Tau Polarization in (anti-)neutrino-nucleon Interactions.

Tau Polarization in (anti-)neutrino-nucleon Interactions.

Mohammad Rafi Alam†

Luis Alvarez-Ruso# Toru Sato††

†Aligarh Muslim University, India#IFIC-CSIC, U.V.,Valencia

††RCNP, Osaka University, Japan

(τ-polarisation) M.R.A. March 17, 2021 1 / 24

Outline

1 Introduction

2 FormalismQuasi-elasticInelastic regionDeep Inelastic Scattering

3 Results

4 Summary


Introduction

1 Introduction


3 Results

4 Summary


Introduction

Motivation

τ leptons were observed in experiments:Atmospheric Experiments : SuperK, IceCUBEAccelerator Experiments : DONOT, OPERA

New Experiments : SHiP, DsTau, DUNENeutrino Oscillation : appearance experiment νµ→ ντ

Observed by the ντ induced (CC)interaction(−)ν τ (k,0) + N(p,M)→ τ±(k′,ml) + X(p′)

Challenges :short lifetime of τ (10−13sec): Practically impossible to detect!

Observe the decay channels

Decay distribution has a strong dependence on τ spin polarization.


Introduction

Motivation

Unpolarized

Polarized

0 |π/4| |π/2| |3π/4| |π|ϕ

0

1

2

3

4

5

6

7

8

Ev

ent/

bin

/4.5

×1

019p

.o.t

./1

.65

kto

n/5

yea

re, µ

πwith τ

without polarization

polarization

NPB727 (2005) 163-175


Formalism

1 Introduction


3 Results

4 Summary


Formalism

(−)ν τ (k,0) + N(p,M)→ τ±(k′,ml) + X(p′)

τ± rest frame

Lab frame QE, Inel, DIS


Formalism

(−)ν τ (k,0) + N(p,M)→ τ±(k′,ml) + X(p′)

k× k′

[k× k′]× k′

k′

Polarization Vector

~ξ = PLξL+PtξT +��HHHPt′ ξP


Formalism

Quasi-elastic[τ±N

]Inelastic

[τ±πN, τ±KN, · · ·

]−→ (Dynamical Coupled Channel)

Deep-Inelastic[τ±+Jet]

Hadronic Tensor

Wµν(p,q) =−gµνW1(p·q,Q2) + pµpν

M2 W2(p·q,Q2)− iεµναβpαqβ

2M2 W3(p·q,Q2)

+ qµqν

M2 W4(p·q,Q2) + pµqν + qµpν

2M2 W5(p·q,Q2),+i pµqν − qµpνM2 W6(p·q,Q2)


Formalism

FPt′ =−2ml

MplW6 sinθ

FPt =−ml sinθ[±(

2W1−W2−m2l

M2W4 + 2ElM

W5

)− EνMW3

]FPl =∓

[(2W1−

m2l

M2W4

)(pl−El cosθ) +W2(pl+El cosθ)−

2m2l

McosθW5

]−W3M

(cosθ(EνEl+p2

l )−pl(Eν +El))

F =(

2W1 +m2l

M2 W4

)[El−pl cosθ

]+W2(El+pl cosθ)

±W3M

[(Eν +El)(El−pl cosθ)−m2

l

]−

2m2l

MW5


Formalism Quasi-elastic

Quasi-elastic

ντ (k) +n(p) −→ τ−(k′) +p(p′),ντ (k) +p(p) −→ τ+(k′) +n(p′),

The hadronic current Jµ is expressed as:

Jµ = u(p′)[Vµ−Aµ

]u(p)

〈N ′(p′)|Vµ|N(p)〉 = u(p′)[γµf1(Q2) + iσµν

qν

Mp+Mnf2(Q2) +

��HHHH

2 qµMp+Mn

f3(Q2)]u(p),

〈N ′(p′)|Aµ|N(p)〉 = u(p′)[γµγ5g1(Q2) +

��

��XXXXXXXXXX

iσµνqν

Mp+Mnγ5g2(Q2) +

2 qµMp+Mn

g3(Q2)γ5

]u(p),


Formalism Inelastic region

Dynamical Coupled Channel

Extension of Sato-Lee modelOnly πN state and works in ∆(1232) region.

The building blocks of DCCstable two-particle channels (πN,ηN,KΛ,KΣ)quasi-stable channels (ρN,σN,π∆)

The T -matrix are constructed for the meson-baryon scattering has beenobtained from the coupled-channel Lippmann-Schwinger equation:

〈α,~p ′|T (W )|β,~p〉 = 〈α,~p ′|V (W )|β,~p〉

+∑γ

∫d3k 〈α,~p ′|V (W )|γ,~k 〉G0

γ(~k,W )〈γ,~k |T (W )|β,~p〉

where α,β and γ are the meson-baryon two-body states.Satisfies two- and three- body unitarity.


Formalism Deep Inelastic Scattering

Structure Function

Wµν =∑

q,q′=u,d,...,u,d,...

∫dξ

ξfq(ξ,Q2)Kµν(pq, q)|Vqq′ |2

Kµν = δ[(pq + q)2−m2

q′]×2{−gµνpq · q+ 2pµq pνq ∓ iεµναβpqαpqβ + (pµq qν + qµpνq )

}.

momentum fraction

Parton distributionfuction

Elements of CKMMatrix

parton momentumpq = ξp, p is nucleonmomentum

parton mass



ξ0 = 2x/λ1 +√

1 +4M2x2/(λQ2)

1≤Q2 ≤ 10GeV 2

00.10.20.30.40.50.60.70.80.9

1

0 0.2 0.4 0.6 0.8 1

𝜉 = 𝑥

𝜉(𝑚𝑞′=0)𝜆(𝑚𝑞′=𝑚𝑐)

𝜉(𝑚𝑞′ = 𝑚𝑐)

𝜉(𝑚𝑞′ = 0)

𝜉(𝑥,

𝜆)

𝑥 →



Structure Function

W1 =∑j

fj(ξ)|Vij |2 = F1(ξ) ,

W2 = M2

p · q+ ξM2

∑j

2ξfj(ξ)|Vij |2 = M2

p · q+ ξM2F2(ξ) ,

W3 = M2

p · q+ ξM2

∑j

2fj(ξ)|Vij |2

= M2

p · q+ ξM2F3(ξ) ,

W4 = M2

p · q+ ξM2

∑j

fj(ξ)|Vij |2 = M2

p · q+ ξM2F4(ξ) ,

W5 = M2

p · q+ ξM2

∑j

2fj(ξ)|Vij |2

= M2

p · q+ ξM2F5(ξ) .



Structure Function

F(N)1(l) = |Vud|2f

(N)d (ξl) + |Vus|2f (N)

s (ξl) +(|Vud|2 + |Vus|2)f (N)u (ξl)

F(N)1(h) = |Vcd|2f

(N)d (ξh) + |Vcs|2f (N)

s (ξh)

F(N)2(l) = 2ξl{|Vud|2f

(N)d (ξl) + |Vus|2f (N)

s (ξl) +(|Vud|2 + |Vus|2)f (N)u (ξl)}

F(N)2(h) = 2ξh{|Vcd|2f

(N)d (ξh) + |Vcs|2f (N)

s (ξh)}

F(N)3(l) = 2{|Vud|2f

(N)d (ξl) + |Vus|2f (N)

s (ξl)− (|Vud|2 + |Vus|2)f (N)u (ξl)}

F(N)3(h) = 2{|Vcd|2f

(N)d (ξh) + |Vcs|2f (N)

s (ξh)}

F(N)4(l,h) = 0

F(N)5(l,h) = 2F (N)

1(l,h)

We use CTEQ6.6 for PDFs.



Structure Function

ξl = 2x1+√

1+4M2x2/(Q2)

ξh = ξ0

ξN = 2x/λ1+√

1+4M2x2/(Q2)

00.10.20.30.40.50.60.70.80.9

1

0 0.2 0.4 0.6 0.8 1

𝜉 = 𝑥

𝜉(𝑚𝑞′=0)𝜆(𝑚𝑞′=𝑚𝑐)

𝜉(𝑚𝑞′ = 𝑚𝑐)

𝜉(𝑚𝑞′ = 0)

𝜉(𝑥,

𝜆)

𝑥 →


Results

1 Introduction


3 Results

4 Summary


Results

Differential cross-section at Eν = 10 GeV & Wcut = 2.1GeV

00.20.40.60.8

11.21.41.61.8

2

0 2 4 6 8 10

𝜈𝜏𝑝 → 𝜏−𝑋

𝐸𝜈 =10 GeV𝜃 = 0∘

𝑑𝜎/𝑑

𝑊𝑑𝑄

2 [10

−38

𝑐𝑚2 /

𝐺𝑒𝑉

3 ]

𝑝𝜏[𝐺𝑒𝑉 ]

DIS DCC

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10

𝜃 = 2.5∘

𝑑𝜎/𝑑

𝑊𝑑𝑄

2 [10

−38

𝑐𝑚2 /

𝐺𝑒𝑉

3 ]


Δ

00.10.20.30.40.50.60.70.80.9

1

0 2 4 6 8 10

𝜃 = 5∘

𝑑𝜎/𝑑

𝑊𝑑𝑄

2 [10

−38

𝑐𝑚2 /

𝐺𝑒𝑉

3 ]


0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

𝜃 = 7.5∘

𝑑𝜎/𝑑

𝑊𝑑𝑄

2 [10

−38

𝑐𝑚2 /

𝐺𝑒𝑉

3 ]


00.20.40.60.8

11.21.41.61.8

0 2 4 6 8 10

𝜈𝜏𝑛 → 𝜏+𝑋

𝐸𝜈 =10 GeV𝜃 = 0∘

𝑑𝜎/𝑑

𝑊𝑑𝑄

2 [10

−38

𝑐𝑚2 /

𝐺𝑒𝑉

3 ]


DIS DCC

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10

𝜃 = 2.5∘

𝑑𝜎/𝑑

𝑊𝑑𝑄

2 [10

−38

𝑐𝑚2 /

𝐺𝑒𝑉

3 ]


Δ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10

𝜃 = 5∘

𝑑𝜎/𝑑

𝑊𝑑𝑄

2 [10

−38

𝑐𝑚2 /

𝐺𝑒𝑉

3 ]


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 2 4 6 8 10

𝜃 = 7.5∘

𝑑𝜎/𝑑

𝑊𝑑𝑄

2 [10

−38

𝑐𝑚2 /

𝐺𝑒𝑉

3 ]



Results

Degree of polarization

00.10.20.30.40.50.60.70.80.9

1

0 1 2 3 4 5 6 7 8 9 10


𝐸𝜈 =10 GeV𝜃 = 0∘

|𝑃|


DIS DCC

0.60.650.7

0.750.8

0.850.9

0.951

0 1 2 3 4 5 6 7 8 9 10

𝜃 = 2.5∘

|𝑃|


Δ

0.80.820.840.860.880.9

0.920.940.960.98

1

0 1 2 3 4 5 6 7 8 9 10

𝜃 = 5∘

|𝑃|


0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

0 1 2 3 4 5 6 7 8 9 10

𝜃 = 7.5∘

|𝑃|


00.10.20.30.40.50.60.70.80.9

1

0 1 2 3 4 5 6 7 8 9 10


𝐸𝜈 =10 GeV𝜃 = 0∘

|𝑃|


DIS DCC

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

𝜃 = 2.5∘

|𝑃|


Δ

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

𝜃 = 5∘

|𝑃|


0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

𝜃 = 7.5∘

|𝑃|



Results

Polarization angle θP = cos−1[PLPT

]Eν = 5GeV

−1

−0.5

0

0.5

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

𝜈𝜏𝑛 → 𝜏−𝑋

𝐸𝜈 =05 GeV𝜃 = 0∘

cos(

𝜃 𝑝)


DIS DCC

−0.6−0.4−0.2

00.20.40.60.8

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

𝜃 = 2.5∘

cos(

𝜃 𝑝)


Δ

−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

𝜃 = 5∘

cos(

𝜃 𝑝)


−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

𝜃 = 7.5∘

cos(

𝜃 𝑝)


−1

−0.5

0

0.5

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


𝐸𝜈 =05 GeV𝜃 = 0∘

cos(

𝜃 𝑝)


DIS DCC

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

𝜃 = 2.5∘

cos(

𝜃 𝑝)


Δ

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

𝜃 = 5∘

cos(

𝜃 𝑝)


−0.4

−0.2

0

0.2

0.4

0.6

0.8

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

𝜃 = 7.5∘

cos(

𝜃 𝑝)



Results

−1

−0.5

0

0.5

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


𝐸𝜈 =05 GeV𝜃 = 0∘

cos(

𝜃 𝑝)


DIS DCC

−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

𝜃 = 2.5∘co

s(𝜃 𝑝

)


Δ

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

𝜃 = 5∘

cos(

𝜃 𝑝)


−0.2

0

0.2

0.4

0.6

0.8

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

𝜃 = 7.5∘

cos(

𝜃 𝑝)


−1

−0.5

0

0.5

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

𝜈𝜏𝑝 → 𝜏+𝑋

𝐸𝜈 =05 GeV𝜃 = 0∘

cos(

𝜃 𝑝)


DIS DCC

−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

𝜃 = 2.5∘

cos(

𝜃 𝑝)


Δ

−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

𝜃 = 5∘co

s(𝜃 𝑝

)


−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

𝜃 = 7.5∘

cos(

𝜃 𝑝)



Results

−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 2 4 6 8 10 12 14 16 18 20

𝜈𝜏𝑛 → 𝜏−𝑋

𝐸𝜈 =20 GeV𝜃 = 0∘

cos(

𝜃 𝑝)


DIS DCC

−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 2 4 6 8 10 12 14 16 18 20

𝜃 = 2.5∘co

s(𝜃 𝑝

)


−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 2 4 6 8 10 12 14 16 18 20

𝜃 = 5∘

cos(

𝜃 𝑝)


−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 2 4 6 8 10 12 14 16 18 20

𝜃 = 7.5∘

cos(

𝜃 𝑝)


−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 2 4 6 8 10 12 14 16 18 20


𝐸𝜈 =20 GeV𝜃 = 0∘

cos(

𝜃 𝑝)


DIS DCC

−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 2 4 6 8 10 12 14 16 18 20

𝜃 = 2.5∘

cos(

𝜃 𝑝)


−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 2 4 6 8 10 12 14 16 18 20

𝜃 = 5∘co

s(𝜃 𝑝

)


−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 2 4 6 8 10 12 14 16 18 20

𝜃 = 7.5∘

cos(

𝜃 𝑝)



Results

−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 2 4 6 8 10 12 14 16 18 20


𝐸𝜈 =20 GeV𝜃 = 0∘

cos(

𝜃 𝑝)


DIS DCC

−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 2 4 6 8 10 12 14 16 18 20

𝜃 = 2.5∘co

s(𝜃 𝑝

)


−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 2 4 6 8 10 12 14 16 18 20

𝜃 = 5∘

cos(

𝜃 𝑝)


−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 2 4 6 8 10 12 14 16 18 20

𝜃 = 7.5∘

cos(

𝜃 𝑝)


−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 2 4 6 8 10 12 14 16 18 20

𝜈𝜏𝑝 → 𝜏+𝑋

𝐸𝜈 =20 GeV𝜃 = 0∘

cos(

𝜃 𝑝)


DIS DCC

−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 2 4 6 8 10 12 14 16 18 20

𝜃 = 2.5∘

cos(

𝜃 𝑝)


−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 2 4 6 8 10 12 14 16 18 20

𝜃 = 5∘co

s(𝜃 𝑝

)


−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1

0 2 4 6 8 10 12 14 16 18 20

𝜃 = 7.5∘

cos(

𝜃 𝑝)



Results

2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0p × cos( )

0.0

0.5

1.0

1.5

2.0

2.5

p×

sin(

)N + X at E = 20GeV

= 0= 2.5= 5= 7.5= 10


Summary

1 Introduction


3 Results

4 Summary


Summary

Summary

τ spin polarization is important for τ lepton detection

The use of Dynamical Couple Channel model helps us to push the invariantmass up to 2.1GeV. In this region, DCC gave an accurate description of theinelastic process.

In the DIS region, we improve the kinematic treatment (especially for low Q2)and write the structure functions in a more consistent way.

Around low invariant mass, the DIS results may need to improve for reasonablematching.

Thanks


Tau Polarization in (anti-)neutrino-nucleon Interactions.

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Transcript of Tau Polarization in (anti-)neutrino-nucleon Interactions.