Distributions of Neutrino Trident Production in gauged Lμ-Lτ model

Takashi Shimomura(Miyazaki U.)

in collaboration withYuichi Uesaka (Saitama U.)

Aug. 8th, 2019 @ 26th New Higgs Working Group

4

IntroductionThe anomalous magnetic moment of muon is a long-standing discrepancy in particle physics.

If there exists a new particle which interacts with muon, the discrepancy can be resolved.

It should be noted that the negative NLO contribution resultsin an anticorrelation between its uncertainty and the uncer-tainty from the LO contribution, consequently resulting in aslight reduction in the overall uncertainty that has beenincorporated into Eq. (3.34).The hadronic LbL contributions, although small compared

to the hadronic vacuum polarization sector, have, in the past,beendetermined throughmodel-dependent approaches.Theseare based on meson exchanges, the large Nc limit, ChPTestimates, short distance constraints from the operator productexpansion, andpQCD.Over time, several different approachesto evaluating ahad;LbLμ have been attempted, resulting in goodagreement for the leading Nc (π0 exchange) contribution, butdiffering for subleading effects. A commonly quoted deter-mination of the LbL contribution is the “Glasgow consensus”estimate of ahad;LbLμ ðGlasgow consensusÞ ¼ ð10.5$ 2.6Þ ×10−10 [101] (alternatively, see [102–105]). However, recentworks [106–108] have reevaluated the contribution to ahad;LbLμ

due to axial exchanges, where it has been found that thiscontribution has, in the past, been overestimated due to anincorrect assumption that the form factors for the axial mesoncontribution are symmetric under the exchange of two photonmomenta [106]. Under this assumption, the determination in[102] previously found the axial vector contribution to beahad;LbL;axialμ ¼ð2.2$0.5Þ×10−10. Correcting this reduces thiscontribution to ahad;LbL;axialμ ¼ð0.8$0.3Þ×10−10 [106,107].Applying this adjustment to theGlasgow consensus result, theestimate in [108] finds

ahad;LbLμ ¼ ð9.8$ 2.6Þ × 10−10; ð3:35Þ

which is the chosen estimate for ahad;LbLμ in this work. Thisresult is notably lower than the previously accepted LbLestimates and will incur an overall downward shift on aSMμ . Itis, however, still within the original uncertainties whencomparing with the original Glasgow consensus estimate.Alternatively, it should be noted that the estimate ofahad;LbLμ ¼ ð10.2$ 3.9Þ × 10−10 [108,109], which is a resultthat is independent of the Glasgow consensus estimate,could be employed here. In addition, the recent work [105]has provided an estimate for the next-to-leading orderhadronic LbL contribution. It has found ahad;NLO-LbLμ ¼ð0.3$ 0.2Þ × 10−10, which does not alter the hadronicLbL contribution significantly, but is taken into accountin the full SM prediction given below.Much work has also been directed at the possibility of a

model independent calculation of ahad;LbLμ to further consoli-date the SM prediction of aμ. One approach involves themeasurement of transition form factors by KLOE-2 andBESIII, which can be expected to constrain the leadingpseudoscalar-pole (π0, η; η0) contribution to a precision ofapproximately 15% [108]. Alternatively, the pion transitionformfactor (π0 → γ%γ%) canbecalculated on the lattice for thesame purpose [110]. New efforts into the prospects of

determining ahad;LbLμ using dispersive approaches are alsovery promising [111–116], where the dispersion relations areformulated to calculate either thegeneral hadronicLbL tensoror to calculate ahad;LbLμ directly. These approaches will allowfor the determination of the hadronic LbL contributions fromexperimental data and, at the very least, will invoke stringentconstraints on future estimates. Last, there has been hugeprogress in developingmethods for a direct lattice simulationof ahad;LbLμ [110,117–123]. With a proof of principle alreadywell established, an estimate of approximately 10% accuracyseems possible in the near future. Considering these develop-ments and the efforts of the Muon g − 2 Theory Initiative[124] to promote the collaborative work of many differentgroups, the determination of ahad;LbLμ on the level of theGlasgowconsensuswill, at thevery least, be consolidated anda reduction of the uncertainty seems highly probable on thetime scales of the new g − 2 experiments.Following Eq. (3.31), the sum of all the sectors of the SM

results in a total value of the anomalous magnetic momentof the muon of

aSMμ ¼ ð11659182.04$ 3.56Þ × 10−10; ð3:36Þwhere the uncertainty is determined from the uncertaintiesof the individual SM contributions added in quadrature.Comparing this with the current experimental measurementgiven in Eq. (1.1) results in a deviation of

Δaμ ¼ ð27.06$ 7.26Þ × 10−10; ð3:37Þcorresponding to a 3.7σ discrepancy. This result is comparedwith other determinations of aSMμ in Fig. 25. In particular, a

160 170 180 190 200 210 220

(aµSM x 1010)−11659000

DHMZ10

JS11

HLMNT11

FJ17

DHMZ17

KNT18

BNL

BNL (x4 accuracy)

3.7σ

7.0σ

FIG. 25. A comparison of recent andprevious evaluations ofaSMμ .The analyses listed in chronological order are DHMZ10 [84], JS11[85], HLMNT11 [9], FJ17 [79], and DHMZ17 [78]. The predictionfrom this work is listed as KNT18, which defines the uncertaintyband that other analyses are compared to. The current uncertaintyon the experimental measurement [1–4] is given by the light blueband. The light grey band represents the hypothetical situation ofthe new experimental measurement at Fermilab yielding the samemean value for aexpμ as the BNL measurement, but achieving theprojected fourfold improvement in its uncertainty [5].

MUON g − 2 AND αðM2ZÞ: A NEW DATA-BASED ANALYSIS PHYS. REV. D 97, 114025 (2018)

114025-23

Keshavariz, Nomura, Teubner, PRD97 (2018)

Discrepancy between the measurements and SM prediction

�aµ = (27.06± 7.26)⇥ 10�10

�aµ = (26.8± 6.3± 4.3)⇥ 10�10

(KNT2018)

(PDG2018) 3.5σ

3.7σ

�aNPµ ⇠

g02

4⇡2

m2µ

m2Z0

⇠ 10�9

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vector boson

Light and feebly interacting vector boson can be a solution to (g-2)μ.

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satisfied with

6

IntroductionOne of the simplest ways to obtain a new vector boson is to impose extra U(1) gauge symmetry, i.e. B-L, Le-Lμ, Lμ-Lτ, …

Minimal gauged Lμ-Lτ model e µ ⌧ ⌫e ⌫µ ⌫⌧

U(1)Lµ�L⌧ 0 1 �1 0 1 �1

TABLE I. The charge assignment of the gauged U(1)Lµ�L⌧ model.

II. MINIMAL Lµ � L⌧ MODEL

We start our discussion with reviewing a minimal gauged Lµ�L⌧ model. The gauge sector

of the SM is extended by adding a gauged U(1)Lµ�L⌧ symmetry under which only muon and

tau flavour leptons among the SM fermions are charged. The charge assignment for leptons

under this symmetry is shown in Table I. In the Table, e, µ and ⌧ represent charged leptons,

and ⌫e, ⌫µ and ⌫⌧ are corresponding left-handed neutrinos of the SM, respectively. Up-type

and down-type quarks as well as the Higgs boson are singlet under the U(1)Lµ�L⌧symmetry.

The relevant interaction Lagrangian for neutrino trident production processes is given by

Lint

= eAµJµem

� 4GFp2[⌫l4�µ⌫l3 ][l2�

µ(gLPL + gRPR)l1] + g0Z 0µJ

µZ0 (2)

where Aµ and Jµem

represent the photon and electromagnetic current of the SM, respectively,

and e and GF are the electric charge and the Fermi coupling constant. In the second term

of Eq. (2), l and ⌫l are a charge lepton and a neutrino with flavour l = e, µ, ⌧ , and PL(R)

is a left-handed (right-handed) projection operator. The constants gL and gR are given by

(these should be checked)

gL =

✓�1

2+ sin2 ✓W

◆�l1,l2�l3,l4 + �l1,l4�l2,l3 , (3a)

gR = sin2 ✓W �l1,l2�l3,l4 , (3b)

where ✓W is the Weinberg angle. The third term of Eq. (2) represents the new gauge

interaction mediated by the Lµ�L⌧ gauge boson Z 0µ with the coupling constant g0, and JµZ0

is the Lµ � L⌧ current given by

JµZ0 = µ�µµ� ⌧�µ⌧ + ⌫µ�

µ⌫µ � ⌫⌧�µ⌫⌧ . (4)

For the minimality of our model, we assume that the gauge kinetic mixing term be-

tween the U(1)Lµ�L⌧and the hypercharge U(1)Y symmetries is vanishing at tree-level. Even

though, such a kinetic mixing can be generated radiatively via muon and tau loops (ref).

3

Lµ�⌧ = �1

4Z0

µ⌫Z0µ⌫

+ g0Z0µJ

µZ0 +

1

2m2

Z0Z0µZ

0µLagrangian

where the gauge current is JµZ0 = µ�µµ� ⌧�µ⌧ + ⌫µ�

µ⌫µ � ⌫⌧�µ⌫⌧

• The Lμ-Lτ symmetry is spontaneously broken appropriately.• Tree-level gauge kinetic mixing is absent.• Loop-induced gauge kinetic mixing is only considered

for constraints.

We assume that

8

IntroductionOne of the simplest ways to obtain a new vector boson is to impose extra U(1) gauge symmetry, i.e. B-L, Le-Lμ, Lμ-Lτ, …

Among these simple and famous extensions, only gauged Lμ-Lτ symmetry can explain muon (g-2).

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mZ0 (GeV)10�4 < g0 < 10�3

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10 < mZ0 < 200 MeV<latexit sha1_base64="8nRqHz/R3g3vg6OxLcDzAcQmF4E=">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</latexit><latexit sha1_base64="8nRqHz/R3g3vg6OxLcDzAcQmF4E=">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</latexit><latexit sha1_base64="8nRqHz/R3g3vg6OxLcDzAcQmF4E=">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</latexit><latexit sha1_base64="8nRqHz/R3g3vg6OxLcDzAcQmF4E=">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</latexit>

Bauer et al, JHEP1807 (2018) 94

11

Neutrino Trident ProductionLow mass region has been constrained by three measurements.

CCFR :

Borexino :

BaBar : e+e� ! µ+µ�Z0 ! 4µ

⌫ie� ! ⌫ie

�

non-standard contrib. from solar neutrino-electron scattering

muonic dark force search

(Neutrino Trident Production or “NTP”)

�

⌫µ ⌫µ/µ

µ/⌫µ

µ

Nucleus/Nucleon

Z/W Z0SM NP

CCFR

Borexino

BaBar

10-3

10-2

0.001 0.01 0.1 1

g'

MZ' (GeV)0.001 0.01 0.1 1

10-3

10-2g0

mZ0 (GeV)

⌫µN ! ⌫µµ+µ�N

The SM process is suppressed by GFg0

g0

g02

m2Z0

⇠ GF g0 ⇠ 10�6

✓mZ0

MeV

◆

NTP is sensitive to small g’ and mZ’

12

NTP search at DUNE NDDeep Underground Neutrino Experiment (DUNE)

Near Detector (ND)

• Intense beam of νμ/anti-νμ from 1.2 MW proton beam.• 1021 POT/year with 60-120 GeV proton. • < ~10 GeV neutrino beam produced.• Large far detector at the Sanford lab. and near detector at Fermi lab.• starting in 2026.

• 75 tonne fiducial mass Liquid Argon detector.• 574m from the target.

DUNE HP at Fermi lab.

13

NTP search at DUNE NDDUNE ND has a potential to measure neutrino tridents.

NTP cross sections in the SM and Lμ-Lτ model

10�3 10�2 10�1 100 101

MZ0 (GeV)

10�5

10�3

10�1

101

103

�/�

SM

Lµ � L⌧ , Vector Z 0, ⌫µ ! ⌫µµ+µ�

g0 = 8 ⇥ 10�4

D.B. BH2

Interf

SMTotal

Ballett et al, 1902.08579

• The NTP cross section increases as Eν becomes larger.• The cross section in Lμ-Lτ model is larger than that of the SM.• The deviation of the cross section is factor of 4-8 to the SM for law mass.

10-44

10-43

10-42

10-41

10-40

1 10

σ (c

m2 )

Eν (GeV)

SMmZ’ = 100MeV, g’=5×10-4

mZ’ = 100MeV, g’=8×10-4

mZ’ = 50MeV, g’=5×10-4

mZ’ = 50MeV, g’=8×10-4

Lµ �L⌧ , Vector Z0, ⌫µ ! ⌫µµ+µ�

Our calculation (BH only)

15

NTP search at DUNE NDDUNE ND has a potential to measure neutrino tridents.

Cross section of NTPs is degenerate in the NP parameters

10�2 10�1 100 101

MZ0 (GeV)

10�4

10�3

10�2

g0

CCFR

LHC

BaBar 4µ

�Ne↵ > 0.5

Borex

ino

" =eg0

12⇡2ln

m2µ

m2⌧

Lµ � L⌧ , DUNE ND, 75 tonnes, 5 y ⌫-mode + 5 y ⌫-mode, 120 GeV p+, �norm = 5%

DUNE 90% C.L.

µ+µ� trident

⌫ � e scattering(g � 2)µ ± 2�(g � 2)µ ± 2�

⌫ ! ⌫µ+µ� trident

⌫ � e scatering

g0

10-2

10-3

10-4

10-2 10-1 100 101

mZ0 (GeV)

Ballett et al, 1902.08579

Contour of cross section

• In Lμ-Lτ model, muon pair trident could provide stringent bound.• 90% C.L. limit covers favored region of (g-2) completely.

Eµ

17

Purpose

We need another information on NP contributions.

In this talk, we show our results

Measurements of the NTP cross section will find the existence of NP.However, the cross section is degenerate in the parameters (mZ’, g’).Therefore, the parameters can not be determined.

To determine the NP parameters,

Distributions of NTPs with respect to• Energy of final state muons•Opening angle between muons• Invariant mass of muons

•Detector resolution•Neutrino energy spectra

Determine the parameters, taking into account

⌫µ

⌫µ

µ�

µ+Eµ̄

✓µµ̄

m2µµ̄

Not yet…

Today, this part

20

Contributions to NTPs

The relevant Lagrangian for NTP in minimal Lμ-Lτ model is

Lint = eAµJµem �

4GFp2

[⌫µ�µ⌫µ][µ�µ(gLPL + gRPR)µ] + g0Z0

µJµZ0

For the NTP,

gL =1

2+ sin2✓W �

p2

4GF

g02

q2Z0 � m2

Z0

gR = sin2✓W �p2

4GF

g02

q2Z0 � m2

Z0

⌫µ ! ⌫µµ+µ� , the NP contributions are included by shifting

q2Z0 = (k0 � k)2 < 0

where

NP contributions appearonly this term.

NP contributions are always positive.

21

Cross SectionThe squared amplitudes are

XXX

spins

|M |2 =e4G2

F

2q4j↵�Lµ⌫

↵�Jµ⌫ ,

where j↵�

Lµ⌫↵�

Jµ⌫

: neutrino spinor part: charged lepton spinor part: hadronic part

Form FactorJµ⌫ = Z2(Q+Q0)µ(Q+Q0)⌫ |F (t)|2

with, for Argon,F (t) = 4⇡

ZZZ 1

0drr2⇢(r)

sinptr

ptr

⇢(r) = ⇢01 +

wr2

c2

1 + exp

�r�cz

�

c = 3.73 fm, z = 0.62 fm, w = �0.19

C. W. de Jager, H. de Vries, & C. de Vries, Atomic Data and Nuclear Data Tables 14, 479-508 (1974)�

22

Example Points Example points for the same cross sections

CCFR

Borexino

BaBar

10-3

10-2

0.001 0.01 0.1 1

g'

MZ' (GeV)

Eν5 GeV

20 GeV40 GeV

E⌫ mZ0 g0 � E⌫ mZ0 g0 �

1 0.020 5.869⇥ 10

�4

0.005546 20 0.020 7.009⇥ 10

�4

3.097

0.10 9.00⇥ 10

�4

0.005546 0.10 9.00⇥ 10

�4

3.097

0.20 1.299⇥ 10

�3

0.005547 0.20 1.111⇥ 10

�3

3.097

1.0 4.972⇥ 10

�3

0.005546 1.0 2.824⇥ 10

�3

3.097

5 0.020 6.584⇥ 10

�4

0.3280 30 0.020 7.11⇥ 10

�4

5.470

0.10 9.00⇥ 10

�4

0.3280 0.10 9.00⇥ 10

�4

5.471

0.20 1.177⇥ 10

�3

0.3279 0.20 1.095⇥ 10

�3

5.470

1.0 3.632⇥ 10

�3

0.3280 1.0 2.64⇥ 10

�3

5.478

10 0.020 6.814⇥ 10

�4

1.085 40 0.020 7.18⇥ 10

�4

8.079

0.10 9.00⇥ 10

�4

1.085 0.10 9.00⇥ 10

�4

8.078

0.20 1.141⇥ 10

�3

1.085 0.20 1.084⇥ 10

�3

8.076

1.0 †3.195⇥ 10

�3

1.085 1.0 2.513⇥ 10

�3

8.079

TABLE II. Parameters for constant cross section. The units of E⌫ and mZ0are GeV, and that of

� is 10

�41

cm

2

, respectively. Symbol ⇤ represents a coupling constant with which (g� 2)µ can not

be explained, † represent a coupling constant that has been already excluded.

8

• 4 example points are chosen for Eν=1-40 GeV.•All points are obtained by numerical calculation with error < 1%.•Points that are already excluded or does not explain (g-2) are included.

23

Eμ-Eμ distributions

•Distributions are not symmetric for Eμ and Eμ due to the charge current.• Eμ is uniformly distributed due to the charged current contributions.•Differences between the parameters are almost the same for each Eν.•Differences can be seen in Eμ, not in Eμ.

10�43 cm2/GeV2

2⇥ 10�43 cm2/GeV2

24

Eμ distributions

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 400

0.51

1.52

2.53

3.54

4.55

0 2 4 6 8 10 12 14 16 18 20

• Eμ is peaked at smaller value.• The peak becomes sharp as Eν increases• Parameter differences appear around the peak, which comes

from the differences in Eμ distribution.

27

Eμ distributions

• Eμ is also peaked at small value.• But parameter differences are smaller than Eμ distribution.

30

Opening angle distributions

•Opening angle is peaked at small value. •Distributions are almost independent of the parameters.• The SM distributions is also the same shape,

which implies the angle distribution is determined by spinor parts.

31

Invariant mass distributions

• Invariant mass is peaked at small value.• Parameter dependence can be seen clearly for higher Eν.

32

Summary and discussions

Eμ and Eμ distributions are not symmetric due to the charged current interactions.Eμ distributions show the difference between the NP parameters.Invariant mass distributions also show differences between the parameters　for higher Eν. Eμ and opening angle distributions show less parameter dependences.

We have considered minimal gauged Lμ-Lτ model, and studied the distributions of NTPs for the same value of the cross section.

Detector resolutionNeutrino energy spectrumDiffractive process (scattering with nucleon) of NTPs

To study the potential to determine the NP parameters, we have to include