Using time-dependent indirect CP asymmetries to measure T and CPT violation in B0–B¯0 mixing

Physics Letters B 781 (2018) 459–463

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Physics Letters B www.elsevier.com/locate/physletb

Using time-dependent indirect CP asymmetries to measure T and CP T violation in B 0 – B¯ 0 mixing Anirban Karan a , Abinash Kumar Nayak a , Rahul Sinha a , David London b,∗ a b

The Institute of Mathematical Sciences, HBNI, Taramani, Chennai 600113, India Physique des Particules, Université de Montréal, C.P. 6128, succ. centre-ville, Montréal, QC, H3C 3J7, Canada

a r t i c l e

i n f o

Article history: Received 14 December 2017 Received in revised form 21 March 2018 Accepted 14 April 2018 Available online 17 April 2018 Editor: B. Grinstein

a b s t r a c t Quantum field theory, which is the basis for all of particle physics, requires that all processes respect CP T invariance. It is therefore of paramount importance to test the validity of CP T conservation. In this Letter, we show that the time-dependent, indirect CP asymmetries involving B decays to a CP eigenstate contain enough information to measure T and CP T violation in B 0 – B¯ 0 mixing, in addition to the standard C P -violating weak phases. Entangled B 0 B¯ 0 states are not required (so that this analysis can be carried out at LHCb, as well as at the B factories), penguin pollution need not be neglected, and the measurements can be made using B d0 or B 0s mesons. © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

CP T invariance is one of the fundamental principles of quantum field theory: all physical processes are expected to respect this symmetry. Indeed, CP T violation would have a profound impact on physics in general, as it would also lead to a violation of Lorentz symmetry [1,2]. Given its importance to the theoretical framework underlying all of particle physics, much attention has been devoted to experimentally testing the validity of CP T invariance. One of the consequences of CP T invariance in quantum field theory is that a particle and its antiparticle should have the same mass and lifetime. However, these quantities are mostly dominated by the strong or electromagnetic interactions. Given that CP T violation, if nonzero, is certainly a very small effect, it is very difficult to test it by measuring the differences of masses or lifetimes. A more promising area for testing CP T violation is in P 0 – P¯ 0 mixing, where P 0 is a neutral pseudoscalar meson (e.g., K 0 , D 0 , B d0 , B 0s ) [3]. Since this mixing is a second-order electroweak process, small C P T -violating effects may be easier to detect. Now, it is known that, in addition to incorporating CP T violation, the most general B 0 – B¯ 0 mixing matrix also involves T and CP violation. As a consequence, studying CP T violation is impossible without discriminating it from the effects of CP and T violation. That is, the effects of C P , T and CP T violation must be considered together.

*

Corresponding author. E-mail addresses: [email protected] (A. Karan), [email protected] (A.K. Nayak), [email protected] (R. Sinha), [email protected] (D. London).

A first step was taken in Refs. [4,5], with followup papers in Refs. [6–11]. The proposed method uses entangled B 0 B¯ 0 states produced in the decay of the ϒ(4S ), with one meson decaying to a CP eigenstate ( J /ψ K S or J /ψ K L ) and the other used to tag the flavor. Using this technique, true T - and C P T -violating asymmetries can be measured. The BaBar Collaboration implemented this strategy [12,13], culminating in the measurement of T violation [14]. At present, all experimental results are consistent with CP T conservation. On the other hand, an important improvement in statistics is expected at LHCb and Belle II, so that it will be possible to measure the C P -, T - and C P T -violating parameters with greater precision. However, the method using entangled states produced in the decay of the ϒ(4S ) cannot be used at LHCb. An alternate approach is needed. In this Letter, we re-examine the possibilities for measuring T and CP T violation in B 0 – B¯ 0 mixing using the decays of B 0 or B¯ 0 to a CP eigenstate. As we will show, the timedependent indirect CP asymmetry contains sufficient information to measure the conventional C P -violating effects and extract the T - and C P T -violating parameters [15,16]. Since no true T - and C P T -violating asymmetries are measured, this is an indirect determination of the T - and C P T -violating parameters. In this sense, this method is complementary to that using entangled states. We focus on B d0 – B¯ d0 mixing but the same approach can be modified and applied to the B 0s system. Note that we restrict the analysis to T and CP T violation arising from the B 0 – B¯ 0 mixing matrix alone. If there are new-physics contributions to B decays, we assume they are C P T -conserving.

https://doi.org/10.1016/j.physletb.2018.04.029 0370-2693/© 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

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A. Karan et al. / Physics Letters B 781 (2018) 459–463

We begin by reviewing the most general formalism for B 0 – B¯ 0 mixing, in which CP T and T violation are incorporated. The 2 × 2 hermitian matrices M and , respectively the mass and decay matrices, are defined in the ( B 0 , B¯ 0 ) basis. When M − (i /2) is diagonalized, its eigenstates are the physical light (L) and heavy (H ) states B L and B H . Now, any 2 × 2 matrix can be expanded in terms of the three Pauli matrices σi and the unit matrix with complex coefficients:

M−

i 2

 = E 1 σ1 + E 2 σ2 + E 3 σ3 − i D1 .

(1)

Comparing both sides of this equation, we obtain

E 1 = Re M 12 −

i 2

E 2 = −Im M 12 + 1

E3 =

2 i

D=

2

Re 12 , i 2

Im 12 , 4 1

( M 11 + M 22 ) + (11 + 22 ) .

(2)

4



E 12 + E 22 + E 32 ,

E 2 = E sin θ sin φ , E 3 = E cos θ .

(3)

The eigenvalues of M − (i /2) are E − i D and − E − i D, with eigenstates

| B L  = p 1 | B 0  + q1 | B¯ 0  , | B H  = p 2 | B 0  − q2 | B¯ 0  ,

(4)

where p 1 = cos 2 , q1 = e sin 2 , p 2 = sin 2 and q2 = e cos 2 . In Ref. [17] (pgs. 349–358), T.D. Lee discusses the CP T and T properties of M and . First, if CP T invariance holds, then, independently of T symmetry, θ

iφ

θ

θ

M 11 = M 22 , 11 = 22 =⇒ E 3 = 0 =⇒ θ =

iφ

π 2

.

θ

(5)

In addition, if T invariance holds, then, independently of CP T symmetry,

∗12 M∗ = 12 =⇒ Im( E 1 E 2∗ ) = 0 =⇒ Im φ = 0 . 12 M 12

(6)

From Eqs. (2) and (3), we have

 e

iφ

=±

∗ − i ∗ /2 M 12 12

M 12 − i 12 /2

.

(7)

Thus, if T is a good symmetry, the left-hand quantity is a pure phase, and the modulus of the square root is one. Note that it is usually said that the absence of CP violation implies |e i φ | = 1. However, strictly speaking, this is due to the absence of T violation. The two reasons are equivalent only if CP T is conserved. Second, defining ζ ≡  S | L  =  B L | B H ,



ζ is

real

if CP T holds,

imaginary

if T holds.

Using Eq. (4), we have

(8)

π 2

, θ2 = 0 , φ1 = −2β mix , φ2 = 0 ,

(10)

where β mix is the weak phase describing B 0 – B¯ 0 mixing. In the standard model, β mix = β for the B d0 meson. Now, if T and CP T violation are present in the mixing, the parameters θ1 , θ2 and φ2 will deviate from these values. We define 1,2,3 via

θ1 →

E 1 = E sin θ cos φ ,

(9)

Now, if CP T is a good symmetry, then θ = π2 [Eq. (5)], so that sinh{Im(θ)} = 0 and  B L | B H  is real. And if T is a good symmetry, then Im(φ) = 0 [Eq. (6)], so that (1 − e −2Im(φ) ) = 0 and  B L | B H  is imaginary. With this, we see that Eq. (8) is verified. (Obviously, if both CP T and T are good symmetries,  B L | B H  = 0, i.e., the states are orthogonal.) It will be useful to define the complex θ and φ in terms of real parameters as θ = θ1 + i θ2 and φ = φ1 + i φ2 . In the absence of both T and CP T violation in B 0 – B¯ 0 mixing, the parameters take the following values:

θ1 =

i

( M 11 − M 22 ) − (11 − 22 ) ,

We can define complex numbers E, θ and φ as follows:

E=

θ θ∗ θ θ∗ ∗  B L | B H  = cos sin − e i φ sin e −i φ cos 2 2 2 2 1 = (1 − e −2Im(φ) ) sin{Re(θ)} 2  −i {1 + e −2Im(φ) } sinh{Im(θ)} .

π 2

+ 1 , θ2 → 2 , φ2 → 3 .

(11)

1 and 2 are C P T -violating parameters, whereas 3 indicates T violation. The values for 1 , 2 and 3 have been reported by the BaBar and Belle Collaborations [18,19]. Their notation is related to ours as follows: cos θ ↔ − z , sin θ ↔ so that



1 − z2 , e i φ ↔

q p

,

(12)

 

q 1 = Re(z) , 2 = Im(z) , 3 = 1 −   . p

(13)

1 and 2 are expected to be very small, as they are C P T -violating parameters. As for 3 , note that |q/ p | has been measured at the ϒ(4S ) using the same-sign dilepton asymmetry, assuming CP T conservation [20]:

q     = 1.0010 ± 0.0008 =⇒ 3 = −(1.0 ± 0.8) × 10−3 . (14) p

Thus, 3 is also very small. Above, we have called 1 and 2 the C P T -violating parameters. But one must be careful about such names. 1 and 2 do not contribute only to observables measuring CP T violation. They also lead to C P - and T -violating effects. Similarly, the T -violating parameter 3 also contributes to C P -violating observables. And the reverse is true: recall that, in Ref. [14], the BaBar Collaboration measured a large true T -violating asymmetry. This does not suggest that 3 is large, as there are also large contributions to the asymmetry coming from C P -violating effects (assuming CP T invariance). The point is that 1 , 2 and 3 are also sources of C P violation, and it is this fact that allows their measurement in the time-dependent indirect CP asymmetry, as we will see below. In the presence of T and CP T violation in B 0 – B¯ 0 mixing, the time evolution of the flavor eigenstates (| B 0  ≡ | B 0 (t = 0) and | B¯ 0  ≡ | B¯ 0 (t = 0)) is given by

| B 0 (t ) = ( g + + g − cos θ)| B 0  + e i φ g − sin θ| B¯ 0  , | B¯ 0 (t ) = e −i φ g − sin θ| B 0  + ( g + − g − cos θ)| B¯ 0  ,

(15)

A. Karan et al. / Physics Letters B 781 (2018) 459–463

where g ± = (e −i H L t ± e −i H H t )/2:





 t , M − i 2 2



  t . g − = e −i ( M −i 2 )t i sin M − i 

g + = e −i ( M −i 2 )t cos

2

(16)

2

Here M ≡ ( M H + M L )/2, M ≡ M H − M L ,  ≡ ( H +  L )/2 and  ≡  H −  L . We consider a final state f to which both B 0 and B¯ 0 can decay. Using Eq. (15), the time-dependent decay amplitudes for uncorrelated or tagged neutral mesons are given by

A( B 0 (t ) → f )=( g + + g − cos θ)A f + e i φ g − sin θ A¯ f ,

¯0

A( B (t ) → f )= e

−i φ

(17)

g − sin θ A f + ( g + − g − cos θ)A¯ f ,

d

( B 0 (t ) → f )  

 1 = e −t sinh ( t /2) 2Re cos θ|A f |2 + e i φ sin θ A∗f A¯ f 2  + cosh ( t /2) |A f |2 + | cos θ|2 |A f |2 + |e i φ sin θ|2 |A¯ f |2

 + 2Re e i φ cos θ ∗ sin θ A∗f A¯ f  + cos( Mt ) |A f |2 − | cos θ|2 |A f |2 − |e i φ sin θ|2 |A¯ f |2

 − 2Re e i φ cos θ ∗ sin θ A∗f A¯ f 

 , (18) − sin( Mt ) 2Im cos θ|A f |2 + e i φ sin θ A∗f A¯ f

dt

d

( B¯ 0 (t ) → f )  1 = e −t sinh ( t /2) 2

  ∗ × 2Re − cos θ ∗ |A¯ f |2 + e i φ sin θ ∗ A∗f A¯ f  + cosh ( t /2) |A f |2 + | cos θ|2 |A f |2 ∗

 + |e −i φ sin θ|2 |A f |2 − 2Re e i φ cos θ sin θ ∗ A∗f A¯ f  + cos( Mt ) |A f |2 − | cos θ|2 |A f |2 − |e −i φ sin θ|2 |A f |2 ∗

 + 2Re e i φ cos θ sin θ ∗ A∗f A¯ f 

 ∗ + sin( Mt ) 2Im − cos θ ∗ |A¯ f |2 + e i φ sin θ ∗ A∗ A¯ f .

dt

f

(19) If we set θ = π2 and Im φ = 0 in the above expressions, we recover expressions for the differential decay rates that are commonly found elsewhere in the literature. We now write θ = θ1 + i θ2 and φ = φ1 + i φ2 , with [see Eqs. (10) and (11)]

θ1 →

π 2

+ 1 , θ2 → 2 , φ1 = −2β mix , φ2 → 3 ,

Below, we illustrate how this can be done in the B d0 system. For B 0s mesons, the procedure is similar, though a bit more complicated. First, as regards d , the value of yd = d /2d has been measured to be small: yd = −0.003 ± 0.015 with the B d0 lifetime of 1.520 ± 0.004 ps [21]. This means that we can approximate sinh( t /2) t /2 = yd d t and cosh( t /2) 1. In principle, for large enough times, this approximation will break down. However, even at time scales of O (10) ps, the approximation holds to ∼ 10−4 , and by this time most of the B d0 s will have decayed. The observable we will use to extract the T - and C P T -violating parameters 1,2,3 is the time-dependent indirect CP asymmetry f

A C P (t ) involving B-meson decays to a CP eigenstate. It is defined as f

where A f ≡  f |H F =1 | B 0  and A¯ f ≡  f |H F =1 | B¯ 0 . The differential decay rates  f ( B 0 (t ) → f ) and  f ( B¯ 0 (t ) → f ) are given by1

(20)

where the 1,2,3 are very small. In order to probe T and CP T violation in B 0 – B¯ 0 mixing, one must measure the parameters 1,2,3 .

1 In Ref. [11] it was pointed out that the coefficient of cos( Mt ) includes a C P T -violating piece.

461

A C P (t ) =

d/dt ( B¯ d0 (t ) → f C P ) − d/dt ( B d0 (t ) → f C P ) d/dt ( B¯ d0 (t ) → f C P ) + d/dt ( B d0 (t ) → f C P )

.

(21)

In the limit of CP T conservation, T conservation in the mixing, and  = 0, one has the familiar expression f

A C P (t ) = S sin( M d t ) − C cos( M d t ),

(22)

where

ϕ ≡ φ1 − arg[A f ] + arg[A¯ f ] , |A f |2 − |A¯ f |2 , |A f |2 + |A¯ f |2  S ≡ 1 − C 2 sin ϕ .

C≡

(23) (24) (25)

Here, C is the direct CP asymmetry and ϕ is the measured weak phase, which differs from the mixing phase φ1 if arg[A f ] = arg[A¯ f ]. If there is no penguin pollution, then ϕ cleanly measures a weak phase and C = 0. But if there is penguin pollution, then neither of these holds. In the presence of T and CP T violation in the mixing, we use Eqs. (18) and (19) to obtain the time-dependent CP asymmetry. We first expand the various functions in the two equations, keeping only terms at most linear in the small quantities 1,2,3 and d :

d

( B¯ d0 (t ) → f ) −

d

( B d0 (t ) → f )   = e −d t (|A f |2 + |A¯ f |2 ) 3 + 1 − C 2 1 cos ϕ    + cos( M d t ) −C − 3 − 1 − C 2 1 cos ϕ    + sin( M d t ) −2 + 1 − C 2 sin ϕ ,

dt

d dt

( B¯ d0 (t ) → f ) +

dt

d dt

( B d0 (t ) → f )

= e −d t (|A f |2 + |A¯ f |2 )   1 1 + C 3 + 1 − C 2 d t cos ϕ − 1 − C 2 2 sin ϕ 2

(26)

(27)

   + cos( M d t ) −C 3 + 1 − C 2 2 sin ϕ    + sin( M d t ) C 2 + 1 − C 2 3 sin ϕ . The denominator [Eq. (27)] has the form A (1 + x), with x small, so we can approximate 1/ A (1 + x) (1 − x)/ A. Combining all the

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A. Karan et al. / Physics Letters B 781 (2018) 459–463

pieces, and again keeping only terms at most linear in yd , we obtain2

1,2,3 and

f

A C P /C P T (t ) c 0 + c 1 cos( M d t ) + c 2 cos(2 M d t )

f

+ s1 sin( M d t ) + s2 sin(2 M d t ) + c 1 d t cos( M d t ) + s 1 d t sin( M d t ) ,

(28)

where the coefficients are given by

1

3 sin2 ϕ , 2 c 1 = −C − 3 − 1 cos ϕ − 2 C sin ϕ , 1 c 2 = 3 sin2 ϕ + 2 C sin ϕ , 2 c 0 = 1 cos ϕ + 3 −



1 − C 2 sin ϕ − 2 cos2 ϕ − 3 C sin ϕ , 1 s2 = − 2 sin2 ϕ + 3 C sin ϕ , 2 c 1 = C yd cos ϕ , s1 =



s1 = −

1 2



(29)

The seven pieces have different time dependences so that, by fitf

ting A C P /C P T (t ) to the seven time-dependent functions, all coefficients can be extracted. The five observables c 0 , c 1 , c 2 , s1 and s2 can be used to solve for the five unknown parameters C , ϕ and 1,2,3 . In practice, a fit will probably be used, but there is an analytical solution. The parameter C is simply given by

C = −(c 0 + c 1 + c 2 ) .

(30)

The solution for sin ϕ is obtained by solving the following quartic equation:

sin

4

ϕ−2 

−4

s1 + 2s2 2 − C2





3

sin

2C 2 (s1 + s2 ) − s2 2 − C2

ϕ + 4C C +



 sin ϕ −



c2

2

2 − C2

8C c 2

sin

ϕ



2 − C2

=0.

(31)

Of course, there are four solutions, but, since the √ i are small, the correct solution is the one that is roughly s1 / 1 − C 2 . Finally, 1 , 2 , 3 are given by

1 = c0 sec ϕ −

(2 − sin2 ϕ )(c 2 sin ϕ + 2C s2 )

(4C 2 + sin2 ϕ ) sin ϕ cos ϕ 2 (2C c 2 − s2 sin ϕ ) 2 = , (4C 2 + sin2 ϕ ) sin ϕ 2 (c 2 sin ϕ + 2C s2 ) 3 = . (4C 2 + sin2 ϕ ) sin ϕ

not change. A C P /C P T (t ) still depends on seven different timedependent functions, a fit can be performed to extract their coefficients, and C , ϕ , 1,2,3 and  S can be found using the measurements of these coefficients. Finally, we have another handle for probing CP T violation in B d0 – B¯ d0 mixing. At present, we know that 3 = −(1.0 ± 0.8) × 10−3 [Eq. (14)]. Now, suppose that there is no CP T violation (i.e., 1 = 2 = 0). In this case, for the time-dependent CP asymmetry [Eq. (28)], we can eliminate C and sin ϕ using Eqs. (30)–(32). The coefficients c 0 , c 2 and s2 can then be expressed in terms of the measured quantities c 1 , s1 and 3 as follows:

c 0 = 3 1 −

yd sin 2ϕ .



Above, the method was described for the B d0 system. In the case of B 0s mesons, s is not that small, so the functions sinh ( s t /2) and cosh ( s t /2) must be kept throughout. This modifies the forms of Eqs. (26), (27) and (28), but the idea does

,

c2 =

2s21

(2 − c 12 + 32 )2

 ,

2s21 3

, (2 − c 12 + 32 )2 2s1 (c 1 + 3 ) 3 . s2 = − (2 − c 12 + 32 )

(33)

The values of c 1 and s1 have been measured for several B d0 decays to CP eigenstates [20], and the value of 3 is independent of the decay mode. Using these values, we can estimate c 0 , c 2 and s2 from Eq. (33), which assumes that CP T is conserved. As an example, for the final state J /ψ K S , we find

c 0 = (−15.18 ± 15.50) × 10−4 , c 2 = (−4.31 ± 4.41) × 10−4 , s2 = (0.29 ± 0.43) × 10−4 .

(34)

Should the measurements of c 0 , c 2 and s2 deviate significantly from the above values, this would indicate the presence of CP T violation in B d0 – B¯ d0 mixing. To sum up, we have shown that the time-dependent, indirect CP asymmetries involving B 0 , B¯ 0 → f C P contain enough information to extract not only the C P -violating weak phases, but also the parameters describing T and CP T violation in B 0 – B¯ 0 mixing. These measurements can be made at the ϒ(4S ) (e.g., BaBar, Belle) or at high energies (e.g., LHCb). There is no need to neglect penguin pollution in the decay, and the method can be applied to B d0 or B 0s -meson decays. Acknowledgements

(32)

This shows that it is possible to measure the parameters describing T and CP T violation in B d0 – B¯ d0 mixing using the time-dependent indirect CP asymmetry, and this can be carried out at LHCb. The parameters c 1 and s 1 depend only on ϕ and yd . Thus, given knowledge of ϕ , the value of yd can be found from measurements of these parameters. Note that, even if the width difference d between the two B-meson eigenstates vanishes, the T -violating parameter 3 can still be extracted. This is contrary to the claim of Refs. [6] and [9].

2 A time-dependent CP asymmetry having a complicated form with higher harmonics in M d t, similar to that in Eq. (28), was noted in Ref. [9].

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