Neutrino-Pair Exchange Long-Range Force Between Aggregate...

Neutrino-Pair Exchange Long-RangeForce Between Aggregate Matter

Alejandro Segarra [email protected]

Taller de Altas Energıas,September 2015

Motivation

e´ e`

νν

e´ e´

A. Segarra Neutrino-Pair Exchange Long-Range Force 2

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(


Motivation

A A

νν

B B

V prq “ ´

ż

d3q

p2πq3ei~q ~rMpq2q


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


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


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

¨

˚

˚

˝

˛

‹

‹

‚

˚¨

˚

˚

˝

˛

‹

‹

‚


S Matrix Unitarity

A A

νν

B B


S Matrix Unitarity

A A

ν ν

B B

ν ν


Summing up...

V prq

FtMptqu

ImtMu

MpAν Ñ Aνq

A A

ν νA A

ν ν

B B


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‰


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


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


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


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


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

“


geV Z ` gpV Z ` g

nV N

J0 “ 2Z ´N ” QeW

MpAνµ,τ Ñ Aνµ,τ q “GF

2?

2Jµ

“


geV Z ` gpV Z ` g

nV N

J0 “ ´N ” Qµ,τW


Weak Flavor Charges of Stable Atoms

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

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● 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


Finally...

V prq

FtMptqu

ImtMu

MpAν Ñ Aνq


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


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

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▲

● Weak

▲ Gravit.

0 20 40 60 80

0

10000

20000

30000

40000

50000

ZA. Segarra Neutrino-Pair Exchange Long-Range Force 21

Perspectives

Interesting ranges:rmin „ 1 nm. a0

rmax „ m´1ν „ p0.1 eVq´1 „ 1 µm. e´2mr

Competitors:Electromagnetic residual interactionsGravitation

Measurement of the absolute mass of the neutrino!

Dirac or Majorana?


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