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CP-Verletzung Und Die CKM-Matrix

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CP-Verletzung Und Die CKM-Matrix Fakultät für Physik Universität Bielefeld Seminarvortrag Elementarteilchenphysik Die Verletzung der CP-Symmetrie Zweiter Vortrag, Benjamin Jurke∗ (Stand: 20. Mai 2005) Die Verletzung der CP-Symmetrie ist für das Verständnis der Elementarteilchenphysik von entscheidender Bedeutung, insbesondere zur Erklärung des Materieüberschusses und der Teilchenentstehung im primordialen Universum (Baryogenese). Die Auswirkungen der verletzten CP-Symmetrie auf Zerfallsprozesse liefern in jüngerer Zeit ein wichtiges Auswahl- kriterium für Theorien der großen Vereinheitlichung (GUT). Im Rahmen des Standardmo- dells lässt sich die CP-Verletzung weitestgehend über eine nichttriviale Phasenverschie- bung der CKM-Quarkmischungs-Matrix beschreiben, die nach einer kurzen Wiederholung der Quark-Massenerzeugung eingeführt wird. Neben der Untersuchung von K-, D- und B- Mesonen-Zerfällen wird auf Unitaritätsdreiecke der CKM-Matrix eingegangen. Die starke CP-Verletzung und supersymmetrische Erweiterungen werden nur kurz behandelt. 1 Das Problem der CP-Verletzung Neben den gewöhnlichen Erhaltungsgrößen wie Energie, Impuls, etc., die nach dem Noether- Theorem zu kontinuierlichen Symmetrien der Lagrange-Dichte gehören, existieren auch diskrete Symmetrien in der Physik. Aus der Raumspiegelung ~x → ~x0 := −~x erhält man die diskrete Symmetrie der Parität P. Die Umwandlung |ψi → |ψ0i := |ψ¯i eines Teilchens in sein zugehö- riges Antiteilchen führt zur Ladungskonjugation C, analog ist die Zeitumkehr T durch die Transformation t → t0 := −t definiert. Die Hintereinanderschaltung dieser drei Symmetrien führt zum CPT-Theorem, nach dem zu jedem physikalischen Vorgang auch der gespiegelte, zeitumgekehrte und durch Austausch von Materie und Antimaterie konstruierte Vorgang denselben Gesetzen folgt (siehe Abbildung 1). Das CPT-Theorem ist ein vorausgesetztes Element aller relativistischen Quantenfeldtheorien, so dass aus einer Verletzung der CP-Symmetrie automatisch eine kompensierende Verletzung der T-Symmetrie aus Sicht der Theorie folgt. Bisher hat sich die CPT-Symmetrie in allen Experimenten als exakt herausgestellt, d.h. CPT ist nicht verletzt. Während Gravitation und die elektromagnetischen Vorgänge diesen drei Symmetrien gehor- chen, ist dies für die starke Wechselwirkung noch nicht eindeutig bekannt - es existieren aller- dings bisher keine experimentellen Hinweise für eine Verletzung. 1956 zeigten Lee und Yang die Verletzung der Raumspiegelungssymmetrie im β-Zerfall über die schwache Wechselwirkung: Die Natur scheint dabei die linkshändigen den rechtshändigen Teilchen vorzuziehen und man ∗E-Mail: [email protected]; Elementarteilchen-Seminar SS 2005; Leiter: Dr. York Schröder 1 v(t) ¡ ¡ v(−t) ¢¡¢¡¢ v(−t) x(t) Tˆ Pˆ Cˆ £¡£¡£¡£¡£ ¤¡¤¡¤¡¤ ¥¡¥¡¥¡¥¡¥¡¥ −x(−t) ¡ ¡ ¢¡¢¡¢ −x(−t) Zeitumkehr Parität Ladungsumkehr x(−t) Das CPT-Theorem besagt, dass sich Materie =⇒ und Antimaterie vollkommen gleich verhalten −v(−t) Abbildung 1: Illustration der Anwendung von Parität P, Ladungsumkehr C und Zeitumkehr T in der Zusammenstellung des CPT-Theorems aus Sicht eines außenstehenden Beobachters. erhält die P-Verletzung. Ähnlich entdeckte man, dass unter Einfluss der schwachen Wech- selwirkung auch die Ladungsumkehr C und Zeitinversion T verletzt wird. In Tabelle 1 findet sich eine Übersicht von physikalische Größen bzw. Symmetrien und ihrer Erhaltung/Verletzung innerhalb der Wechselwirkungen. Kombiniert man aber Ladungsumkehr C und Raumspiegelung P so erhält man eine Sym- metrie, die aus Sicht der Theorie von der starken und schwachen Wechselwirkung respektiert wird. Doch schon 1964 entdeckten die amerikanischen Physiker Christenson, Cronin, Fitch und Turlay eine winzige Unregelmäßigkeit beim K-Mesonen-Zerfall, d.h. eine Verletzung der CP-Symmetrie im Zerfallsprozess, wie später explizit hergeleitet wird - die beiden letzteren der Genannten erhielten für diese Entdeckung 1980 den Nobelpreis. Auf weitere Arten der CP- Verletzung wird in Abschnitt 3 eingegangen. Bereits in den 60er-Jahren schlug Sakharov vor, dass die CP-Verletzung eine entscheidende Rolle in der Entstehungsgeschichte des Universums spielen könnte. Gemäß der Urknalltheorie brachte die primordiale Singularität die gleiche Menge an Teilchen wie auch Antiteilchen hervor, die sich gegenseitig hätten annihilieren sollen. Durch die sogenannte Baryogenese, die sich möglicherweise durch die CP-Verletzung erklären lässt, entstand das heute zu beobachtende Ungleichgewicht, in dem nur noch Materie übrig blieb. Erhaltungsgröße / Symmetrie stark el.-magn. schwach Energie E, Impuls ~p, Drehimpuls ~l ••• elektrische Ladung q, Leptonen-/Baryonen-Zahl nL/nB ••• Strangeness s • • × Charmness (Charm) c • • × Bottomness (Beauty) b, Topness (Truth) t • • × Isospin Is • × × Parität Pˆ • • × Ladungskonjugation Cˆ • • × Zeitumkehr Tˆ • • × Raum-/Ladungumkehr-Symmetrie CˆPˆ • • × Zeit-/Raum-/Ladungsumkehr-Symmetrie CˆPˆTˆ ••• Tabelle 1: Übersicht der Erhaltung „•“ bzw. Verletzung „ד von Symmetrien und physikalischen Größen im Zusammenhang mit den einzelnen Wechselwirkungen. 2 K¯ 0 0 Λ p+ K K+ s d¯ s¯ s u d d d u s¯ u u u u d d ¯ u u¯ u u d d + − π + π p+ p Abbildung 2: Erzeugung neutraler K0- und K¯ 0-Mesonen durch die starke Wechselwirkung, wobei nur diejenigen Gluonen mit punktierten Vertizes am Erzeugungsprozess beteiligt sind. 2 CP-Verletzung in Mesonen-Zerfällen 2.1 Eigenschaften der Kaonen Der Begriff „Meson“ (griech. Mitte) wurde in den 30er Jahren für Teilchen mittlerer Masse (im Vergleich zu leichten Leptonen und schweren Baryonen) eingeführt. Das Kriterium der Masse erwies sich allerdings schnell als unzureichend als sich heraus stellte, dass das erste nachgewiesene Meson eigentlich ein Lepton ist. Zudem ist etwa das Tauon-Lepton (1.777 GeV) wesentlich schwerer als das Proton (938.2 MeV) oder Neutron (939.5 MeV). Das positiv geladene Kaon K+ wurde 1947 über den Prozess π+ + n → K+ + Λ ent- deckt, ihre besondere Bedeutung erlangten die Kaonen aber erst durch die Entdeckung der CP-Verletzung in ihren Zerfallsprozessen. Die Auswirkungen der CP-Verletzung werden explizit an diesem historischen Beispiel der neutralen Kaonen K0 bzw. K¯ 0 gezeigt. Da dem K0-Meson eine Strangeness s = +1 zugeschrieben wird, kann es nicht sein eigenes Antiteilchen sein, so dass es zwei Zustände K0 und K¯ 0 des neutralen K-Mesons gibt. Die Erzeugung neutraler Kaonen findet durch π− +p+ → K0 +Λ und π+ +p+ → K+ +K¯ 0 +p+ über die starke Wechselwirkung statt, die zugehörigen Feynman-Diagramme sind in Abbildung 2 zu sehen. Eine bemerkenswerte Eigenschaft ist die Kaonen-Mischung K0 ↔ 2π ↔ K¯ 0 zwischen K0 und K¯ 0, zumeist über den Zwischenzustand 2π. Diese Mischung geschieht durch einen schwachen Prozess zweiter Schleifen-Ordnung, so dass die Übergangsrate sehr klein ist. 2.2 Zerfallsprozesse in der Weisskopf-Wigner-Approximation Empirisch zeigt sich, dass nahezu alle Zerfallsprozesse der Natur in exponentieller zeitlicher Ab- hängigkeit stattfinden. Trotzdem ist die Beschreibung eines exponentiellen Zerfallspro- zesses im Rahmen der Quantenmechanik kein triviales Problem. Die Schrödinger-Gleichung d s d s u, c, t W − WW u,¯ c,¯ t¯ u, c, t u,¯ c,¯ t¯ W + s¯ d¯ s¯ d¯ Abbildung 3: K0-K¯ 0-Kaonen-Mischung bzw. -Oszillation durch Einfluß der schwachen Wechselwirkung. 3 Pionen / Λ K-Mesonen D-Mesonen B-Mesonen π0 = uu¯ − dd¯ K0 = ds¯ K¯ 0 = sd¯ D0 = cu¯ D¯ 0 = uc¯ B0 = d¯b B¯0 = bd¯ π+ = ud¯ K+ = us¯ K− = su¯ D+ = cd¯ D− = dc¯ B+ = u¯b B− = bu¯ − + − + ¯ − π = du¯ Ds = cs¯ Ds = sc¯ Bc = cb Bc = bc¯ + ¯ − Λ = uds Bs = sb Bs = bs¯ Tabelle 2: Quark-Zusammensetzung der im Weiteren verwendeten Teilchen. impliziert für einen stationären Zustand durch die formale Lösung ˆ |ψ(t)i = e−iHt|ψ(0)i = e−iE0t|ψ(0)i =⇒ A(t) = hψ(0)|ψ(t)i = hψ(0)|e−iE0t|ψ(0)i für den exponentiellen Abfall der Amplitude einen komplexwertigen Energie-Eigenwert E0. Ein komplexer Eigenwert kann aber nicht zu einem hermiteschen bzw. selbstadjungierten Hamilton- Operator gehören. Die Weisskopf-Wigner-Approximation ignoriert dieses Problem, wie im weiteren Verlauf zu sehen ist. Allerdings ist es auch möglich, neutrale Kaonen ohne diese Nä- herung zu beschreiben (siehe [Azi95]). Daher ist ein plausibler Ansatz zur Beschreibung eines exponentiellen Zerfalls für Zeiten t > 0 (negative Zeiten würden zu Wahrscheinlichkeitsamplituden größer als Eins führen) durch den Zustandsvektor Γ Γ |ψ(τ)i := exp −i m − i τ |ψ(0)i = exp −im − τ |ψ(0)i 2 2 gegeben, wo m die Masse, Γ die totale Zerfallsrate und τ die Eigenzeit ist. Diese zeitliche Evolution führt zu dem gewünschten exponentiellen Abklingen der Wahrscheinlichkeit, da hψ(τ)|ψ(τ)i = e−Γτ gilt. Dieser Zustandsvektor |ψ(τ)i genügt dann einer effektiven Schrödinger-Gleichung ∂ Γ i |ψ(τ)i = m − i |ψ(τ)i ∂τ 2 Γ mit der komplexen Masse m − i 2 . + − 0 π− π π+ π π0 π ¯ u¯ d u u u¯ d d u¯ u d d¯ d¯ W + W − Z0 d s¯ d¯ s d¯ s K0 K¯ 0 K¯ 0 Abbildung 4: Feynman-Diagramme der Zerfallprozesse K0 → π−π+ und K¯ 0 → π−π+ der neutralen Kaonen durch die schwache Wechselwirkung. 4 2.3 Beschreibung des Kaonen-Zerfallsprozesses Die zeitliche Entwicklung der Kaonen-Oszillation zwischen K0 und K¯ 0 lässt sich durch die Superposition |ψ(t)i := α(t)|K0i + β(t)|K¯ 0i beschreiben. Eine Untersuchung der starken Prozesse der neutralen K-Mesonen zeigt, dass K0 und K¯ 0 pseudoskalare Teilchen sind. Unter Anwendung des Raumspiegelungsoperators verhalten sie sich also gemäß Pˆ|K0i = −|K0i bzw. Pˆ|K¯ 0i = −|K¯ 0i . Über die Ladungskonjugation wird das neutrale Kaon K0 in sein Antiteilchen K¯ 0 überführt. Durch eine passende Phasenwahl gilt Cˆ|K0i = |K¯ 0i bzw. Cˆ|K¯ 0i = |K0i , so dass für die Zusammenschaltung
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