On the feasibility of Bell Inequality violation
at ATLAS experiment with avor entanglementof B meson pairs from p-p collisions
Tsubasa ICHIKAWA (QIQB, Osaka U)
30th Sept. 2021 @ Flavor Physics Workshop 2021
based on Physical Review D 104, 056004 (2021) [arXiv:2106.07399]
Collaborators: Y. Takubo, S. Higashino, Y. Mori, K. Nagano, I. Tsutsui  

fl1

実験形而上学（Experimental Metaphysics）-

科学哲学者・物理学者A. Shimonyの造語（1989）
指導教官：R. Carnap, E. Wigner---

主に量子力学の基礎的な問題に実験でアプローチする分野-

Einstein, Podolsky, Rosen（EPR）のパラドックス-ベル不等式とその実験的検証など

主な実験系は低エネルギー系
フォトン
原子分子
イオン など-本トークでのtake-home message-

高エネルギー系でも実験形而上学が行える見込みがありそう-

特にLHCのATLAS実験は実施に有望な実験系である
2

本トークのターゲット：ベル不等式+1~a+1粒子源
B

A~b1-

1

2準位2体系の測定実験-

隠れた変数（系の実在的な記述）の確率分布

相関関数を以下で記述できると仮定
Z
C(~a, ~b) = d A(~a, )B(~b, )P ( ).

局所的情報のみから測定値が決定-

測定軸を自由に選択して相関関数の組み合わせを計算
S = C(~a, ~b) + C(a~0 , ~b) + C(~a, b~0 )-C(a~0 , b~0 )ベル不等式|S|  23量子力学はベル不等式を破る+1~a+1粒子源BA~b1-

1
シングレット状態：| i = p (| "i| #i2-

オブザーバブル：(~a · ~ ) ⌦ (~b · ~ )-相関関数： C(~a, ~b) =ベル不等式の破れ⇨| #i| "i)|~a| = |~b| = 1

a~0~b~acos ✓S = C(~a, ~b) + C(a~0 , ~b) + C(~a, b~0 )-1✓ ✓C(a~0 , b~0 )✓b~0

量子力学は実在性・局所性・自由選択を同時には満足しない
4

スピン測定とフレーバー測定のアナロジー-

メソンペア（BB, KK etc）のフレーバーを用いたベル不等式検証--【意義１】既存の実験と異なる物理系での実験
11【意義２】ベル不等式を検証したエネルギースケールが拡大

Simulation study【実装のアイデア】スピン測定とフレーバー測定のアナロジー• The simulation was performed for truth level study with PYTHIA8.245.
スピンのアップダウンをフレーバーに、測定軸を崩壊時刻に読み替える• The low-μ (μ~1) operation with 1 fb-1 of data was assumed.
Am. J. Phys. 69, 264 (2001)• The sensitivity to Bell inequality violation was evaluated withത
𝐵0 𝐵ത 0 events
from+1+1~a 「gg/qq→𝑏 𝑏」(69μb) and 「gg/qq→jj」(319μb).
粒子源• 「𝐵0 → 𝐷∗− μ+ ν (𝐷∗− → 𝐷0 π− , 𝐷0 → 𝐾 + π− )」 events were selected.
➢➢

A

B

B meson flavor can be identified with charge sign of decay objects.
The decaytime (𝑡𝑎 , 𝑡𝑏 ) can be measured, reconstructing~b11 vertex withμ+ and π− from 𝐷∗− decay (σ𝑡 ~0.11 ps)

5

アナロジーの不完全な部分
1. 粒子崩壊で状態ベクトルのノルム収縮112. 測定軸が固定⇨⇨

相関が弱まる PLA 332, 355 (2004)

Simulation study

Freewill loophole

本当に隠れた変数に依存せず、ランダムに測定軸を選べるか• The simulation was performed for truth level study with PYTHIA8.245.
• 慣習的に量子乱数を用いることで対処• The low-μ (μ~1) operation with 1 fb-1 of data was assumed.
•• The sensitivity to Bell inequality violation was evaluated withത𝐵0 𝐵ത 0 events
from+1+1~a 「gg/qq→𝑏 𝑏」(69μb) and 「gg/qq→jj」(319μb).
粒子源• 「𝐵0 → 𝐷∗− μ+ ν (𝐷∗− → 𝐷0 π− , 𝐷0 → 𝐾 + π− )」 events were selected.
➢➢

AB

B meson flavor can be identified with charge sign of decay objects.
The decaytime (𝑡𝑎 , 𝑡𝑏 ) can be measured, reconstructing~b
11 vertex withμ+ and π− from 𝐷∗− decay (σ𝑡 ~0.11 ps)

6

3つのloophole-3つの抜け穴（loophole）を閉じた実験が望ましい
1. Detection loophole• 低い検出効率 ⇨ サンプリングの偏り•

検出効率の下限 = 2

2 − 2 ≈ 0.824•

検出効率が下限より低い場合⇨⇨

ベル不等式の上限が増加（＞2）
PRA 57, 3304 (1998)

fair sampling assumption を課して対処

2. Locality loophole• 局所性を実験的に担保しないと意味のある実験にならない•

部分系の測定をspacelikeな２地点で行うことで対処

3. Freewill loophole• 本当に隠れた変数に依存せず、ランダムに測定軸を選べるか• 慣習的に量子乱数を用いることで対処

7

論点は4つ1. 粒子崩壊による相関の弱まり
2. Freewill loophole
3. Detection loophole
4. Locality loophole-

LHC Run 3の実験セットアップを仮定して以下の方法で検証
-

理論分析
-

-

CPの破れは無視

シミュレーション-

使用ソフトウェア：PYTHIA 8.245-

ビーム衝突あたりの陽子・陽子反応数が約1の運転(low-μ run)-

1fb^{-1}のデータを取得

8

【参考】既存の素粒子を用いた実験の問題点[PLA373 (2008) 39-44]

9Previous experiments|S||S| in QM and HVT only with Δtmeasurement
HVT

B meson (Belle) [arXiv:0310192 (2003)]• A measurement only for Δt, where(𝑡𝑎 , 𝑡𝑏 ) was not measured separately.

QM• 𝑆 has the maximum value of 4 andincludes that of QM.

ΔMΔt

K meson (CPLEAR) [PLB422, 339-348]
A measurement only for 𝐶 𝑄 𝑡𝑎 , 𝑡𝑏 andis not Bell test.

Fraction of space-like events of2 B mesons in Belle

Both experiments
Most of 2 mesons are not space-like andthe locality is not ensured.

Lower limit for |S|≦𝟐 𝟐Belle

The previous experiments were not enoughfor Bell test on the flavor entanglement.

- ATLASは個々の崩壊時刻を特定できるため、上記問題は解決できている9• The sensitivity to Bell inequality violation was evaluated withത
𝐵0 𝐵ത 0 events from 「gg/qq→𝑏𝑏」(69μb)
and 「gg/qq→jj」(319μb).
• 「𝐵0 → 𝐷∗− μ+ ν (𝐷∗− → 𝐷0 π− , 𝐷0 → 𝐾 + π− )」 events were selected.
➢➢-

検討１：粒子崩壊による相関の弱まり

B meson flavor can be identified with charge sign of decay objects.

The【回答】崩壊時刻に関する条件付き確率を使って相関関数を再定義することで補正
decay time (𝑡𝑎 , 𝑡𝑏 ) can be measured, reconstructing vertex withμ+ and π− from 𝐷∗− decay (σ𝑡 ~0.11 ps)：崩壊幅
M ：mass eigenstateの質量差t = t a tb-

【量子力学側】Am. J. Phys. 69, 264 (2001) & PRD 104, 056004 (2021)
時刻 ta , tb でフレーバーが B 0 , B 0 である確率(ta +tb )e
P (B 0 , B 0 , ta , tb ) =(1 cos( M t))4--

条件付き確率100
P (B , B |ta , tb ) = (14cos( M t))【隠れた変数側】PLA 373, 39 (2008) & PRD 104, 056004 (2021)-

スピン測定とフレーバー測定のアナロジーが成立 ⇨ ベル不等式は導出可アナロジーが不成立 ⇨ 仮定を追加すればベル不等式は導出可
10検討２：Freewill loophole-

PRD 104, 056004 (2021)【回答】既存の実験の慣習に従う
既存の実験の慣習：量子乱数を用いて測定軸選択11
粒子崩壊：量子力学的なランダムな現象 ⇨ 慣習に合致

Simulation study-【メリット】スピン測定とフレーバー測定のアナロジーがランダム性を含めて成立• The simulation was performed for truth level study with PYTHIA8.245.
• The low-μ (μ~1) operation with 1 fb-1 of data was assumed.
• The sensitivity to Bell inequality violation was evaluated withത
𝐵0 𝐵ത 0 events
from「gg/qq→𝑏𝑏」(69μb)
and 「gg/qq→jj」(319μb).
+1+1~a Random
粒子源• 「𝐵0 → 𝐷∗− μ+ ν (𝐷∗− → 𝐷0 π− , 𝐷0 → 𝐾 + π− )」 events were selected.
➢➢

BA

B meson flavor can be identified with charge sign of decay objects.
The decay
time (𝑡𝑎 , 𝑡𝑏 ) can be measured, reconstructing~b
1
1 vertex withμ+ and π− from 𝐷∗− decay (σ𝑡 ~0.11 ps)

Random11

検討３：Detection loophole-【回答】検出効率2%-fair sampling assumptionを課して対処⇨（検出効率）＝（イベント再構成効率）×（トリガー効率）×（イベント選択効率）
13Event selection

PRD 104, 056004 (2021)Efficiency Comment
Track reconstruction (ε𝑟𝑒𝑐𝑜 )0.483 From 「𝐷 ∗+ μ− X」 analysis

Trigger (ε𝑡𝑟𝑖𝑔𝑔𝑒𝑟 )0.429 (0.819 × 0.8)2 was assumed.
• 0.819 is efficiency for single-μ
trigger with pT>6GeV.

Selection criteria (ε𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛 )Total eff.

Comment

pT > 1 GeV for π+ /𝐾 − in 𝐷 0 candidates

0.510

pT > 250 MeV for π+ from 𝐷 ∗+0.452•|m(𝐾 − π+ ) – m(𝐷 0 )| < 64 MeV (pT(𝐾 − π+π+ ) > 12 GeV, |η(𝐾 − π+ π+ )| > 1.3)•|m(𝐾 − π+ ) – m(𝐷 0 )| < 40 MeV elsewhere

0.209 Assume σ2 cut (0.46)

2.5 GeV < m(𝐷 ∗+ μ− ) < 5.4 GeV

0.097 Assume σ2 cut (0.46)

ε𝑡𝑜𝑡𝑎𝑙 (ε𝑟𝑒𝑐𝑜 ×ε𝑡𝑟𝑖𝑔𝑔𝑒𝑟 ×ε𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛 ) = 0.020

12Simulation study• The simulation was performed for truth level study with PYTHIA8.245.
• The low-μ (μ~1) operation with 1 fb-1 of data was assumed.

検討４：Locality loophole• The sensitivity to Bell inequality violation was evaluated withത
𝐵0 𝐵ത 0 events from 「gg/qq→𝑏𝑏」(69μb)
and 「gg/qq→jj」(319μb).

• 「𝐵0 → 𝐷∗− μ+ ν (𝐷∗− → 𝐷0 π− , 𝐷0 → 𝐾 + π− )」 events were selected.
【回答】ATLASではLocality loopholeは閉じている➢ B meson flavor can be identified with charge sign of decay objects.
➢The decay time (𝑡𝑎 , 𝑡𝑏 ) can be measured, reconstructing vertex withFEASIBILITY OF BELL INEQUALITY VIOLATION AT THE …
μ+ and π− from 𝐷∗− decay (σ𝑡 ~0.11 ps)

PRDREV.
104,056004
PHYS.
D 104,056004 (2021)(2021)As the acceptance cut, the existence of two muons withpT being above 4 GeV and pseudorapidity jηj below 2.4 arerequired. In addition, the requirement on two D!" mesonswith pT > 2.25 GeV and jηj < 2.5 were applied. With
these cuts, the acceptance A and cross section times−3A become
× 10056004
and 369
pb, respecLATION AT THE … acceptance σ ×PHYS.
REV.1.49
D 104,(2021)
tively, for the signal events.
Summing up, the number of the signal events left aftertor (RNG)
the acceptance and selection cuts can be evaluated as
may accept
ðσ × AÞ × ϵ × L ¼ 7.4L, where L is an integrated lumies in theirnosity in unit of pb−1 . With this formula, by assumingthat such
L ¼ 1 × 103 pb−1 , we obtained the expected number of thesource of
signal events 7.4 × 103 .

he fact thatment is thespace ofther hand,determined
he system,
pped withr part. Thisa par withreedom ofhas appathe Belle

SpacelikeD. Background estimation
We have considered two possible sources of the background on our analysis. First, two neutral B mesons can becreated from different gluons and they may contribute asirreducible background. Our simulation showed that suchbackground is less than 0.1% with respect to the signal and,therefore, negligible.
FIG. 1. Minkowski
for backgroundthe spacetime events
toThe otherdiagrams
possibleis therelated
combinatorial
the decayof a pair of Bthat
mesonsconsidered
the Bell test. After signals
backgroundis causedbyformisreconstructed
the pp collision emerge a pair of B mesons, which subsequently
with particles from different origins. 0 In measurements ofdecay at t2 (or t02 ) in theregion (A) and at t1 (or t1 ) in the region
decays to D!þ μ− X final states in
the ATLAS experiment

FIG. 2. Distributions of the squared proper distances s2 of theB0 B̄0 decay events before and after the acceptance and selectioncuts. The events are spacelike when s2 > 0.

the events are spacelike (s2 > 0) even without any cuts, andmore than 99% events are spacelike after the cuts. The
13
locality condition is thus perfectly satisfied in our analysis.
Q

結論
YOSUKE TAKUBO et al.
-LHC Run 3でATLAS検出器を

exmaantimtheBe
usbefun

使えば、ベル不等式の破れは検証できる-

3つのloopholeのうち、freewillloopholeとlocality loopholeを閉じた実験-meprotes
stamoforadepro

最高エネルギースケールでの実験（〜14TeV）-

素粒子実験で最初のベル不等式検証実験
FIG. 3. jSQ ðjΔtjÞj after the acceptance and selection cuts in thecase of QM with the theory line. The dotted line at jSðjΔtjÞj ¼ 2
shows the upper bound of the Bell inequality.
14assof