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

Light gauge boson in rare K decay Chuan-Hung Chen a , Takaaki Nomura b,∗ a b

Department of Physics, National Cheng-Kung University, Tainan 70101, Taiwan School of Physics, KIAS, Seoul 130-722, Republic of Korea

a r t i c l e

i n f o

a b s t r a c t We study the production of a light gauge boson in K − → π − X decay, where the associated new charge current is not conserved. It is found that the process can be generated by the tree-level W -boson annihilation and loop-induced s → d X. We ﬁnd that it strongly depends on the SU (3) limit or the unique gauge coupling to the quarks, whether the decay amplitude of K − → π − X in the W -boson annihilation is suppressed by m2X X · p K ; however, no such suppression is found via the loop-induced s → d X. The constraints on the relevant couplings are studied. © 2016 The Author(s). 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 .

Article history: Received 15 August 2016 Received in revised form 25 October 2016 Accepted 26 October 2016 Available online 2 November 2016 Editor: J. Hisano

A light gauge boson X has been studied widely for various reasons [1–10]. In particular, a mass of around 17 MeV boson with more than 5σ signiﬁcance is indicated by the measurement of e + e − angular correlations in the 8 Be transition [11], where the implications of this light gauge boson are investigated [12–15]. However, two conclusions associated with the X emission in rare K decay appear in the literature, where some authors [16, 17] concluded that the longitudinal component in the K − → π − X decay is enhanced by X · p K ∼ m2K /m X , but others [4,5, 18] showed that this decay amplitude should be suppressed by X · p K m2X /m2K ∼ m X . In general, the decay amplitude for the K − → π − X process μ can be written as A λ = X π − | H I | K − = M μ X (k, λ), where H I is the involved interaction; M μ is the transition matrix element μ for K + → π + , and X (λ) denotes the X polarization vector with the momentum k and helicity λ. Thus, the spin-average amplitude square can be expressed as:

λ

2

| Aλ| = Mμ Mν

kμ kν − g μν + m2X

.

(1)

If the charge, which is associated with gauge symmetry for the X -gauge boson, is conserved, following the current conservation kμ M μ = 0, it can be seen that the term kμ kν /m2X vanishes. Clearly, the 1/m X enhancement for a light gauge boson is associated with the charge nonconservation, i.e., kμ M μ = 0. Accordingly, the 1/m X

*

Corresponding author. E-mail addresses: [email protected] (C.-H. Chen), [email protected] (T. Nomura).

factor indeed is suppressed when the associated current is conserved, such as the case of a dark photon that mixes with the photon through the kinetic term [20,21]. Furthermore, due to gauge invariance, the decay amplitude for K + → π + X in such cases should vanish at the tree level according to chiral perturbation theory [19]; the main contributions then are from the loop effects. A detailed analysis about the dark-photon case can be found in Refs. [4,5]. To obtain a further understanding the properties of K − → π − X in the model with charge nonconservation, in this study, we analyze this issue by exploring the situations with and without the SU (3) limit and unique gauge coupling when the K − → π − transition arises from the W -boson annihilation. In addition, we also study the contributions from the loop-induced ﬂavor-changing neutral current (FCNC) process s → d X . Since our purpose is to investigate the properties of a light gauge boson emission from the K − → π − transition, we do not focus on a speciﬁc gauge model. Instead, we study a case in which the X -boson vectorially couples to the standard model (SM) quarks and the interactions are dictated by:

Lqq X = gqq q¯ γμ q X μ .

(2)

In general, the couplings gqq are ﬂavor-dependent, and the FCNCs at the tree level are then induced. Since we have little knowledge on the ﬂavor mixings, the associated FCNC parameters are completely free. We thus skip discussions on the tree-level FCNC effects in this work by assuming that they are small, or that they can always be constrained by low energy physics. In the following analysis, we focus on couplings with q = q by writing gq ≡ gqq for simplicity.

http://dx.doi.org/10.1016/j.physletb.2016.10.063 0370-2693/© 2016 The Author(s). 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 .

C.-H. Chen, T. Nomura / Physics Letters B 763 (2016) 304–307

305

1 A j ≈ g j C K π X · pK

dxR j ( p K , p π , x)φ j (x) ,

(8)

0

√

∗ V f f , g where C K π = G F /(2 2) V ud us K π 1, 3 = g u , g 2 = g s , g 4 = g d , φ1,2 = φ K , φ3,4 = φπ , and R j are deﬁned as: Fig. 1. Flavor diagrams for the tree-level K − → π − X decay, where the square boxes and numbers denote the possible places to emit the X -boson.

In order to demonstrate the characteristics of the K − → π − X decay, we analyze the hadronic effects with leading-twist parton distribution amplitudes (DAs) for the π and K mesons. As usual, the twist-2 DA of a pseudoscalar meson is deﬁned by [22,23]:

1

0|¯q (x)γ5 γμ q(−x)| P ( p ) = −i f P p μ

due

i ξ p ·x

φ P (u ) ,

(3)

0

1

where ξ = 2u − 1, 0 du φ P (u ) = 1, and f P is the decay constant of a meson P . The DA can be expanded by Gegenbauer polynomials as:

φ P (u ) = 6u (1 − u ) 1 +

aiP (

3/2 )C i (2u

μ

− 1) ,

(4)

i =1

where the Gegenbauer moments aiP for the π and K mesons at μ = 1 GeV are aπ2i+1 = 0, aπ2 = 0.44, aπ4 = 0.25, a1K = 0.17, and

a2K = 0.2 [22–24]. It can be seen that due to breaking of the SU (3), the odd moments in the K meson do not vanish. To calculate the X emission from the K and π mesons, we adopt the spin structure for incoming meson as [25,26]:

0|¯q1β (0)q2α ( z)| P ( p ) =

−i f P 4N c

1

dxe −ixp ·z [ p / γ5 ]α β φ P (x) ,

(5)

0

where N c = 3 is the number of colors, and the spin structure for outgoing meson can be obtained by using γ5 p / instead of p/ γ5 . According to Eq. (5), the f P can be obtained by taking the trace in spinor space as:

0|¯q2 γμ γ5 q1 | P ( p ) = −i

Nc f P 4N c

1 T r (γμ γ5 p / γ5 )φ P = i f P p μ , (6) 0

where the N c in the numerator is from the sum of color charges of quark line q¯ 2α [...]q1α . With the couplings in Eq. (2), the K − → π − X decay can arise from the tree and one-loop diagrams. Since the hadronic effects from the tree level are more copious than those from one-loop, we ﬁrst discuss the tree contributions in detail. The ﬂavor diagrams for the K − → π − X decay are shown in Fig. 1, where the square boxes and numbers denote the possible places to emit the X -boson. According to Eq. (5), the decay amplitude for the X -boson emitting from place-1 can be derived as:

i A1 ≈

1 ×

−ig 2 2m2W

∗

V ud V us

dx T r ig u / X

0

μ

i f π pπ

2 i

− p/ u + p/ X − mu

γμ P L p/ K γ5

−iN c f K φ K (x) 4N c

.

(7) It is found that the decay amplitude for the X -boson emitting from place- j can be formulated as:

R 1 ( p K , p π , x) =

−2xp 2π − 2(1 − x)m2X m2X − 2xp X · p K

,

R 2 ( p K , p π , x) = − R 1 ( p K , p π , 1 − x) , R 3 ( p K , p π , x) = R 1 (− p π , p K , x) , R 4 ( p K , p π , x) = − R 1 (− p π , p K , 1 − x).

(9)

In order to understand the SU (3) limit and gq dependence of A j , we study the cases by requiring an exact/partial SU (3) limit and different/same gq . (I) SU (3) limit: m K = mπ ≡ m P , φ K = φ p ≡ φ P : Since f K and f π are the multiplier factors, they are irrelevant to the discussions of the SU (3) limit; therefore, we do not need to set f K = f π . Due to the kinematics, p K = p π + p X has to be satisﬁed; thus, we need to leave the factors p X · p K and p X · p π , which appear in the denominators of Eq. (9), alone. From Eqs. (8) and (9), we then get:

A 1 + A 3 = − g u C kπ X · p K m2X

1 ×

dxφ P (x) 0

4(1 − x) xm2P + (1 − x)m2X

(m2X + 2xp X · p π )(m2X − 2xp X · p K )

, (10)

and A 2 + A 4 = −( gd + g s )/(2g u )( A 1 + A 3 ). We consider A 1 + A 3 because A 1 and A 3 involve the same gauge coupling g u . One can alternatively use A 1 + A 2 ( A 3 + A 4 ) according to the convenience. It is clearly seen that the decay amplitude for the K − → π − X is proportional to m2X . This result matches the conclusions given in two earlier works [4,18]. We note that when calculating A 2 + A 4 , we have used the property φ P (x) = φ P (1 − x), where this condition is suitable for the π meson, and it is violated in the K meson due to the breaking of SU (3). As a result, the leading-twist contributions can not lead to an interesting result in the SU (3) limit. Furthermore, if we further set g s + gd = 2g u , it can be found that A i = 0. (II) Partial SU (3) limit: φ K = φπ ≡ φ P : When we release the condition m K = mπ , the ﬁrst term in the numerator of Eq. (10) inside the integral becomes m2X (m2K + m2π )(1 − x) − (m2K − m2π )2 x, and the denominator is m4X (1 − x)2 − (m2K − m2π )2 x2 . With m X = 0, we ﬁnd:

A 1 + A 3 = −2g u C kπ X · p K , A2 + A4 = 2

gd m2K − g s m2π m2K − m2π

C K π K · p K ,

(11)

where the x dependence of the numerator and denominator in the 1 integral is cancelled, and 0 φ P (x)dx = 1 is applied. By this analysis, it is clear that when we put back the SU (3) breaking effect with m K = mπ , the decay amplitude is not proportional to m2X anymore. This result conﬁrms the conclusions given in two earlier studies [16,17]. Furthermore, if we take all gauge couplings to be the same and m2X = 0, we ﬁnd that i A i = 0 is still satisﬁed. We can understand the cancellations from another viewpoint: by using φ P (x) = φ P (1 − x), from Eqs. (8) and (9) it can be easily found that A 1 + A 2 = 0 for g u = g s and A 3 + A 4 = 0 for g u = gd . (III) SU (3) breaking: With a partial SU (3) limit, which is conditioned by φ K = φπ , it can be seen that the decay amplitude for the K − → π − X is not

306

C.-H. Chen, T. Nomura / Physics Letters B 763 (2016) 304–307

Table 1 Magnitude with some chosen values of couplings and m X under the SU (3) assumptions, where M ( g u , g s , gd )m X is deﬁned in Eq. (14). M ( g u , g s , gd )m X

SU (3) I

SU (3) I I

SU (3) I I I

M ( g , g , g )10 MeV M (2g , g , g )10 MeV M ( g , g , g )100 MeV M (2g , g , g )100 MeV

0 0.35g 0 0.79g

0 2.35g 0 2.58g

−9 · 10−4 g 2.35g −0.1g 2.43g

suppressed by m2X ; however, it diminishes when g s ∼ gd ∼ g u . It is intriguing to see whether the cancellations work or not when the SU (3) breaking effects are taken into account in the DA of the K meson. From Eqs. (8) and (9), we can easily get the result A 4 = − gd / g u A 3 when the g u and x in A 3 are replaced by the gd and 1 − x. The connection between A 3 and A 4 is based on the property φπ (x) = φπ (1 − x), where the odd Gegenbauer moments vanish. That is, if gd = g u , the cancellation between A 3 and A 4 still works. Unlike the DA of pion, φ K (x) = φ K (1 − x) due to nonvanishing odd Gegenbauer moments, e.g., a1K = 0.17. Hence, we have:

1 A1 + A2 = C K π X · p K

R 1 ( p K , p π , x) [g u φ K (x) − g s φ K (1 − x)] 0

(12) In order to examine Eq. (12), we simplify the analysis by taking the limit m X → 0. Thus, with g s = g u , we ﬁnd:

1 A1 +

A 2 ∝ a1K

x2 (1 − x)(1 − 2x) = 0 .

(13)

0

According to the result, it can be seen that amplitude the decay K 2 for the K − → π − X with g u = g s = gd is A ∝ a f ( m i i X ), where 1 2 the function f (m X ) is from the integration in x and only depends on the m2X . We then conclude that if the X gauge couplings to the quarks are the same, the decay amplitude of the K − → π − X process from the leading-twist DA is suppressed by m X . To illustrate the relative magnitude under the SU (3) assumptions, we show the numerical values with some chosen values of couplings and m X in Table 1, where the function M m X is deﬁned as:

M ( g u , g s , gd )m X =

1 C K π X · pK

4

Aj .

(14)

j =1

SU (3) I , I I , I I I denote the cases for the SU (3) limit, the partial SU (3) limit, and the breaking of SU (3). It is of interest to examine the X emission from the W -boson propagator shown in Fig. 1. To estimate the contribution, we parametrize the Lorentz covariant gauge coupling W W X to be:

L ⊃ g W W X g α β ( p − − p + )μ + g β μ ( p + − p X )α

+ g μα ( p X − p − )β W −α W +β X μ ,

(15)

where g W W X is the trilinear gauge coupling; p − , p + , and p X are the momenta of the W − , W + , and X gauge bosons, respectively, and the momenta are chosen to ﬂow into the vertex. Accordingly, the decay amplitude for the K − → π − X via the trilinear coupling can be obtained as: ∗ A W W X = V ud V us

G F f K f π m2X gW W X X · p K . √ 2 m2W

(16)

It is clear that the contribution from the coupling W W X is suppressed by m X . This result is nothing to do with the SU (3) limit.

Fig. 2. Contours for B R ( K − → π − X ) in units of 10−7 as a function of g u and gd − g u in units of 10−3 , where the numbers on the lines denote the values of B R ( K − → π − X ), and dashed lines are the central value of B R exp ( K − → π − e+ e− ).

Without the SU (3) limit, it can be seen from Eq. (9) that the numerators of R 1,2 are related to m2π and that those of R 3,4 are associated with m2K . Since m K is around 3.5 times larger than mπ , numerically, the values of A 1,2 are one order of magnitude smaller than those of A 3,4 . As a result, the B R ( K − → π − X ) is sensitive to the difference between gd and g u . To present the numerical analysis, we adopt g u and gd − g u as the free parameters and set g s = gd . We show the contours for B R ( K − → π − X ) (in units of 10−7 ) as a function of g u and gd − g u (in units of 10−3 ) in Fig. 2, where we have used f K = 0.16 GeV and f π = 0.13 GeV, and the numbers on the lines denote the values of B R ( K − → π − X ). With the assumption of B R ( X → e + e − ) ∼ 1, we obtain B R ( K − → π − e+ e− ) ≈ B R ( K − → π − X ), where the current measurement is B R exp ( K − → π − e + e − ) = (3.00 ± 0.09) × 10−7 [28]. Therefore, the dashed lines in the plot can be regarded as the central value of the experimental measurement. From the ﬁgure it can be seen that | gd − g u | cannot be larger than 10−4 . In addition to the tree-level W -boson annihilation, the FCNC coupling sd X , which is induced from one-loop and is depicted in Fig. 3, can also contribute to the K − → π − X process. According to the interactions in Eq. (2), the effective coupling of sd X from each up-type quark loop can be derived as:

A q = C q d¯ γμ (1 − γ5 )s X μ , Cq =

g m2 ∗ 4G F q q V qs V qd Iq √ (4 )2

π

2

1 I q (r ) ≈

dy 0

y 1 − (1 − r ) y

mq2 m2W

(17)

,

,

where we have dropped the small effects from ms,d ; the factor mq2 is from the mass inserted twice in q-quark propagator, and I q (r ) is the loop integral. From Eq. (17), the associated Cabibbo– Kobayashi–Maskawa (CKM) matrix elements for top-quark loop are ∗ V , and due to the enhancement of m2 , its contribution is comV td ts t parable to that from the charm-quark, in which the essential factor ∗ V . The contribution from the u-quark loop can be igis mc2 V cd cs nored because of the m2u suppression. Additionally, it can be clearly seen that although the X couplings to quarks are vector-like, the induced coupling sd X indeed is chiral. Unlike the W -boson annihilation shown in Fig. 1, the dominant hadronic effect for the K − → π − X decay from sd X interaction is formulated by [27]:

π − |d¯ / X s| K − = 2 X · p K f + (q2 ) ,

(18)

C.-H. Chen, T. Nomura / Physics Letters B 763 (2016) 304–307

307

π − e+ e− can be generated from both tree and penguin diagrams. It is found that the decay amplitude for K − → π − X from W -boson annihilation can be directly proportional to m2X X · p K when the

SU (3) limit is applied, or when the gauge couplings satisfy g u = gd = g s . When these conditions are relaxed, it is found that if g u is of the O (10−3 ), | gd − g u | has to be less than 10−4 . By contrast, the m2X suppression is not found in the loop-induced K − → π − X process. We show that the loop effects indeed produce more severe bounds on the gauge couplings gt and g c . Our results are consistent with the conclusions given in two previous studies [16,17].

Fig. 3. One-loop Feynman diagram for s → d X .

Acknowledgements This work was partially supported by the Ministry of Science and Technology of Taiwan R.O.C., under grant MOST-103-2112M-006-004-MY3 (CHC). References

Fig. 4. Loop-induced B R ( K − → π − X ) as a function of gq (in units of 10−5 ), where the solid and dashed lines denote the contributions from top- and charm-quark loops, respectively. The horizontal dotted line is the central value of B R exp ( K − → π − e + e − ).

where f + ≈ 0.971 at q2 = 0. As a result, the corresponding transition amplitude and BR are respectively given by:

π − X | A q | K − = 2C q f + (m2X ) X · p K , B R(K − → π − X ) =

|C q f + (m2X )|2 τ K + |p X |3 2π

m2X

.

(19)

It can be seen that Eq. (19) is not suppressed by m X but rather is enhanced by 1/m X . This result is consistent with the dark Z model in Ref. [5] if we set gq ∼ δ m X /m Z . To show the bounds on the gauge couplings gt and g c independently, we present the numerical values of Eq. (19) as a function of gt and g c in Fig. 4, where m X = 17 MeV is used, the solid (dashed) line stands for the result of gt ( g c ) in units of 10−5 , and the horizontal dotted line is the central value of B R exp ( K − → π − e + e − ). With B R ( X → e + e − ) ∼ 1, the bounds from the penguin diagrams are stronger than those from the W -boson annihilations. Therefore, we conﬁrm the conclusion given in a previous work [17], where the effective coupling arising from the penguin diagrams obtains a stricter bound. We note that the X -boson emitting from the W propagator shown in Fig. 3 can also contribute to K − → π − X ; however, due to the m2K /m2W suppression, we neglect its contribution in the numerical analysis. In summary, a light gauge boson predominantly decaying to e + e − in rare K decay is studied. The process K − → π − X →

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