Eur. Phys. J. C (2014) 74:2856 DOI 10.1140/epjc/s10052-014-2856-9

Regular Article - Theoretical Physics

New physics in Bd,s – B¯ d,s mixings and Bd,s → µ+ µ− decays Jong-Phil Leea Division of Quantum Phases and Devices, School of Physics, Konkuk University, Seoul 143-701, Korea

Received: 24 December 2013 / Accepted: 9 April 2014 / Published online: 30 April 2014 © The Author(s) 2014. This article is published with open access at Springerlink.com

Abstract A new way of probing new physics in the B meson system is provided. We define double ratios for the observables of Bd,s – B¯ d,s mixings and Bd,s → μ+ μ− decays, and find simple relations between the observables. By using the relations we predict the yet-to-be-measured branching ratio of Bd → μ+ μ− to be (0.809–1.03)×10−10 , up to the new physics models.

1 Introduction The recent discovery of a Higgs boson at the large hadron collider (LHC) opened a new era of high energy physics. It may take time to confirm whether the new particle is really the Higgs boson of the standard model (SM), but it looks more and more like the SM Higgs. The discovery of the Higgs boson would mean a completion of the SM. On the other hand, we have many reasons to believe that there must be new physics (NP) beyond the SM. Unfortunately, the LHC up to now has not reported any clues of NP. But it is too early to say that there is no NP at all. Bd,s mesons are good test beds for NP. Especially, Bd,s – B¯ d,s mixings and Bd,s → μ+ μ− decays are loop-induced phenomena in the SM and very sensitive to NP effects. The current status of the experiments is well compatible with the SM predictions. For example, the LHCb and the CMS collaboration reported [1,2] −9 Br(Bs → μ+ μ− ) = (2.9+1.1 −1.0 ) × 10 ,

Br(Bd → μ+ μ− ) < 7.4 × 10−10 (LHCb),

(1)

+1.0 Br(Bs → μ+ μ− ) = (3.0−0.9 ) × 10−9 ,

Br(Bd → μ+ μ− ) < 1.1 × 10−9 (CMS).

(2)

The measured value is slightly smaller than the previous LHCb measurements [3]: a e-mail:

[email protected]

+1.5 Br(Bs → μ+ μ− ) = (3.2−1.2 ) × 10−9 ,

Br(Bd → μ+ μ− ) < 9.4 × 10−10 .

(3)

For comparison: the SM predictions are [4,5] Br(Bs → μ+ μ− ) = (3.25 ± 0.17) × 10−9 , + −

Br(Bd → μ μ ) = (1.07 ± 0.10) × 10

−10

(4) .

(5)

But there is still some room for NP, as discussed in [4,6,7]. In this paper, we provide a very simple and quick way to probe NP in Bd,s – B¯ d,s mixings and Bd,s → μ+ μ− decays. The idea is that a double ratio for one observable between different flavors extracts the relevant couplings for NP, and they are directly related to the other observable. Schematically, for a physical observable Oia with flavor a, Riab

≡

a a Oi,exp /Oi,SM −1 b b Oi,exp /Oi,SM −1

  fi

 ca , cb

(6)

where the ca are the new couplings and f i is some function of ca/cb . For another observable O j we can define a simia b lar quantity, R ab j , which would behave  f j (c /c ). Conab ab sequently, Ri and R j are related through the functions f i and f j , and the relations are remarkably simplified when the new couplings belong to the category of the minimal flavor violation (MFV). In this way, we can establish simple relations between the observables of Bd,s – B¯ d,s mixings and Bd,s → μ+ μ− decays. The relations are very useful because Riab and R ab j are directly connected, and the relations are different for various NP models. For example, we can predict Br(Bd → μ+ μ− ) from other known observables such as M of Bd,s – B¯ d,s mixings, without knowing the values of the new couplings. Or if we measure the branching ratio Br(Bd → μ+ μ− ), we can find from the double ratio relations which NP is realized in B physics. In this paper we specifically consider flavor changing scalar (un)particles and vector boson (Z  ) scenarios. Actually it is already known that Mq and Br(Bq → μ+ μ− ) can be related to each other

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Eur. Phys. J. C (2014) 74:2856

[8–10]. In our approach, the Riab are directly proportional to the NP effects, so the resulting relations are solely those of NP. The relations might be different for various models, which makes it easier to see which kind of NP is realized. The NP couplings adopted in this analysis are summarized as follows [4,11]:

LU =

(7) L R sb sb  ¯ [ L (H )(¯s PL b)+ R (H )(¯s PR b)+ L (H )(PL ) ¯ (8) +  R (H )(PR )]H, bs  cU L c ¯ μ (1−γ5 ) ∂ μ OU , s¯ γμ (1−γ5 )b ∂ μ OU + UdL γ d UU

UU

(9) where PL ,R = (1 ∓ γ5 )/2. In LU one can also include the right-handed couplings, but here (and in [11]) only the minimal extension of the SM is considered for simplicity. First consider the Bd,s – B¯ d,s mixing. The mixing effect is parametrized as the following quantity: Mq =

G 2F 2 M m B |V ∗ Vtq |2 FB2q η B |S(Bq )|, 6π 2 W q tb

(10)

Bq

S(Bq ) = S0 (xt ) + S(Bq ) ≡ |S(Bq )|eiθ S ,

(11)

and xt = m 2t /m 2W . Here the loop function 4xt − 11xt2 + xt3 3xt2 log xt − , 4(1 − xt )2 2(1 − xt )3

(12)

(18)

[ L (H )]2 2 T (Bq )2M H

× [C1S L L (μ H )Q 1S L L (μ H , Bq ) + C2S L L (μ H )Q 2S L L (μ H , Bq ) ], [S(Bq )] S R R = [S(Bq )] S L L (L → R), bq

[S(Bq )] S L R = −

(19) (20)

bq

 L (H ) R (H ) 2 T (Bq )M H

×[C1S L R (μ H )Q 1S L R (μ H , Bq ) + C2S L R (μ H )Q 2S L R (μ H , Bq ) ].

(21)

The expectation values of the operators Q ia are 1 m B F 2 P a (μ M , Bq ). 3 q Bq i

(22)

bq

For the case of  R = 0,   [S(Bq )]V L L  Mq (Z  )  = 1 + MqSM S0 (xt )   bq 2  L (Z  ) 1 4˜r  1+ Re , 2 S0 (xt ) Vtb∗ Vtq M Z2  gSM

(23)

bq

up to the leading order of  L . Now we define a double ratio Z  as RM

and S(Bq ) = [S(Bq )]V (S)LL + [S(Bq )]V (S)RR + [S(Bq )]V (S)LR ,

(13)

where the subscript V (S) stands for Z  (H ) contributions. Explicitly [6,7],  bq 2  L (Z  ) 4˜r [S(Bq )]VLL = , (14) ∗ 2 2 Vtb Vtq M Z  gSM  bq 2  R (Z  ) 4˜r , (15) [S(Bq )]VRR = ∗ 2 2 Vtb Vtq M Z  gSM  L (Z  ) R (Z  ) bq

[S(Bq )]VLR =

bq

T (Bq )M Z2  × [C1VLR (μ Z  )Q VLR 1 (μ Z  , Bq ) + C2VLR (μ Z  )Q VLR 2 (μ Z  ,

123

(17)

bq

[S(Bq )] S L L = −

Q ia (μ M , Bq ) ≡

where

S0 (xt ) =

α 4G F , gS M ≡ √ 2 2π sin2 θW G 2F 2 F Bˆ B m B M 2 (V ∗ Vtq )2 η B , T (Bq ) ≡ 12π 2 Bq q q W tb and r˜ = 0.985 for M Z  = 1 TeV. For the scalar field,

  s γμ PL b) + sb s γμ PR b) L Z  = [sb L (Z )(¯ R (Z )(¯   ¯   ¯ +  (Z )(γμ PL ) +  (Z )(γμ PR )]Z μ ,

LH =

where

Bq ) ],

(16)

Z RM

2   Re bs Ms (Z  )/MsSM − 1 L (Z )/Vts ≡ = 2 ,   Md (Z  )/MdSM − 1 Re bd L (Z )/Vtd (24)

where the result of Eq. (23) is applied. Similarly, for the scalar bq contribution (with  R = 0),

H RM ≡

2  Ms (H )/MsSM − 1 Bˆ Bd Re bs L (H )/Vts = 2 .  Md (H )/MdSM − 1 Bˆ Bs Re bd (H )/V td L (25)

We assumed here that the light-quark dependence on Pia (μ H , Bq ) is negligible [12], and thus Pia (μ H , Bd )  Pia (μ H , Bs ).

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In the scalar unparticle scenario [11], MqU MqSM

− 1 ≡ |U | − 1 =

where

bq q Re[(cU L )2 f U

bq

cot dU π ]

q

4 + Im[(cU L )2 f U ] + O(cU L ).

(26)

Here 5 fU = A SM dU 24M12



FB2q

q

where

SM M12

A dU ≡



m Bq

m 2Bq

dU ,

U

(27)

is the SM contribution and

(dU + 1/2) 16π 5/2 , (2π )2dU (dU − 1)(2dU )

(28)

d

(29) where we put cU L ≡ c˜U L · Vtb∗ Vtq . For real c˜U L , one has bq

U = RM

Bˆ Bd Bˆ Bs



bq

m 2Bs m 2Bd

dU −1

bq

bs c˜U

bd c˜U L

2 .

(30)

bq

If c˜U L is purely imaginary, one gets a similar result. Now we move to Bd,s → μ+ μ− decays. The relevant effective Hamiltonian is given by

Heff

(37)

R H − RL , R H + RL

(38)

  10,S,P  GFα ∗   = −√ (Ci Oi + Ci Oi ) + h.c. , Vts Vtb 2π i

where the operators Oi are  ¯ μ γ5 ), O10 ¯ μ γ5 ), O10 = (¯s γμ PL b)(γ = (¯s γμ PR b)(γ

(32) ¯ ¯ OS = m b (¯s PL b)(), O S = m b (¯s PR b)(), ¯ 5 ), OP = m b (¯s PL b)(γ ¯ 5 ). O P = m b (¯s PR b)(γ

(33) (34)

For Bs decay it is convenient to define [13,14] 1 Br(Bs → μ+ μ− )th , r (ys )

Br(Bs →

μ+ μ− )

=

SM

1 + ys A (|P|2 + |S|2 ), 1 + ys

(39)

where P≡

 C10 − C10 SM C10

+

m b C P − C P ≡ |P|eiϕ P , SM 2m μ m b + m s C10 (40) m 2Bs



4m 2μ m 2Bs m b C S − C S S ≡ 1 − 2 ≡ |S|eiϕ S . (41) SM m Bs 2m μ m b + m s C10 The standard model contribution is SM C10 =−

1 sin2 θ

ηY Y0 (xt ),

(42)

W

with ηY = 1.012 and xt Y0 (xt ) = 8



 xt − 4 3xt log xt . + xt − 1 (xt − 1)2

(43)

For the Z  model, (31)

Br(Bs → μ+ μ− ) ≡

where R H (L) exp[− H (L) t] is the decay rate of the heavy (light) mass eigenstate. Here Br(Bs → μ+ μ− )th is a theoretical prediction, while Br(Bs → μ+ μ− ) would be directly compared with the experimental results. In general, Br(Bs → μ+ μ− )

MsU /MsSM − 1 ≡ MdU /MdSM − 1

2 dU −1 bs )2 cot d π +Im(c˜bs )2 m Bs Re(c˜U Bˆ Bd U L UL ,  2 bd bd )2 2 ˆ mB Re(c˜U L ) cot dU π +Im(c˜U B Bs L



A ≡

(36)

(s)

with dU being the scaling dimension of the scalar unparticle operator. The double ratio for the scalar unparticle is U RM

1 − ys2 , 1 + ys A s = 0.088 ± 0.014, ys ≡ τ Bs 2 and we have the asymmetric parameter

r (ys ) ≡

(35)

μμ

  1 sb L (Z ) A (Z ) , 2 M2 Vts∗ Vtb gSM Z (44) μμ sb   1 1  R (Z ) A (Z )  sin2 θW C10 (Z  ) = − 2 , (45) Vts∗ Vtb gSM M Z2 

sin2 θW C10 (Z  ) = −ηY Y0 (xt )−

1

 while the other coefficients are vanishing. Using sb L ,R (Z ) = bs  ∗  L ,R (Z ) , one has

Br(Bs → μ+ μ− )

ys [cos(2θYBs + θ SBs ) − 1] 1 + ys   μμ bs∗ (bs∗ 1 1 1 L −  R ) A + 2Re , 2 1 + ys ηY Y0 (xt ) M Z2  gSM Vts∗ Vtb

Br(Bs → μ+ μ− )SM

−1

(46)

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Br(Bd → μ+ μ− ) Br(Bd → μ+ μ− )SM

  μμ bd∗ (bd∗ 1 1 L −  R ) A −1  2Re , 2 ηY Y0 (xt ) M Z2  gSM Vtd∗ Vtb (47)

bq ˜ bq Vtq where up to O(ys  L ,R  A ). For  R = 0 and  L =  L bq ˜ is real, the double ratio  L   Br(Bs → μ+ μ− ) Z  Z −1 Rμμ ≡ Br(Bs → μ+ μ− )SM   Br(Bd → μ+ μ− ) Z  −1 (48) Br(Bd → μ+ μ− )SM

remarkably reduces to Z Rμμ

1 = 1 + ys



˜ bs  L . ˜ bd 

(49)

L

Z RM

In this case we have the ratio = one arrives at the very simple relation 

Z Rμμ (1 + ys ) =



˜ bs / ˜ bd )2 , ( L L



Z . RM

and thus

(50)

For neutral scalar H , the coefficients are SM , C10 (H ) = C10

(51)

C S (H ) =

1 1 1 2 2 2 m b sin θW gSM M H

C S (H ) =

1 1 1 2 2 2 m b sin θW gSM M H

C P (H ) =

1 1 1 2 2 2 m b sin θW gSM M H

1 1 1 C P (H ) = 2 2 2 m b sin θW gSM M H

μμ sb R (H ) S (H ) , Vts∗ Vtb μμ sb L (H ) S (H ) , Vts∗ Vtb μμ sb R (H ) P (H ) , Vts∗ Vtb μμ sb L (H ) P (H ) . ∗ Vts Vtb

(52) (53) (54) (55)

H similar to Eq. (48). For One can define a double ratio Rμμ bq ˜ bq Vtq with simplicity we assume that  R = 0 and  L =  L ˜ bq . Note that in this case real  L

H RM

 Br(Bd → μ+ μ− ) H − 1 Br(Bd → μ+ μ− )SM 2

2

m Bs m b + m d 1 Bˆ Bs H = . RM ˆ 1 + ys m 2B m b + m s B Bd 

and

Bˆ Bd = Bˆ Bs



˜ bs  L ˜ bd 

2 .

(56)

L

μμ

For the case of  S (H ) = 0, the double ratio reduces to   Br(Bs → μ+ μ− ) H H Rμμ ≡ −1 Br(Bs → μ+ μ− )SM

123

(57)

d

μμ

On the other hand if  P = 0,

  1 − 4m 2μ /m 2Bs 1 − 2ys H Rμμ = 1 + ys 1 − 4m 2μ /m 2Bd 2

2

m Bs m b + m d Bˆ Bs H . × RM m 2B m b + m s Bˆ Bd

(58)

d

For scalar unparticles [15], √   sin2 θW 2π AdU m Bs 2dU P = 1− ηY Y0 (xt ) αG F m 2B U s   bs  ∗ cU L cU L mb × (cot dU π + i), mb + ms Vtb∗ Vts S = 0,

(59) (60)

and thus A = cos(2ϕ P − φsU ). Here φsU is the phase of bq  , cos(2ϕ − φ U )  1 up U in Eq. (26). For real c˜U L , cU P s L 4 to O(cU L ) , and the double ratio is   + μ− ) Br(B → μ s U U ≡ −1 Rμμ Br(Bs → μ+ μ− )SM   Br(Bd → μ+ μ− )U − 1 Br(Bd → μ+ μ− )SM     bs

c˜U L m Bs 2dU −2 m b + m d 1 = bd 1 + ys m Bd mb + ms c˜U L  dU −1   ˆ  m Bs mb + md 1  B Bs R U , = M 1 + ys m Bd mb + ms Bˆ Bd (61) where the result of Eq. (30) is used. Our results are summarized as follows:  Z Z , (1 + ys ) = RM (62) Rμμ 2

2 m Bs m b + m d H Rμμ (1 + ys ) = m 2Bd m b + m s

Bˆ Bs μμ H × (if  S = 0), (63) RM ˆ B Bd ⎛ ⎞ 4m 2μ

2 2 1 − ⎜ m 2Bs ⎟ m Bs m b + m d H ⎟ Rμμ (1 + ys ) = (1 − 2ys ) ⎜ ⎝ 4m 2 ⎠ m 2 m b + m s Bd 1 − m2 μ Bd

Bˆ Bs μμ H × (if  P = 0), (64) RM Bˆ Bd

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Table 1 Predictions for Br(Bd → μ+ μ− ) for various Br(Bs → μ+ μ− ) measurements. For unparticles, the branching ratio is given at a reference point dU = 1.5 μμ

μμ

New physics

Z

H ( S = 0)

H ( P = 0)

U (dU = 1.5)

Br(Bs → μ+ μ− ) = 2.9 × 10−10 [1] Br(Bs → μ+ μ− ) = 3.0 × 10−10 [2] Br(Bs → μ+ μ− ) = 3.2 × 10−10 [3]

0.799 × 10−10 0.837 × 10−10 0.913 × 10−10

0.775 × 10−10 0.816 × 10−10 0.900 × 10−10

0.716 × 10−10 0.766 × 10−10 0.868 × 10−10

0.803 × 10−10 0.840 × 10−10 0.915 × 10−10

U Rμμ (1 + ys ) =



m Bs m Bd

dU −1 

 ˆ  mb + md  B Bs R U . M mb + ms Bˆ Bd (65)

H ∼ R H is that in R H , Br/Br The reason why Rμμ SM − 1 μμ M μμ 2 is non-vanishing only at O(c ), due to the fact that  P is purely imaginary [4]. Numerically, Eqs. (62)–(65) are  Z Z  = 0.775, (66) Rμμ = 0.919 × RM H, S =0 H Rμμ = 0.993 × RM = 0.707, H, P =0 Rμμ

= 0.818 ×

(67)

= 0.583, (68)  U U = 0.780 × (1.02)dU −1 , Rμμ = (1.02)dU −1 × 0.925 RM H RM

(69) where RM = 0.712 is used. The above results can be used to predict the yet-to-be-measured branching ratio, Br(Bd → μ+ μ− ). Table 1 shows the predicted values of Br(Bd → μ+ μ− ). Note that the values of Table 1 are all far below the current upper bound, Br(Bd → μ+ μ− ) < 7.4 × 10−10 by the LHCb [1] and Br(Bd → μ+ μ− ) < 1.1 × 10−9 by the CMS [2], and slightly smaller than the SM prediction, Br(Bd → μ+ μ− )SM = 1.05 × 10−10 . This is because Br(Bs → μ+ μ− ) < Br(Bs → μ+ μ− )SM = (3.56 ± 0.18) × 10−9 [4] and RM = 0.712 > 0. Note also that the predictions are made without knowing any numerical details of the new couplings, except that they are small enough to neglect higher orders. In this way, by measuring Br(Bd → μ+ μ− ) we can easily figure out which kind of NP is realized in B systems. In conclusion, we derived new relations between Bd,s observables. The relations are valid only when NP exists in Bd,s systems, which is a very plausible assumption. The relations are different in specific models. In this analysis we only consider flavor changing scalar (un)particles and vector bosons. For other models one can define similar double ratios as given in this work. The double ratios become very simple when there are only left- (or right-) handed couplings, and the couplings are MFV-like. If this were not the case, then our simple relations would not hold any more. In other words, if we confirm that the simplified double ratio relations really

hold, then we may conclude that NP is realized in a minimal way. One point to be mentioned is that our double ratio becomes meaningless if there were no NP at all. In this case both numerator and denominator are vanishing and one cannot take a ratio. Thus the double ratio is not adequate to check whether there is any NP or not, but to see which kind of NP is involved once the observables turn out to be quite different from the SM predictions. The current status of NP searches in the case of the B meson is not so pessimistic. According to [16], the relative size of NP in Md,s (= h d,s ) is currently 0.2–0.3, and would be 0.1 in the near future (“Stage I” where the LHCb will end). As for Bd → μ+ μ− , the current upper bound is almost an order of magnitude larger than the SM prediction. It is predicted in [17] that at 2σ , 0.3×10−10  Br(Bd → μ+ μ− )  1.8×10−10 . If the measured branching ratio does not lie within this window, it would be a clear indication of NP. It is also found in [17] that although the measured value of Br(Bs → μ+ μ− ) provides constraints on NP, there are still sizable regions allowed for C S –C S and C P –C P parameter space. Besides the current status of NP searches, we need NP for various reasons (dark matter for example). Although there have been no smoking-gun signals for NP up to now, we believe that the SM is not (and should not be) the full story of particle physics. In this context the double ratio analysis might be very promising with the coming flavor precision era, and it can also be applied to K meson systems. Acknowledgments This work is supported by WCU program through the KOSEF funded by the MEST (R31-2008-000-10057-0). Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. Funded by SCOAP3 / License Version CC BY 4.0.

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