Absorptive and Dispersive CP violation in D 0 D 0 mixing Alex Kagan - PowerPoint PPT Presentation

Absorptive and Dispersive CP violation in D 0 − D 0 mixing Alex Kagan University of Cincinnati Based on Yuval Grossman, A. K., Zoltan Ligeti, Gilad Perez, Alexey Petrov, Luca Silvestrini in preparation

Plan Introduction Formalism for absorptive and dispersive CPV in D 0 − D 0 mixing Indirect CPV phenomenology U-spin and Approximate Universality How large is indirect CPV in the SM? How large is the current window for New Physics (NP)? Prospects for measuring SM indirect CPV

Introduction In the SM, CP violation (CPV) in D 0 − D 0 mixing and D decays enters at O ( V cb V ub /V cs V us ) ∼ 10 − 3 , due to weak phase γ , yielding all 3 types of CPV: direct CPV (dCPV) CPV in pure mixing (CPVMIX): due to interference between dispersive and absorptive mixing amps CPV in the interference of decays with and without mixing (CPVINT) Primary interest is in CPVMIX and CPVINT, both of which result from mixing, and which we refer to as “indirect CPV" upper bounds suggest dCPV is already in the SM QCD “brown muck"

We are interested in the following questions: How large are the indirect CP asymmetries in the SM? What is the appropriate minimal parametrization of indirect CPV? How large is the current window for new physics (NP)? Can this window be closed at HL-LHCb and Belle-II? To answer, we develop the description of CPVINT in terms of generally final state dependent dispersive and absorptive CPV phases φ M and φ Γ f for CP conjugate final f states f , ¯ f . - introduced by Branco, Lavoura, Silva (’99) f parametrize CPVINT contributions from interference of D 0 decays with φ M and φ Γ f and without dispersive mixing, and with and without absorptive mixing These are separately measurable CPV effects

NP is most likely to appear in dispersive short distance mixing amplitudes SM dispersive and absorptive mixing amplitudes are due to long distance off-shell and on-shell intermediate states. subleading O ( V cb V ub /V cs V us ) decay amplitudes ∝ e iγ yield indirect CPV can not currently be calculated from first principles QCD f , φ Γ meaningful SM estimates of φ M f can be made using SU (3) F flavor symmetry arguments, yielding a minimal parametrization of indirect CPV: approximate universality estimates of the SM indirect CP asymmetries

Formalism time-evolution of linear combination a | D 0 � + b | D 0 � follows from Schrodinger equation,        a  a  a i d  = H  ≡ ( M − i  . 2 Γ) dt b b b transition amplitudes � D 0 | H | D 0 � = M 12 − i 12 − i � D 0 | H | D 0 � = M ∗ 2Γ ∗ 2 Γ 12 , 12 M 12 is the dispersive mixing amplitude Γ 12 is the absorptive mixing amplitude 0 � : Mass eigenstates | D 1 , 2 � = p | D 0 � ± q | D mass and width differences expressed in terms of x , y x = m 2 − m 1 y = Γ 2 − Γ 1 , Γ D 2Γ D

introduce three“theoretical" physical mixing parameters x 12 ≡ 2 | M 12 | / Γ D , y 12 ≡ | Γ 12 | / Γ D , φ 12 ≡ arg( M 12 / Γ 12 ) φ 12 is the CPV phase responsible for CPVMIX CP conserving observables: | x | = x 12 + O (CPV 2 ) , | y | = y 12 + O (CPV 2 ) ( D 0 (0) = D 0 , D 0 (0) = D 0 ) Time-evolved meson solutions, for t � τ D : e.g. mixed components � M D − i Γ D � � D 0 | D 0 ( t ) � = e − i t � � e − iδ M M ∗ 12 − 1 2 Γ ∗ t, ... 2 12 δ M = π/ 2 is a CP-even “dispersive strong phase", originating from the time derivative. Contributes to strong phase differences in indirect CPV

The dispersive and absorptive CPV phases φ M f , φ Γ f CPVINT observables for CP -conjugate final states f = ¯ f : � � � � A f A f A f A f M 12 Γ 12 � e i φ M � e i φ Γ � � � � f , λ Γ f . λ M = η CP = η CP ≡ f ≡ � � � � f f f � � � � | M 12 | | Γ 12 | A f A f A f A f � � A f = � f |H| ¯ ¯ with the decay amplitudes A f = � f |H| D 0 � , D 0 � pairs of CPVINT observables for non- CP conjugate final states f � = ¯ f : � � � � A f A f A f A f M 12 Γ 12 � e i ( φ M f − ∆ f ) , � e i ( φ Γ � � � � f − ∆ f ) λ M λ Γ ≡ = f ≡ = � � � � f | M 12 | � � | Γ 12 | � � A f A f A f A f � � , � � � � A ¯ A ¯ A ¯ A ¯ M 12 Γ 12 � � � e i ( φ M � � e i ( φ Γ � f +∆ f ) . f f f f f +∆ f ) λ M λ Γ ≡ = f ≡ = � � � � ¯ ¯ f | M 12 | � � | Γ 12 | � � A ¯ A ¯ A ¯ A ¯ f � f f � f ∆ f is the strong phase difference between A f and A f , and between A ¯ f and A ¯ f φ 12 = arg( M 12 / Γ 12 ) = φ f M − φ f relation to the CPVMIX phase: Γ

Hadronic D 0 ( t ) and D 0 ( t ) decay amplitudes sum over contributions with and without mixing, A ( D 0 ( t ) → f ) = � D 0 | D 0 ( t ) � ¯ A f + � D 0 | D 0 ( t ) � A f , A ( D 0 ( t ) → f ) = � D 0 | D 0 ( t ) � A f + � D 0 | D 0 ( t ) � ¯ A f . f , λ Γ Time-dependent decay rates given in terms of CPVINT observables λ M f, ¯ f, ¯ f φ M 12 and φ Γ 12 are the CPV phase differences between mixed and unmixed decay amps The strong phase differences are sum of δ M = π/ 2 and ± ∆ f (dispersive mixing), ± ∆ f (absorptive mixing)

In SM Cabibbo favored/ doubly Cabibbo suppressed decays (CF/DCS) the CPVINT phases are universal, e.g. D 0 → Kπ, K ∗ π ,... φ M cfds ≡ φ M φ Γ cfds ≡ φ Γ f , f , f ∈ CF / DCS also true under the well motivated assumption that CF/DCS decays do not contain NP weak phases, NP with non-negligible direct CPV in DCS/CF decays, which evades ǫ K bounds, must be very exotic or tuned Bergmann, Nir In SM singly Cabibbo suppressed decays (SCS), e.g. D 0 → π + π − , K + K − ,... the CPVINT phases have small final state dependence due to the subleading QCD penguin decay amplitudes.

The more familiar general CPV observables � � q � � CPVMIX : � − 1 � � p � � � A f q , for f = ¯ CPVINT : φ λ f = arg f p A f Relation to absorptive and dispersive CPVINT phases ( φ 12 = φ M f − φ Γ f ) � � q � − 1 = x 12 y 12 sin φ 12 � � [1 + O (sin φ 12 )] � � x 2 12 + y 2 p � 12 x 2 f + y 2 12 sin 2 φ Γ � 12 sin 2 φ M � f tan 2 φ λ f = − . x 2 f + y 2 12 cos 2 φ Γ 12 cos 2 φ M f same number of CPV quantities in each description

Indirect CPV phenomenology CPV requires non-trivial CPV “weak phase" differences ( φ w ) and CP conserving “strong phase" differences ( δ s ) between interfering amplitudes ⇒ CP asymmetries ∝ sin δ s sin φ w this dependence is manifest in the absorptive/dispersive CPV phase formalism Examples: The CPVMIX “wrong sign" semileptonic CP asymmetry a SL ≡ Γ( D 0 ( t ) → ℓ − X ) − Γ( D 0 ( t ) → ℓ + X ) , Γ( D 0 ( t ) → ℓ − X ) + Γ( D 0 ( t ) → ℓ + X ) 2 x 12 y 12 sin δ M sin φ 12 = x 2 12 + y 2 12 − 2 cos φ 12 cos δ M = 2 x 12 y 12 sin φ 12 . x 2 12 + y 2 12 note the importance of the dispersive “strong phase" δ M = π/ 2

time-dependent CP asymmetries in SCS decays to CP conjugate final states ( f = ¯ f ), e.g. D 0 → K + K − , π + π − to good approximation, the decay widths take the exponential forms Γ( D 0 ( t ) → f ) = | A f | 2 exp[ − ˆ A f | 2 exp[ − ˆ Γ( D 0 ( t ) → f ) = | ¯ Γ D 0 → f τ ] , Γ D 0 → f τ ] Γ D 0 → f − ˆ ˆ Γ D 0 → f CP asymmetry : ∆ Y f ≡ 2 f sin δ M + a d = η f CP ( − x 12 sin φ M f y 12 cos φ Γ f ) = η f CP ( − x 12 sin φ M + a d f y 12 ) , f ∆ Y f depends on φ M f , but not φ Γ f : for f = ¯ f , no strong phase difference between A f , A f . Thus, the only available CP-even strong phase is δ M = π/ 2 ⇒ asymmetry purely dispersive in origin! up to subleading dCPV contribution (second term), where � ¯ � = − 2 r f sin δ f sin φ f . a d � � f = 1 − A f /A f

f , e.g. D 0 → K ± π ∓ : the wrong sign D 0 ( t ) → f and CF/DCS decays for f � = ¯ D 0 → f decay widths expressed as � � � f ) = e − τ | A f | 2 Γ( D 0 ( t ) → ¯ R + R + f c + f τ + c ′ + f τ 2 f + , � � � Γ( D 0 ( t ) → f ) = e − τ | ¯ R − R − f c − f τ + c ′− f | 2 f τ 2 A ¯ f + , and R ± f are the DCS to CF ratios R + R − f = | ¯ A f / ¯ f /A f | 2 , f | 2 f = | A ¯ A ¯ linear time dependence yields the CPVINT asymmetry (assuming no NP weak phases in CF/DCS) ∆ c f = x 12 sin φ M cfds cos ∆ f − y 12 sin φ Γ cfds sin ∆ f the cos ∆ f and sin ∆ f dependence originates from the strong phase differences ∆ f − δ M (dispersive), and ∆ f (absorptive)

U-spin decomposition and Approximate Universality using CKM unitarity ( λ i = V ci V ∗ ui ) Γ 3 + λ 2 λ i λ j Γ ij = ( λ s − λ d ) 2 Γ 5 + ( λ s − λ d ) λ b � b Γ 12 = − 4 Γ 1 4 2 i,j = d,s M 3 + λ 2 λ i λ j Γ ij = ( λ s − λ d ) 2 M 5 + ( λ s − λ d ) λ b � b M 12 = − 4 M 1 4 2 i,j = d,s the Γ i , M i are ∆ U 3 = 0 elements of U-spin multiplets. They enter at different orders in ǫ , which characterizes SU (3) F breaking. Nominally, ǫ = O (0 . 2) . dd ) 2 ⇒ ∆ U = 2 (5 plet) ⇒ O ( ǫ 2 ) , CF / DCS / SCS ss − ¯ Γ 5 = Γ ss +Γ dd − 2Γ sd ∼ (¯ ss − ¯ ss + ¯ Γ 3 = Γ ss − Γ dd ∼ (¯ dd )(¯ dd ) ⇒ ∆ U = 1 (3 plet) ⇒ O ( ǫ ) , SCS Γ 5 , M 5 dominate and yield ∆ M , ∆Γ , or y 12 , x 12 δ Γ 12 ∝ Γ 3 , δM 12 ∝ M 3 ⇒ CPV via γ = arg( λ b ) neglect O ( λ 2 b ) effects of Γ 1 , M 1