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Neutrino-Pair Exchange Long-RangeForce Between Aggregate Matter
Alejandro Segarra [email protected]
Taller de Altas Energıas,September 2015
Motivation
ż
d3q
p2πq3ei~q ~r
1
q2“´1
4πr
A A1
QA
γpqq
B B1
QB
“ ie2QAQB ru1Aγ
µuAs1q2ru1BγµuBs
ùñ Mpq2q “ e2QAQB1q2
V prq “e2
4π
QAQBr
V prq “ ´F
Mpq2q(
A. Segarra Neutrino-Pair Exchange Long-Range Force 3
Motivation
A A
νν
B B
V prq “ ´
ż
d3q
p2πq3ei~q ~rMpq2q
A. Segarra Neutrino-Pair Exchange Long-Range Force 4
Low-Energy Dispersion Relation
Rettu
Imttu
t0s
C8
Cauchy Formula: fpzq “1
2πi
ż
Cdz1
fpz1q
z1 ´ z
Physical Region:
´s ď t ď t0
´M2A À t ď 4m2
ν
Mpt; sq “1
π
ż 8
t0
dt1ImtMpt1qu
t1 ´ t`
1
π
ż ´s
´8
dt1ImtMpt1qu
t1 ´ t`
ż
C8
A. Segarra Neutrino-Pair Exchange Long-Range Force 6
Low-Energy Dispersion Relation
Rettu
Imttu
t0s
C8
Cauchy Formula: fpzq “1
2πi
ż
Cdz1
fpz1q
z1 ´ z
Physical Region:
´s ď t ď t0
´M2A À t ď 4m2
ν
Mpt; sq “1
π
ż 8
t0
dt1ImtMpt1qu
t1 ´ t` Short Range
A. Segarra Neutrino-Pair Exchange Long-Range Force 7
S Matrix Unitarity
S:S “ 1S” 1`iT
ÝÝÝÝÝÝÝÝÑ ´ipT ´ T :q “ T :T
2 Im txf |T |iyu “ xf |T : 1T |iy
“ÿ
n
xf |T :|ny xn|T |iy
MpiÑ fq „ xf |T |iy
Im
$
’
’
’
’
&
’
’
’
’
%
i n f,
/
/
/
/
.
/
/
/
/
-
„ÿ
n
¨
˚
˚
˝
˛
‹
‹
‚
˚¨
˚
˚
˝
˛
‹
‹
‚
A. Segarra Neutrino-Pair Exchange Long-Range Force 8
Summing up...
V prq
FtMptqu
ImtMu
MpAν Ñ Aνq
A A
ν νA A
ν ν
B B
A. Segarra Neutrino-Pair Exchange Long-Range Force 11
Low-Energy Effective Neutral Currents
ψj ψj
Z
ν ν
ˇ
ˇq2ˇ
ˇ !M2Z
ψjψj
ν νgV “ 2T3 ´ 4Q sin2 θW
gA “ 2T3
LNC “1
2M2ZZµZ
µ ´e
4 sin θW cos θWZµ ψj γ
µ`
gVj ´ gAjγ5
˘
ψj
0 “BLNC
BZµ“M2
ZZµ ´
e
4 sin θW cos θWψj γ
µ´
gVj´ gAj
γ5
¯
ψj
L effNC “
GF
2?
2rν γµ p1´ γ5q νs
“
ψj γµ`
gVj ´ gAjγ5
˘
ψj‰
A. Segarra Neutrino-Pair Exchange Long-Range Force 12
Low-Energy Effective Charged Currents
νe
e
W
νe
e
q2 !M2W
νe
e
νe
e
Fierz
ee
νe νe
LCC “M2WW
:µW
µ ´e
2?
2 sin θWWµ e γ
µp1´ γ5q νe ` h.c.
L effCC “ ´
GF?
2rνe γ
µ p1´ γ5q es re γµ p1´ γ5q νes
L effCC “
GF?
2rνe γ
µ p1´ γ5q νes re γµ p1´ γ5q es
A. Segarra Neutrino-Pair Exchange Long-Range Force 13
Low-Energy Effective Weak Theory
ψj
ψj
GF2?
2γµ
´
gjV ´ gjAγ5
¯
νi
νi
γµ p1´ γ5q
ψj “ e, p, n
νi “ νe, νµ, ντ
NC:
NC:
NC:
NC+ CC:
p νe,µ,τ ÝÑ gpV “ 1´ 4 sin2 θW
n νe,µ,τ ÝÑ gnV “ ´1
e νµ,τ ÝÑ geV “ ´1` 4 sin2 θW “ ´gpV
e νe ÝÑ geV “ 2` geV “ 2´ gpV
A. Segarra Neutrino-Pair Exchange Long-Range Force 14
Scattering Amplitude Aνe Ñ Aνe
A A1
νe ν 1e
q A “
$
’
&
’
%
Z electrons
Z protons
N neutrons
Tνe “
ż
d4x xA1 ν 1e|L pxq |Aνey “GF
2?
2
ż
d4x jµpxq Jµpxq
jµpxq ““
u1 γµ p1´ γ5qu‰
e´ipk´k1qx
A. Segarra Neutrino-Pair Exchange Long-Range Force 15
Molecular matrix element
Jµpxq “ geV xA1| epxqγµepxq |Ay ´ g
eA xA
1| epxqγµγ5epxq |Ay
Scalar, γ0: number operator, e:e.COHERENTPseudo-scalar, γ0γ5: matrix element „ ~σ~q{M .INCOHERENT, RELATIVISTICPolar vector, ~γ: matrix element „ ~q{M .RELATIVISTICAxial vector, ~γγ5: matrix element „ ~σ.INCOHERENT
J0pxq “ Z geV eiqx
A. Segarra Neutrino-Pair Exchange Long-Range Force 16
Scattering Amplitude Aν Ñ Aν
A A1
νe,µ,τ ν 1e,µ,τ
q A “
$
’
&
’
%
Z electrons
Z protons
N neutrons
MpAνe Ñ Aνeq “GF
2?
2Jµ
“
u1 γµ p1´ γ5qu‰
geV Z ` gpV Z ` g
nV N
J0 “ 2Z ´N ” QeW
MpAνµ,τ Ñ Aνµ,τ q “GF
2?
2Jµ
“
u1 γµ p1´ γ5qu‰
geV Z ` gpV Z ` g
nV N
J0 “ ´N ” Qµ,τW
A. Segarra Neutrino-Pair Exchange Long-Range Force 17
Weak Flavor Charges of Stable Atoms
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲
▲ ▲ ▲ ▲▲ ▲ ▲
▲ ▲ ▲▲ ▲ ▲
▲ ▲▲ ▲ ▲
▲ ▲▲ ▲
▲ ▲▲ ▲
▲ ▲▲ ▲
▲▲ ▲
▲ ▲▲▲ ▲
▲▲ ▲
▲▲ ▲
▲▲▲ ▲
▲▲▲ ▲
▲▲▲▲▲ ▲
▲▲▲▲▲▲▲▲▲ ▲
▲▲▲▲▲▲▲
● QW (e)
▲ -QW (μ, τ)
0 20 40 60 80
0
20
40
60
80
100
120
140
Z
QeW “ 2Z ´N QµW “ QτW “ ´N
A. Segarra Neutrino-Pair Exchange Long-Range Force 18
Conclusion: Long-Range Weak Interaction
Interaction Potential:
V prq “G2F
8π3
¨
˝
ÿ
f
QfW,AQfW,B
˛
‚
1
r5
Force:
F “5G2
F
8π3
¨
˝
ÿ
f
QfW,AQfW,B
˛
‚
r
r6
A. Segarra Neutrino-Pair Exchange Long-Range Force 20
Conclusion: Long-Range Weak Interaction
Interaction Potential:
V prq “G2F
8π3
¨
˝
ÿ
f
QfW,AQfW,B
˛
‚
1
r5
Gravit « pZ `Nq2
Weak “ p2Z ´Nq2 ` 2N2
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●
●●●●
●●●●
●●●●●●
●●●●●●●●●●●●●●●●●
▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲
▲▲ ▲
▲ ▲▲▲ ▲
▲▲ ▲
▲▲ ▲
▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲
▲
● Weak
▲ Gravit.
0 20 40 60 80
0
10000
20000
30000
40000
50000
ZA. Segarra Neutrino-Pair Exchange Long-Range Force 21