Investigating Violations of Some Fundamental Symmetries of Nature via Dalitz Plots and Dalitz Prisms

Transcription

1 1 Investigating Violations of Some Fundamental Symmetries of Nature via Dalitz Plots and Dalitz Prisms Dibyakrupa Sahoo Department of Physics, and Institute of Physics and Applied Physics, Yonsei University, Seoul, South Korea 13 October 2016

2 2 The take home sketch of the full story Three-body decays Dalitz plots Symmetries CP Phase space Event distribution asymmetries CPT Bose Multi-body processes Amplitude Dalitz prisms SU(3) flavor

3 3 Layout of the talk Various aspects Mathematics Classification Usefulness Symmetry Testing various symmetries Condition Choice of mode Signatures Feasibility Kinematics Decay rate Types of Dalitz plot Usefulness 3-body decay & Dalitz plot Dalitz prism Concept Construction Features Usefulness

4 A gentle introduction to Symmetry in the Physical Laws 4

5 Symmetry is a simple but vast subject to explore. Symmetry q pervades the whole of Nature, q underlies our notions of beauty, aesthestics, harmony and regularity, q has a firm mathematical structure, and q forms the foundation of our theoretical formulation of physical laws. 5

6 6 Symmetry has four vital aspects to it.... a thing is symmetrical if there is something that you can do to it so that after you have finished doing it, it looks the same as it did before. RICHARD P. FEYNMAN, The Character of Physical Laws All considerations of symmetry must include an object whose symmetry is being studied, an operation called symmetry operation/transformation which is applied on the object, an observable which is a measure of the effect of symmetry operation, an observer who assures that the measurements are repeatable and correct, and testifies that the symmetry operation has not affected the object.

7 6 Symmetry has four vital aspects to it.... a thing is symmetrical if there is something that you can do to it so that after you have finished doing it, it looks the same as it did before. RICHARD P. FEYNMAN, The Character of Physical Laws All considerations of symmetry must include Object an object can bewhose any living symmetry or non-living is beingbeing, studied, a mathematical construct, an operation or even called as abstract symmetry as a operation/transformation physical law. which is applied on the object, an observable which is a measure of the effect of symmetry operation, an observer who assures that the measurements are repeatable and correct, and testifies that the symmetry operation has not affected the object.

8 6 Symmetry has four vital aspects to it.... a thing is symmetrical if there is something that you can do to it so that after you have finished doing it, it looks the same as it did before. RICHARD P. FEYNMAN, The Character of Physical Laws All considerations of symmetry must include Object an object can bewhose any living symmetry or non-living is beingbeing, studied, a mathematical construct, an A operation symmetry or even called transformation as abstract symmetry as a operation/transformation is physical a change law. in our point of view which is applied that does on the notobject, change the results of possible experiments. an observable which is a measure of the effect of symmetry STEVEN WEINBERG, operation, The Quantum Theory of Fields, Vol. I Foundations an observer who assures that the measurements are repeatable and correct, and testifies that the symmetry operation has not affected the object.

9 6 Symmetry has four vital aspects to it.... a thing is symmetrical if there is something that you can do to it so that after you have finished doing it, it looks the same as it did before. RICHARD P. FEYNMAN, The Character of Physical Laws All considerations of symmetry must include Object an object can bewhose any living symmetry or non-living is beingbeing, studied, a mathematical construct, an A operation symmetry or even called transformation as abstract symmetry as a operation/transformation is physical a change law. in our point of view which is applied that does on the notobject, change the results of possible experiments. Observable is some quantity that can be experimentally an observable which is a measure of the effect of symmetry measured. STEVEN WEINBERG, operation, The Quantum Theory of Fields, Vol. I Foundations an observer who assures that the measurements are repeatable and correct, and testifies that the symmetry operation has not affected the object.

10 6 Symmetry has four vital aspects to it.... a thing is symmetrical if there is something that you can do to it so that after you have finished doing it, it looks the same as it did before. RICHARD P. FEYNMAN, The Character of Physical Laws All considerations of symmetry must include Object an object can bewhose any living symmetry or non-living is beingbeing, studied, a mathematical construct, an A operation symmetry or even called transformation as abstract symmetry as a operation/transformation is physical a change law. in our point of view which is applied that does on the notobject, change the results of possible experiments. Observable is some quantity that can be experimentally an observable which is a measure of the effect of symmetry measured. STEVEN operation, Observer must be unbiased. When possible, it WEINBERG, is desirable to dispense The the Quantum human Theory aspect with of Fields, actual Vol. quantitative I Foundations measurements (the data). Of course, to interpret the data a hu- an observer who assures that the measurements are repeatable and correct, and testifies that the symmetry operation has not man experimenter is necessarily required. affected the object.

11 7 Symmetry in the physical laws has two views, one is experimental and the other is theoretical. A physical law is said to have a certain symmetry, if all unbiased, competent measurements agree that the experimental observable remains invariant under the appropriate symmetry operation. Experimentalist s point of view A physical law is said to have certain symmetry, if the equation describing the law retains its form (i.e. it is covariant) under the appropriate symmetry transformation. Theorist s point of view

12 7 Symmetry in the physical laws has two views, one is experimental and the other is theoretical. A physical law is said to have a certain symmetry, if all unbiased, competent measurements agree that the experimental observable remains invariant under the appropriate symmetry operation. Experimentalist s point of view The two points of view must match with each other. A physical law is said to have certain symmetry, if the equation describing the law retains its form (i.e. it is covariant) under the appropriate symmetry transformation. Theorist s point of view

13 8 Symmetry operations exhibit the same properties as those of elements of a group. All valid symmetry operations obey the following properties: 1. application of two symmetry operations is also another symmetry operation (i.e. symmetry operations close), 2. application of the same set of three symmetry operations gives the same result whether two symmetry operations follow one symmetry operation, or one symmetry operation follows two symmetry operations, as long as the same sequence is maintained (i.e. symmetry operations are associative), 3. there exists a symmetry operation, called identity transformation, which does not affect anything, 4. for every symmetry operation there exists another symmetry operation, its inverse transformation, which nullifies (or reverses) its effect. It is important to note that the sequence of combination of two symmetry operations is very important, and, in general, the final symmetry operation is different when the sequences are different.

14 8 Symmetry operations exhibit the same properties as those of elements of a group. All valid symmetry These properties operationsare obey the the foundational following properties: basis of group theory. 1. application of two symmetry operations is also another symmetry operation (i.e. symmetry operations close), Closure 2. application of the same set of three symmetry operations gives the same result whether two symmetry operations follow one symmetry operation, or one symmetry operation follows two symmetry operations, as long as the same Associativity sequence is maintained (i.e. symmetry operations are associative), 3. there exists a symmetry operation, called identity Existence of transformation, which does not affect anything, identity element 4. for every symmetry operation there exists another symmetry operation, its inverse transformation, which nullifies (or reverses) its effect. Existence of inverse element It is important to note that the sequence of combination of two symmetry operations is very important, and, in general, the final symmetry operation is different when the sequences are different.

15 8 Symmetry operations exhibit the same properties as those of elements of a group. All valid symmetry These properties operationsare obey the the foundational following properties: basis of group theory. 1. application of two symmetry operations is also another symmetry operation (i.e. symmetry operations close), 2. application of the same set of three symmetry operations gives the same result whether two symmetry operations follow one symmetry operation, or one symmetry operation follows two symmetry operations, as long as the same Associativity sequence is maintained (i.e. symmetry operations are associative), So symmetries are described mathematically in the language of group theory 3. there exists a symmetry operation, called identity Existence of transformation, which does not affect anything, identity element 4. for every symmetry operation there exists another symmetry operation, its inverse transformation, which nullifies (or reverses) its effect. Existence of inverse element It is important to note that the sequence of combination of two symmetry operations is very important, and, in general, the final symmetry operation is different when the sequences are different. Closure

16 9 Symmetries are classified in many ways, based on the properties of the symmetry transformations. internal spacetime gauge symmetry local permutation symmetry global exact/perfect approximate/imperfect broken discrete continuous Symmetry Symmetry transformation explicit spontaneous anomalous doesn t change covariant affects Experimental observable Physical phenomenon Governing equation / Lagrangian variables Mathematical model / theory

17 10 Symmetries of physical laws are based on non-observables leading to many conservation laws or selection rules. Symmetry operation Non-observable Conservation Law / Selection Rule permutation difference between B.E. or F.D. statistics identical particles translation in space absolute space momentum translation in time absolute time energy rotation in space absolute direction angular momentum in space Lorentz transformation absolute velocity generators of Lorentz group space-inversion absolute left/right parity time reversal absolute time reversal past/future particle antiparticle exchange absolute sign of charge conjugation electric charge quantum mechanical relative phase charge phase between quantum states charged leptonneutrino relative phase beton symmetry tween charged lep- and neutrino Symmetry permutation continuous, spacetime discrete, spacetime discrete, internal U(1), gauge lepton number SU(2) L, gauge

18 11 If a symmetry gets violated, there always exists a physical observable which can be measured. If a previously considered non-observable can indeed be observed (due to technological advancement in the experimental sector, or adoption of some new methodology), it can always be used to test the extent of breaking of the underlying symmetry under consideration. The very structure of modern physics is based on the foundation of symmetry considerations which in turn have got enormous support from the mathematical framework of group theory. But the litmus test for any symmetry of Nature is an experimental verification of it.

19 A gentle introduction to Three-body decays and Dalitz plots

20 13 Like any other process, a three-body decay requires both amplitude and phase-space for a complete description. Decay mode: X Decay rate transition amplitude 2 (phase space volume) }{{}}{{} probability of the total volume transition to certain of all possible final final configuration state configurations in the phase space dγ = S 2 2E X 3 d 3 p i i=1 (2π) 3 2E i (2π) 4 δ 4 (p X p 1 p 2 p 3 ), where S = (1/n a!) is permutation symmetry factor a with n a being the number of particles of type a in the final state.

21 14 Kinematics of any three-body decay can be completely described by only two parameters. X Decay plane X 1 3 No. of parameters to describe 12 final state No. of constraints from conservation of 4-momentum 4 on-shell nature 3 rotational symmetry of decay 1 plane orientation of axis of rotational 2 symmetry No. of independent parameters to describe final state = 2.

22 15 Two independent invariant mass squares completely describe the phase space for any three-body decay. Three invariant mass squares possible: m 2 ij = 2 p i + p j, = m 2 ij = m2 X + m2 1 + m2 2 + m2 3 = M2 (say), i,j=1,2,3 i<j = Only two invariant mass squares are independent. Notation à la Mandelstam: s = p X p 1 2 = p2 + p 3 2, t = p X p 2 2 = p1 + p 3 2, u = p X p 3 2 = p1 + p 2 2. Absolute limits on s, t, u: 0 s, t, u M 2. Exchanges: s t u

23 16 The collection of all events contained inside the ternary plot of s, t, u gives the Dalitz plot. 2 s 1 y 0 1 u x 3 3 3s (x, y) (t u), M2 M 1 2 t Phase space for the area of X the Dalitz plot

24 17 The Dalitz plot distribution can be described in terms of barycentric rectangular and polar coordinate systems. 2 s 1 The barycentric rectangular (x, y) and polar (r, θ) coordinates are related to one another by: y 0 1 u θ θ θ x 3 θ = θ + 2π 3 = θ + 4π 3. t s = M2 1 + y 3 t = M2 6 = M2 3 u = M2 6 = M2 3 = M x y 2π 1 + r cos 3 + θ 2 3x y 1 + r cos 1 + r cos θ, 2π 3 θ,.

25 18 The sextants of the Dalitz plot are related to one another via exchange of a pair of the Mandelstam-like variables. s 1 IV t u III s u s t IV V VI III II I V s t II s u u 3 2 t VI t u I

26 19 Distribution of events inside Dalitz plot depends on properties of the amplitude. Phase space s 1 Amplitude s u t Flat distribution of events 3 2 u t Variation in distribution of events Dalitz plot would to be symmetric under exchange of two particles, say 2 and 3, i.e. also under t u, iff particles 2 and 3 are equally massive, and they exist in even partial wave states, i.e. with orbital angular momentum = 0, 2, 4,....

27 20 Dalitz plots and CP violation D.S., Rahul Sinha, N.G. Deshpande and Sandip Pakvasa, PRD 89, (R) (2014).

28 Dalitz plot distribution gets affected by direct CP violation in the initial particle(s). s +0 = (p + + p 0 ) 2 s +0 = s 0 s 0 = (p + p 0 ) 2 Direct CP violation in untagged, neutral B meson decays to certain self-conjugate, hadronic final states (e.g. π + π π 0, K + K π 0, D + D π 0 ), would appear as an asymmetry in the Dalitz plot distribution, across the mirror-line. S. Gardner, PLB 553, (2003). S. Gardner and J. Tandean, PRD 69, (2004).

29 22 Dalitz plot distribution is affected if there is direct CP violation in final particle decays. The central idea If CP is violated in neutral D mesons, it would exhibit its signature on the B K D 0 D 0 Dalitz plot when the neutral D mesons are reconstructed from daughter particles of definite CP. D.S., Rahul Sinha, N.G. Deshpande and Sandip Pakvasa, PRD 89, (R) (2014).

30 23 The neutral D mesons can be described in terms of mass, flavor and CP eigenstates. Flavor eigenstates: D 0 (cū) and D 0 (u c). Mass eigenstates: D 1 and D 2. D1,2 = N1,2 p 1 z D 0 ± q 1 ± z D 0, p, q (in general, complex) lead to CP violation in mixing, z (also complex) leads to CPT violation in mixing, 1 N 1,2 = p 2 + q 2 z p 2 q 2, and no CPT violation z = 0 = p 2 + q 2 = 1 = N 1,2 = 1. CP eigenstates: D ± = 1 2 D 0 ± D 0. No CP violation = p = q = D 1,2 D ±.

31 24 In the final state D 0 and D 0 are entangled. K D 0 f 1 B D 0 t 1 t 2 f 2 Time How to reconstruct D 0 and D 0? Remarks from flavor sensitive modes we can distinguish between D 0 and D 0 from flavor insensitive modes can not distinguish D 0 and D 0 from flavor insensitive modes both D 0 and D 0 are not only indistinguishable, but also completely of same CP Bose symmetric (assuming no CP violation)

32 25 The time evolution of the entangled state is easy to study if we write the state in terms of mass eigenstates. The time evolution of mass eigenstates is given by D1,2 (t) = e iµ 1,2t D1,2 e i(µ± µ)t D1,2, where µ = M i (Γ /2) and µ = (x i y) (Γ /2) with M and Γ being the average mass and decay width of D 1 and D 2, xγ, 2 y Γ being the differences in masses and decay widths of D 1 and D 2 respectively. Experimentally [PDG 2014, HFAG results] x(%) = , y(%) = 0.63 ± 0.09 No CP violation CP violation allowed.

33 26 The decay B(p) K(p 1 ) D 0 (p 2 ) D 0 (p 3 ) is best analysed in the Gottfried-Jackson frame. Before Decay B(p) z Variables à la Mandelstam: s = (p 2 + p 3 ) 2, t = (p 1 + p 3 ) 2 = a + b cos Θ, D 0 (p 2 ) p 2 + p 3 = 0 u = (p 1 + p 2 ) 2 = a b cos Θ, where Θ a = 1 M 2 B 2 + M2 K + 2M2 D s, After Decay K(p 1 ) z b = s 4M 2 D λ(m 2 B, M2 K, s) 2, s D 0 (p 3 ) with λ(x, y, z) being the Källén function λ(x, y, z) = x 2 + y 2 + z 2 2(xy + yz + zx).

34 27 The kinematically allowed region for the traditional Dalitz plot in case of B K D 0 D 0 looks as follows. t u cos Θ cos Θ u (in GeV 2 ) t < u region t = u line t (in GeV 2 ) t > u region cos Θ

35 28 The kinematically allowed region for the triangular Dalitz plot in case of B K D 0 D 0 looks as follows. t = u line s The s, t, u axes run from 0 to 1 such that 1 on any axis corresponds to the value M 2 = M 2 B + M2 K + 2M2 D = s + t + u. Thus the full range 0 s, t, u M 2 is covered t < u region t > u region u t

36 29 The entangled state of D 0 D 0 can be written in terms of mass eigenstates. K(p1 )D 0 (p 2 )D 0 (p 3 ) K D 0 D 0 even = K D 0 D 0 odd 4N 2 1 N2 2 pq, K(p1 )D 0 (p 3 )D 0 (p 2 ) K D 0 D 0 even = + K D 0 D 0 odd 4N 2 1 N2 2 pq. K D 0 D 0 = 1 z N even K D 1(p 2 ) D 1 (p 3 ) N 2 1 K D 2(p 2 ) D 2 (p 3 ) + zn 1 N 2 K D 1 (p 2 ) D 2 (p 3 ) + K D 2 (p 2 ) D 1 (p 3 ), K D 0 D 0 = N odd 1N 2 K D 1 (p 2 ) D 2 (p 3 ) K D 2 (p 2 ) D 1 (p 3 ). K D 0 D 0 even p 2 p 3 == t u + K D 0 D 0 even, K D 0 D 0 odd p 2 p 3 == t u K D 0 D 0 odd.

37 30 Assuming no CPT violation, the state K D 0 D 0 is fully even Bose symmetric. If there is no CPT violation, i.e. z = 0, then K D 0 D 0 even = K D 1(p 2 ) D 1 (p 3 ) K D 2 (p 2 ) D 2 (p 3 ), K D 0 D 0 odd = K D 1(p 2 ) D 2 (p 3 ) K D 2 (p 2 ) D 1 (p 3 ). Thus, in the absence of CPT violation, the state K D 0 D 0 is fully Bose even symmetric. Since p 2 p 3 t u, the Bose symmetry is realised as a symmetry under t u exchange on the Dalitz plot, if and only if there is no CPT violation in the decay. CPT violation also leaves its signature in Dalitz plot. We shall come to this later. But for the time being we shall assume no CPT violation.

38 31 The neutral D mesons when reconstructed from distinct final states of definite CP yield signals of CP violation. Reconstruct the two neutral D mesons from the following set of definite CP final states (denoted by f CP ): CP-even: f + = {K + K, π + π }, CP-odd: f = {K 0 S π0, K 0 S ω, K0 S φ}. Any asymmetry in the Dalitz plot must arise from the K D 0 D 0 odd state and must imply direct CP violation, since it would mean that both D 1 and D 2 both decay to the same CP final states. B K D 0 / D 0 f CP 1 D 0 /D 0 f CP 2 Time

39 32 The direct CP violation can be parametrized in terms of the ε s defined as below. Define the amplitudes for the decay: D ± f CP i (p i+1 ) as follows: Decay D + f + i D f + i D f i D + f i Amplitude A + i ε + i A+ i A i ε i A i Here ε ± i quantifies the amount of direct CP violation in the decays of the neutral D mesons, and i = 1, 2. The amplitudes for D 1,2 f ± are given by Amp(D 1,2 f + i ) = 1 (p ± q) + (p q)ε + i A + i, 2 Amp(D 1,2 f i ) = 1 2 (p q) + (p ± q)ε i A i.

40 33 Amplitudes for B K D 0 D 0 (where how D 0 and D 0 are reconstructed is immaterial) must also be defined. Let us also define the amplitudes for B K D 0 D 0 : Amp(B K D 0 (p 2 ) D 0 (p 3 )) = K D 0 (p 2 ) D 0 (p 3 ) B = A(t, u), Amp(B K D 0 (p 3 ) D 0 (p 2 )) = K D 0 (p 3 ) D 0 (p 2 ) B = A(u, t). Also define the following: A e = 1 2 A(t, u) + A(u, t), Ao = 1 A(t, u) A(u, t). 2 cos Θ Note: Both A e and A o are even under t u.

41 34 Amplitudes for B K D 0 D 0, where D 0 and D 0 are properly reconstructed, are as given below. D(p 2 ) f s 1 1 at time t 1, and D(p 3 ) f s 2 2 at time t 2 with s 1,2 = {+, }. Amp(B K(f + 1 ) D(f + 2 ) D) = e iµ(t 1+t ) 2 A 1 A 2 A e (1 ε +1 ε+2 ) A o(ε +1 ε+2 ) cos Θ, Amp(B K(f + 1 ) D(f 2 ) D) = e iµ(t 1+t ) 2 A 1 A 2 A e (ε +1 ε 2 ) + A o(1 ε +1 ε 2 ) cos Θ, Amp(B K(f 1 ) D(f + 2 ) D) = e iµ(t 1+t ) 2 A 1 A 2 A e (ε 1 ε+2 ) A o(1 ε 1 ε+2 ) cos Θ, Amp(B K(f 1 ) D(f 2 ) D) = e iµ(t 1+t ) 2 A 1 A 2 A e (1 ε 1 ε 2 ) + A o(ε 1 ε 2 ) cos Θ.

42 34 Amplitudes for B K D 0 D 0, where D 0 and D 0 are properly reconstructed, are as given below. D(p 2 ) f s 1 1 at time t 1, and D(p 3 ) f s 2 2 at time t 2 with s 1,2 = {+, }. Amp(B K(f + 1 ) D(f + 2 ) D) = e iµ(t 1+t ) 2 A 1 A 2 A e (1 ε +1 ε+2 ) A o(ε +1 ε+2 ) cos Θ, Amp(B K(f + 1 ) D(f 2 ) D) = e iµ(t 1+t ) 2 A 1 A Here we have neglected 2 the mass and width differences between Athe e (ε +1 D ε 2 ) + A o(1 ε +1 1 and D 2 mesons. ε 2 ) cos Θ, Later Amp(B K(f 1 ) we D(f + 2 ) will discuss the effect of including D) = e iµ(t 1+t ) 2 these differences on A 1 Aour 2 results. A e (ε 1 ε+2 ) A o(1 ε 1 ε+2 ) cos Θ, Amp(B K(f 1 ) D(f 2 ) D) = e iµ(t 1+t ) 2 A 1 A 2 A e (1 ε 1 ε 2 ) + A o(ε 1 ε 2 ) cos Θ.

43 35 Dalitz plot distribution for B K (f + 1 ) D (f + 2 ) D shows asymmetry under t u. The Dalitz plot distribution for the process B K (f + 1 ) D (f + 2 ) D is given by D ++ dγ B K (f + 1 ) D (f + 2 ) D dt du = Br+ 1 Br π 3 M 3 B 1 + ε ε ε + 1 ε+ 2 2 A e 2 + ε + 1 ε+ 2 2 A o 2 cos 2 Θ 2 Re (ε + 1 ε+ 2 ) (1 ε+ 1 ε+ 2 )A e A o cos Θ, where Br + 1 Br(D0 f + 1 ) and Br+ 2 Br(D0 f + 2 ). When f + 1 f + 2, there is no observable asymmetry in the Dalitz plot. So the modes of reconstruction must be different.

44 36 Dalitz plot distribution for B K (f + 1 ) D (f 2 ) D shows asymmetry under t u. The Dalitz plot distribution for the process B K (f + 1 ) D (f 2 ) D is given by D + dγ B K (f + 1 ) D (f 2 ) D dt du = Br+ 1 Br π 3 M 3 B 1 + ε ε 2 2 ε + 1 ε 2 2 A e ε + 1 ε 2 2 A o 2 cos 2 Θ 2 Re (1 ε + 1 ε 2 ) (ε+ 1 ε 2 )A e A o cos Θ, where Br + 1 Br(D0 f + 1 ) and Br 2 Br(D0 f 2 ).

45 37 Dalitz plot distribution for B K (f 1 ) D (f + 2 ) D shows asymmetry under t u. The Dalitz plot distribution for the process B K (f 1 ) D (f + 2 ) D is given by D + dγ B K (f 1 ) D (f + 2 ) D dt du = Br 1 Br π 3 M 3 B 1 + ε ε ε 1 ε+ 2 2 A e ε 1 ε+ 2 2 A o 2 cos 2 Θ 2 Re (1 ε 1 ε+ 2 ) (ε 1 ε+ 2 )A e A o cos Θ, where Br 1 Br(D0 f 1 ) and Br+ 2 Br(D0 f + 2 ).

46 38 Dalitz plot distribution for B K (f 1 ) D (f 2 ) D shows asymmetry under t u. The Dalitz plot distribution for the process B K (f 1 ) D (f 2 ) D is given by D dγ B K (f 1 ) D (f 2 ) D dt du = Br 1 Br π 3 M 3 B 1 + ε ε ε 1 ε 2 2 A e 2 + ε 1 ε 2 2 A o 2 cos 2 Θ 2 Re (ε 1 ε 2 ) (1 ε 1 ε 2 )A e A o cos Θ, where Br 1 Br(D0 f 1 ) and Br 2 Br(D0 f 2 ). When f 1 f 2, there is no observable asymmetry in the Dalitz plot. So the modes of reconstruction must be different.

47 39 Relaxing all assumptions does modify the Dalitz plot distributions by adding more terms. D ++ D + D + D A e 2 1 ε + 1 ε φ 1 (ε + 1, ε+ 2 ) + A o 2 ε + 1 ε+ 2 2 cos 2 Θ 2Re 1 ε + 1 ε+ 2 + ψ 1(ε + 1, ε+ 2 ) ε + 1 ε+ 2 Ae Ao cos Θ, A e 2 ε + 1 ε 2 2 φ 2 (ε + 1, ε 2 ) + A o 2 1 ε + 1 ε 2 2 cos 2 Θ 2Re (ε + 1 ε 2 ψ 2(ε + 1, ε 2 )) 1 ε + 1 ε 2 Ae Ao cos Θ, A e 2 ε 1 ε+ 2 2 φ 2 (ε 1, ε+ 2 ) + A o 2 1 ε 1 ε+ 2 2 cos 2 Θ 2Re (ε 1 ε+ 2 ψ 2(ε 1, ε+ 2 )) 1 ε 1 ε+ 2 Ae Ao cos Θ, A e 2 1 ε 1 ε φ 1 (ε 1, ε 2 ) + A o 2 ε 1 ε 2 2 cos 2 Θ 2Re 1 ε 1 ε 2 + ψ 1(ε 1, ε 2 ) ε 1 ε 2 Ae Ao cos Θ.

48 40 The additional terms are sensitive to CP violation in mixing. The additional terms are proportional to the mass and width differences: φ 1 (ε 1, ε 2 ) = 4 Γ Re i µ f 1 (ε 1, ε 2 ) (1 ε 1 ε 2 ), φ 2 (ε 1, ε 2 ) = 4 Γ Re i µ f 2 (ε 1, ε 2 ) (ε 1 ε 2 ), ψ 1 (ε 1, ε 2 ) = 2 Γ i µ f 1(ε 1, ε 2 ), ψ 2 (ε 1, ε 2 ) = 2 Γ i µ f 2(ε 1, ε 2 ), p f 1 (ε 1, ε 2 ) = q 2 p 2 q 2 z 2 (1 + ε 1 ε 2 ) + (ε 1 + ε 2 ), 2pq 2pq p f 2 (ε 1, ε 2 ) = 1 2 q 2 p 2 + q 2 z 2 (1 + ε 1 ε 2 ) + (ε 1 + ε 2 ). 2pq 2pq

49 41 Neglecting the additional terms in the derivation is justified. In the additional terms we have product of direct CP violating ε s, mixing CP violating terms and µ which has the mass and width differences. Such terms are, therefore, expected to be smaller in comparison to the direct CP violating terms. Hence, we can safely neglect the additional terms at the current level of experimental accuracy.

50 42 The expressions for Dalitz plot distributions can be further simplified. Since the ε s are much smaller than unity, we can safely neglect them when compared to unity, and we can also neglect higher powers of ε s as these are much smaller than ε s themselves. Keeping only those terms that are linear in ε s, we have D ++ = Br+ 1 Br π 3 M 3 B D + = Br+ 1 Br π 3 M 3 B D + = Br 1 Br π 3 M 3 B D = Br 1 Br π 3 M 3 B A e 2 2Re ε + 1 ε+ 2 Ae Ao cos Θ, A o 2 cos 2 Θ 2Re ε + 1 ε 2 Ae A o cos Θ, A o 2 cos 2 Θ 2Re ε 1 ε+ 2 Ae A o cos Θ, A e 2 2Re ε 1 ε 2 Ae Ao cos Θ.

51 43 The Dalitz plot distributions can be clearly separated into an even and an odd part under t u exchange. All the Dalitz distributions can be split into two parts: even E and odd O under t u exchange, D s 1s 2 = E s 1s 2 O s 1s 2. A special case: When f s 1 1 = f s 2 2 (we call this a symmetric case), then D s 1s 1 = E s 1s 1 = (Brs 1 1 )2 256π 3 M 3 A e 2. B Defining E sym = E s 1s 1 and Brsym = Br s 1 1 A e 2 = 256π 3 M 3 B in this case, we have Esym Br 2 sym Using E + and E + one can also extract A o 2..

52 44 Lower limits for differences in the ε s can be obtained from Dalitz distributions. ε + ε + ε ε 1 ε+ 2 1 ε 2 1 ε+ 2 1 ε 2 O ++ 2Br + 1 Br+ 2 O + 2Br + 1 Br 2 O + 2Br 1 Br+ 2 O 2Br 1 Br 2 Brsym Esym Br ± 1 Br 2 E ± Brsym Esym Br ± 1 Br 2 E ± Brsym Esym Br ± 1 Br 2 E ± Brsym Esym Br ± 1 Br 2 E ±,,,.

53 45 The difference in ε s is related to the difference in the usual CP asymmetries A CP. Āf A f s2 2 A s CP,i = i s i Āf A f s2 2 = 2 Re(εs i ) + s 1 + ε s 2 2 Re(εs i ), i i i where A f s = Amp(D 0 f s i i ) and Ā f s = Amp(D 0 f s i i ). A s 1s 2 CP,ij = As 1 CP,i As 2 CP,j 2 Re ε s 1 ε s 2 2 i j ε s 1 ε s 2 i j = ε s 1 ε s 2 i j 1 2 As 1s 2 CP,ij. Thus if we were to look for A CP = A CP (K + K ) A CP (π + π ), we must also look for t u exchange asymmetry in the Dalitz plot for B K (K + K ) D (π + π ) D.

54 46 The parent particle plays no role in the kind of direct CP violation under consideration. Since in the kind of direct CP violation under consideration, the reconstructions of D 0 and D 0 are the only necessary and sufficient conditions to observe the Dalitz plot asymmetry, the parent particle has hardly any role. Therefore, instead of concentrating on a single decay mode, such as B 0 K 0 D 0 D 0, or B + K + D 0 D 0, we can look at non-resonant processes such as e + e K 0 D 0 D 0 also. It is also possible to look for Dalitz plot asymmetry for the decays B ± K ± D 0 D 0 K ± (D 0 π 0 )(D 0 π 0 ), where the D 0 and D 0 have been reconstructed properly. The best strategy is to look for Dalitz plot asymmetry under t u exchange for any final state YD 0 D 0 (where Y can be one or many particle(s)) at any center-of-momentum energy.

55 47 This new search for direct CP violation in neutral D mesons is data rich. Decay modes: X Y(p 1 ) D 0 (p 2 ) D 0 (p 3 ), where X, Y can be many things. This means the final Dalitz plot is a superposition of many individual Dalitz plots. s Y IV III V II VI I Reconstruct D 0 and D 0 from distinct final states of definite CP, e.g. D 0 (p 2 ) K + K and D 0 (p 3 ) π + π. The freedom over the parent particle and the choice for the third particle in final state, render the method very powerful. u D 0 D 0 t

56 48 Dalitz plots and testing Bose symmetry D.S., Rahul Sinha and N.G. Deshpande, PRD 91, (R) (2015)

57 49 If two final particles are fully Bose symmetric, the Dalitz plot must be symmetric under their exchange. s 1 Particles 2 and 3 identical to one another (but reconstructed from distinct final states), e.g. K +, D +, D + s u V IV VI III II 3 2 I t π + (p 1 ) }{{} µ + ν µ π 0 (p 2 ) }{{} e + e γ π 0 (p 3 ) }{{} γγ = the Dalitz plot must be left-right symmetric. All particles identical (but two are reconstructed from distinct final states), e.g. B 0 K 0 S (p 0 1) KS }{{} (p 0 2) KS }{{} (p 3) }{{} π + π π + π π 0 π 0 = half of the Dalitz plot can be reconstructed. The three sextants of that half must be symmetrical to one another.

58 50 Dalitz plot asymmetry under t u exchange would probe Bose symmetric nature of particles 2 and 3. The amplitude A(r, θ) can be written in terms of a Fourier series: A(r, θ) = n=0 S n (r) sin(nθ) + C n (r) cos(nθ), where S n (r) and C n (r) are Fourier coefficients. The amplitude can also be denoted as follows: A(t, u) A(r, θ) and A(u, t) A(r, θ). Bose symmetry between particles 2 and 3 would imply that A(t, u) = A(u, t). Breaking of Bose symmetry would imply that the amplitude can be written as A(t, u) = A S + A N, where A S = 1 2 A(t, u) + A(u, t) and A N = 1 2 A(t, u) A(u, t). And A N leads to Dalitz plot asymmetry under t u exchange.

59 51 Dalitz plots and CPT violation D.S., Rahul Sinha and N.G. Deshpande, PRD 91, (R) (2015)

60 52 Dalitz plot distribution is also affected by CPT violation. X is a self-conjugate process with no CP violation, i.e. it occurs via strong or electromagnetic interactions. Moreover, 2 and 3 are CP conjugate of one another. Amplitude: A(r, θ) = s n (r) sin(nθ) + c n (r) cos(nθ), where the n=0 Fourier coefficients s n (r) and c n (r) are complex, in general. Under CPT: θ θ, s n (r) s n (r) and c n(r) c n (r). When CPT is exact: A(r, θ) = A (r, θ). If both CP and CPT were exact, s n (r) = 0 and c n (r) = real. This says that the Dalitz plot is symmetric under θ θ. However, if CP is exact, but not CPT, then there must be an observable asymmetry under θ θ.

61 53 CPT violation can be parametrized as shown below. The amplitude Ā(r, θ) for the CP conjugate process, assuming CPT violation, is given by Ā(r, θ) = n=0 s n (r) sin(nθ) + c n (r) cos(nθ), where s n (r) and c n (r) necessarily differ from s n (r) and c n (r): s n (r) = ( s n (r) + ε s n (r))eiδs n, sn (r) = ( s n (r) ε s n (r))eiδs n, c n (r) = ( c n (r) + ε c n (r))eiδc n, cn (r) = ( c n (r) ε c n (r))eiδc n, with δ s,c n being strong phases and εs,c n (r) being CPT violating parameters. No explicit weak phase dependence is shown as we assume CP to be conserved.

62 54 CPT violation gives rise to an observable asymmetry in Dalitz plot distribution. Since in our case the process and its CP conjugate process are the same, the amplitude which comes into picture is the average of both A(r, θ) and Ā(r, θ): A = 1 2 = n=0 A(r, θ) + Ā(r, θ) ε s n (r) sin(nθ)eiδs n + cn (r) cos(nθ)e iδc n. In the Dalitz plot distribution, which is proportional to A 2, the term which is odd under θ θ is proportional to n,m=0 c n (r) ε s m (r) cos δ c n δs m cos(nθ) sin(mθ). This term survives only if CPT is violated and leads to an asymmetry in Dalitz plot under θ θ t u.

63 55 Three-body decays via strong interaction are ideal for study of CPT violation. Decay mode: J/ψ Nπ + π, where N can be π 0, ω, η or φ. s N IV III V II VI I Amplitude and the Dalitz plot distribution have one part even and another part odd under t u. The odd part exists if and only if there is any direct CPT violation contribution in the decay. The odd part leads to a left-right asymmetry in the Dalitz plot. π π + u t

64 56 Introduction to the Dalitz prism D.S., Rahul Sinha and N.G. Deshpande, PRD 91, (R) (2015)

65 57 When details of the parent particle do not matter, Dalitz plot can evolve into a Dalitz prism. Our discussion is unaffected by what the parent particle X is. X can be replaced by anything leading to the same final states and with 4-momentum precisely known, e.g. e + e. m X = the center-of-momentum mass = it can vary continuously = plenty of Dalitz plots in the way. Pile up these Dalitz plots with increasing m X to get the Dalitz prism.

66 58 The Dalitz prism can handle gargantuan amount of data enabling precise measurements of the violations. t II Dalitz prism (hypothetical) I VI u mx V IV III s Very precise measurements essential to study violations of CP, CPT and Bose symmetries require analysis of a huge number of events. Dalitz prism combines data from the continuum with data from many resonances. This enhances the statistics immensely. u z We just need the projection of the Dalitz prism at its base to do our analysis. y s t x

67 59 The Dalitz prism helps in considering multi-body data. Treating a multi-body decay as an effective three-body decay we can construct a Dalitz prism, e.g. J/ψ Nπ + π where N can be K + K, π 0 K + K, ηk + K, ωπ 0, p p, p pπ 0, n n. Dalitz prism is helpful even when initial state radiation (ISR) or final state radiation (FSR) are present. In case of both ISR and FSR, the 4-momenta of the two particles whose exchanges are most important and the total initial 4-momentum need to be precisely known, in order to make an entry into the Dalitz prism. When events with ISR and FSR enter the Dalitz prism, its horizontal slices no longer provide Dalitz plots, as not all the events correspond to only three-body processes.

68 60 Dalitz plots and violation of SU(3) flavor symmetry D.S., Rahul Sinha and N.G. Deshpande, PRD 91, (2015)

69 61 We concentrate on the SU(3) meson octet of light pseudo-scalars to study SU(3) flavor symmetry breaking. The SU(3) flavor symmetry subsumes three non-commuting SU(2) symmetries: Isospin (or T-spin), U-spin, V-spin. U-spin V-spin +1 Y 0 1 π K 0 K + π 0 η 0 K K 0 T 3 π + Isospin We shall study Dalitz plots of those three-body decays of which two pairs of the final particles are related by two different SU(2) symmetries, e.g. K 0 π 0 π +, K + π 0 π, K + π 0 K 0, π + π 0 K

70 62 Implementation of two SU(2) symmetries simultaneously dictates the behaviour of the decay amplitude. Decay mode: X Particles 1 and 2 are related by one SU(2) symmetry, particles 2 and 3 are related by another SU(2) symmetry. By explicitly working out the decomposition of the final state under each of the SU(2) symmetries, one can show that the decay amplitude would be either symmetric or anti-symmetric under both 1 2 s t and 2 3 t u exchanges. Thus the amplitude can be described by four independent functions: Exchanges s t t u Amplitude symmetric symmetric SS (s, t, u) symmetric anti-symmetric SA (s, t, u) anti-symmetric symmetric AS (s, t, u) anti-symmetric anti-symmetric AA (s, t, u)

71 63 Only the fully symmetric and the fully anti-symmetric amplitudes are allowed. SS (s, t, u) is fully symmetric under s t u: SS (s, t, u) s t == SS (t, s, u) t u === SS (u, s, t) s t == SS (u, t, s). AA (s, t, u) is fully anti-symmetric under s t u: AA (s, t, u) s t == AA (t, s, u) t u === + AA (u, s, t) s t == AA (u, t, s). SA (s, t, u) is identically zero: SA (s, t, u) s t == SA (t, s, u) t u === SA (u, s, t) s t == SA (u, t, s) t u === + SA (t, u, s) s t == + SA (s, u, t) t u === SA (s, t, u) = 0. AS (s, t, u) is identically zero: AS (s, t, u) s t == AS (t, s, u) t u === AS (u, s, t) s t == + AS (u, t, s) t u === + AS (t, u, s) s t == AS (s, u, t) t u === AS (s, t, u) = 0.

72 64 Implementation of two SU(2) symmetries dictates the Dalitz plot distribution. The decay amplitude, after implementing the two SU(2) symmetries, is given by (s, t, u) = SS (s, t, u) + AA (s, t, u). Since the Dalitz distribution is proportional to (s, t, u) 2, the distribution function has a part fully symmetric and another fully anti-symmetric under s t u exchange: f S (s, t, u) SS (s, t, u) 2 + AA (s, t, u) 2, f A (s, t, u) 2 Re SS (s, t, u) AA (s, t, u). Thus the various sextants of the Dalitz plot have a characteristic alternate distribution pattern: f I = f III = f V = f S (s, t, u) + f A (s, t, u), f II = f IV = f VI = f S (s, t, u) f A (s, t, u).

73 65 The concept of G-parity can be generalized to U-spin and V-spin cases. The G-parity operator G I (or G U or G V ) is defined as a rotation through π radian (180 ) around the second axis of isospin (or U-spin or V-spin) space, followed by charge conjugation ( ): G I = e iπi 2 = e iπτ 2 /2, where I 2 is the second generator of SU(2) isospin (or U-spin or V-spin) group, and τ 2 is the second Pauli matrix. G I G U G V π + π 0 π = π + π 0 π, G I K + K 0 K 0 K 0 π 0 = π 0 K +, G U K 0 K 0 π + K + K + π 0 = π 0 π +, G V K K K 0 = K 0 K 0 K + = K, G I K = K 0, π π K + = K, G U K = π +, K 0 K 0 π + π, G V π = K 0,

74 66 Generalized G-parity consideration can make the Dalitz plot distribution fully symmetric. In addition to the two exchanges 1 2 and 2 3, if generalized G-parity can be used to write the state 1 3 in terms of G-parity even and G-parity odd states, then the Dalitz plot distribution would be fully symmetric, as G-parity even and G-parity odd amplitudes do not interfere. In all other cases the alternate sextants of the Dalitz plot would have identical distributions. Any deviation from this would constitute observation of SU(3) flavor violation.

75 67 SU(3) flavor symmetry violation can be quantified by using some Dalitz plot asymmetries. We define two more quantities: Σ i j (r, θ) = f i + f j, i j (r, θ) = f i f j, where i and j are two sextants and i j. In terms of these we can define the following asymmetries that quantify the extent of SU(3) flavor symmetry breaking: Σ 1 = I VI ΣIII IV Σ I VI + + Σ III IV ΣV II ΣIII IV Σ III IV + + Σ V II ΣI VI ΣV II Σ V II + + I VI III IV ΣI VI I VI + + III IV V II III IV III IV + + V II I VI V II V II +, I VI Σ 2 = V IV ΣI II Σ V IV + + Σ I II ΣIII VI ΣI II Σ I II + + Σ III VI ΣV IV ΣIII VI Σ III VI + + V IV I II ΣV IV V IV + + I II III VI I II I II + + III VI V IV III VI III VI +, V IV Σ 3 = I IV ΣIII II Σ I IV + + Σ III II Σ V VI ΣIII II Σ III II + Σ V + Σ V VI ΣI IV VI Σ V VI + + I IV III II ΣI IV I IV + + III II V VI III II III II + V + V VI I IV VI V VI +. I IV

76 Summary 68

77 69 Summary of the decay modes under consideration: Symmetry X What is particle 1? Important criteria [Reference] CP X Y D 0 D 0. Y = {K, π}. Both D 0 & D 0 be reconstructed from distinct final states having definite CP. [PRD 89, (R) (2014)] CPT X Yπ + π. Y = {π 0, ω, η, φ, K + K, π 0 K + K, ηk + K, ωπ 0, p p, p pπ 0, n n}. Bose X Yπ 0 π 0, Yπ + π +, Y K S K S. SU(3) flavor {B +, D + s } K0 π 0 π +, {B 0 d, B 0 s } K+ π 0 π, {B +, D + } K + π 0 K 0, D + π + π 0 K 0. Y is different from the other two particles. Self-conjugate process with no weak interaction. [PRD 91, (R) (2015)] Particles 2 & 3 be reconstructed from distinct final states. [PRD 91, (R) (2015)] Pairs of final particles, (12) and (23), belong to two different SU(2) symmetries. [PRD 91, (2015)] Here X can be a suitable meson or even e + e, and Y can include both initial state radiation and final state radiation.

78 70 Conclusion The Dalitz plot and the new concept of Dalitz prism, provide a unified and powerful method to study violations of CP, CPT and Bose symmetries. Dalitz plots can, also, be used profitably for better estimation of the extent of breaking of the SU(3) flavor symmetry.

79 Investigating Violations of Some Fundamental Symmetries of Nature via Dalitz Plots and Dalitz Prisms Thank you. 71