and H. P. Simanjuntak,
111901(1999). Solar Phys.151, 205 (1994).
[12] K.Enqvist,P.KeraenenandJ.Maalampi,Phys.Lett.B [26] J. Morgan, Phys. Lett. B102, 247 (1981); M. Fukugita
438, 295 (1998). and S.Yazaki, Phys. Rev.D 36, 3817 (1987).
[13] K.J.F.Gaemers,R.Gandhi,J.M.Lattimer,Phys.Rev.
D 40, 309 (1989).