Probing the Effects of New Physics in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.344299pt" id="M1" height="21.4604pt" version="1.1" viewBox="-0.0657574 -17.1161 84.9374 21.4604" width="84.9374pt"><rect height="0.8612076" width="10.3518096" x="0" y="-14.778456"/><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-67" d="M578 512C578 619 494 650 387 650H140L134 622C219 615 223 609 210 542L127 119C112 40 104 34 23 28L17 0H235C317 0 387 7 444 33C515 65 564 122 564 195C564 289 495 335 412 352V354C505 373 578 422 578 512ZM486 510C486 422 423 367 314 367H257L294 565C299 591 305 602 315 608C324 614 339 617 362 617C421 617 486 595 486 510ZM466 200C466 100 393 35 296 35C222 35 198 51 212 127L250 333H303C388 333 466 303 466 200Z"/></g><g transform="matrix(.012,0,0,-0.012,10.351,-11.36)"><path id="g50-43" d="M486 158C486 177 478 202 466 220C413 228 386 236 336 262C386 288 413 297 466 304C478 323 486 347 485 366C470 376 444 381 422 380C389 338 368 319 321 288C323 345 329 372 349 422C339 442 322 461 305 470C289 461 271 442 262 422C281 372 287 345 290 288C243 319 222 338 189 380C167 381 142 376 125 366C125 347 133 322 145 304C198 296 225 288 275 262C225 236 198 227 145 220C133 201 125 177 126 158C141 148 167 143 189 144C222 186 243 205 290 236C288 179 282 152 262 102C272 82 289 63 306 54C322 63 340 82 350 102C330 152 324 179 321 236C368 205 390 186 422 144C444 143 470 148 486 158Z"/></g><g transform="matrix(.017,0,0,-0.017,23.17,0)"><path id="g117-149" d="M567 230V280H69V230H567Z"/></g><g transform="matrix(.017,0,0,-0.017,30.794,0)"><path id="g117-148" d="M901 255C861 290 790 361 716 442L694 427L796 280H69V230H796L694 83L716 68C790 149 861 220 901 255Z"/></g><g transform="matrix(.017,0,0,-0.017,52.061,0)"><path id="g113-81" d="M600 480C600 590 528 650 384 650H143L137 622C222 614 225 607 210 531L130 127C113 41 106 36 23 28L17 0H294L300 28C204 36 195 42 212 127L243 284L314 263C327 263 339 263 352 264C465 271 600 337 600 480ZM508 481C508 351 402 304 329 304C289 304 265 311 250 317L295 559C302 594 310 606 323 611C335 616 350 619 367 619C455 619 508 573 508 481Z"/></g><g transform="matrix(.017,0,0,-0.017,62.655,0)"><path id="g113-127" d="M442 569C442 628 407 660 362 660C328 660 292 648 255 616C200 568 165 508 123 334L106 264C80 242 52 222 23 202L30 176C54 191 76 205 100 222L87 166C73 103 79 43 95 19C108 0 139 -12 169 -12C242 -12 318 38 369 106L354 129C322 93 260 49 215 49C173 49 147 81 170 187L189 274C293 344 442 452 442 569ZM380 565C380 478 267 377 196 322L216 406C254 556 282 623 332 623C364 623 380 596 380 565Z"/></g><rect height="0.8612076" width="7.8198516" x="70.637952" y="-11.488606800000001"/><g transform="matrix(.017,0,0,-0.017,70.638,0)"><path id="g185-47" d="M213 67C239 136 257 225 255 300C253 353 242 447 164 447C112 447 54 389 41 319L67 309C90 362 117 377 134 377C184 377 189 277 190 239C193 153 164 43 146 -12H176C278 73 450 253 450 390C450 439 410 463 396 463C384 463 380 459 382 449C385 440 389 415 389 402C389 284 312 146 216 67H213Z"/></g><g transform="matrix(.012,0,0,-0.012,78.458,4.134)"><path id="g50-127" d="M443 572C443 635 407 665 362 665C328 665 293 654 255 623C199 577 159 509 120 338L105 272C80 250 53 230 24 211L32 182C54 195 77 210 98 226L86 167C72 99 77 49 95 24C112 1 141 -12 174 -12C246 -12 320 36 371 105L354 129C324 97 265 54 218 54C175 54 148 87 170 191L188 277C294 349 443 451 443 572ZM380 569C380 486 267 383 198 330L218 413C256 564 284 626 332 626C362 626 380 601 380 569Z"/></g></svg> Decays

Chang, Qin; Zhu, Jie; Wang, Na; Wang, Ru-Min

doi:https://doi.org/10.1155/2018/7231354

Advances in High Energy Physics

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Abstract Introduction Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 7231354 | https://doi.org/10.1155/2018/7231354

Probing the Effects of New Physics in Decays

Qin Chang,¹Jie Zhu,^1,2Na Wang,¹and Ru-Min Wang²

Academic Editor: Enrico Lunghi

Received04 Jun 2018

Accepted02 Aug 2018

Published27 Sept 2018

Abstract

The significant divergence between the SM predictions and experimental measurements for the ratios, with , implies possible hint of new physics in the flavor sector. In this paper, motivated by the “ puzzle” and abundant data samples at high-luminosity heavy-flavor experiments in the future, we try to probe possible effects of new physics in the semileptonic decays induced by transitions in the model-independent vector and scalar scenarios. Using the spaces of NP parameters obtained by fitting to the data of and , the NP effects on the observables including branching fraction, ratio , lepton spin asymmetry, and lepton forward-backward asymmetry are studied in detail. We find that the vector type couplings have large effects on the branching fraction and ratio . Meanwhile, the scalar type couplings provide significant contributions to all of the observables. The future measurements of these observables in the decays at the LHCb and Belle-II could provide a way to crosscheck the various NP solutions to the “ puzzle”.

1. Introduction

Thanks to the fruitful running of the factories and Large Hadron Collider (LHC) in the past years, most of the mesons decays with branching fractions have been measured. The rare -meson decays play an important role in testing the standard model (SM) and probing possible hints of new physics (NP). Although most of the experimental measurements are in good agreement with the SM predictions, several indirect hints for NP, the tensions or the so-called puzzles, have been observed in the flavor sector.

The semileptonic decays are induced by the CKM favored tree-level charged current, and therefore, their physical observables could be rather reliably predicted in the SM and the effects of NP are expected to be tiny. In particular, the ratios defined by are independent of the CKM matrix elements, and the hadronic uncertainties canceled to a large extent; thus they could be predicted with a rather high accuracy. However, the BaBar [1, 2], Belle [3–5], and LHCb [6] collaborations have recently observed some anomalies in these ratios. The latest experimental average values for reported by the Heavy-Flavor Average Group (HFAG) are [7]which deviate from the SM predictions(see [8] in the first equation of (2) and [9] in the second equation of (2)) at the levels of and errors, respectively. Moreover, when the correlations between and are taken into account, the tension would reach up to level [7]. Besides, the ratio has recently been measured by the LHCb collaboration [10], which also shows an excess of about from the central value range of the corresponding SM predictions . In addition, another mild hint of NP in the induced decay has been observed by the BaBar and Belle collaborations [11–14]; the deviation is at the level of [15].

The large deviations in and possible anomalies in the other decay channels mentioned above imply possible hints of NP relevant to the lepton flavor violation (LFV) [15]. The investigations for these anomalies have been made extensively both within model-independent frameworks [16–37] and in some specific NP models where the transition is mediated by leptoquarks [16, 17, 38–46], charged Higgses [16, 47–59], charged vector bosons [16, 60, 61], and sparticles [62–65].

In addition to mesons, the vector ground states of system, mesons, with quantum number of and [66–69], also can decay through the transitions at quark-level. Therefore, in principle, the corresponding NP effects might enter into the semileptonic decays as well. The decay occurs mainly through the electromagnetic process , and the weak decay modes are very rare. Fortunately, thanks to the rapid development of heavy-flavor experiments instruments and techniques, the weak decays are hopeful to be observed by the running LHC and forthcoming SuperKEK/Belle-II experiments [70–72] in the near future. For instance, the annual integrated luminosity of Belle-II is expected to reach up to ~13 and the weak decays with branching fractions are hopeful to be observed [70, 73, 74]. Moreover, the LHC experiment also will provide a lot of experimental information for weak decays due to the much larger beauty production cross-section of collision relative to collision [75].

Recently, some interesting theoretical studies for the weak decays have been made within the SM in [73, 74, 76–82]. In this paper, motivated by the possible NP explanation for the puzzles, the corresponding NP effects on the semileptonic decays will be studied in a model-independent way. In the investigation, the scenarios of vector and scalar NP interactions are studied, respectively; their effects on the branching fraction, differential branching fraction, lepton spin asymmetry, forward-backward asymmetry, and ratio () of semileptonic decays are explored by using the spaces of various NP couplings obtained through the measured .

Our paper is organized as follows. In Section 2, after a brief description of the effective Lagrangian for the transitions, the theoretical framework and calculations for the decays in the presence of various NP couplings are presented. Section 3 is devoted to the numerical results and discussions for the effects of various NP couplings. Finally, we give our conclusions in Section 4.

2. Theoretical Framework and Calculation

2.1. Effective Lagrangian and Amplitudes

We employ the effective field theory approach to compute the amplitudes of decays in a model-independent scheme. The most general effective Lagrangian at for the () transition can be written as [19, 21, 40, 46]where is the Fermi coupling constant, denotes the CKM matrix elements, and is the negative/positive projection operator. Assuming the neutrinos are left-handed and neglecting the tensor couplings, the effective Lagrangian can be simplified aswhere and are the effective NP couplings (Wilson coefficients) defined at . In the SM, all the NP couplings will be zero.

We use the method of [83–87] to calculate the helicity amplitudes. The square of amplitudes for the decay can be written as the product of leptonic () and hadronic () tensors,where the superscripts and refer to four operators in the effective Lagrangian given by (4) (the tensors related to the scalar and pseudoscalar operators can be understood through the relations given by (21) and (22)); in the SM, corresponds to the operator . For convenience in writing, these superscripts are omitted below. Inserting the completeness relationthe product of and can be further expressed asHere, is the polarization vector of the virtual intermediate states, which is boson in the SM and named as in this paper for convenience of expression. The quantities and are Lorentz invariant and therefore can be evaluated in different reference frames. In the following evaluation, and will be calculated in the -meson rest frame and the center-of-mass frame, respectively.

2.2. Kinematics for Decays

In the -meson rest frame with daughter -meson moving in the positive -direction, the momenta of particles and areFor the four polarization vectors, , one can conveniently choose [83, 84]where and , with and being the momentum transfer squared, are the energy and momentum of the virtual . The polarization vectors of the initial -meson can be written as

In the center-of-mass frame, the four momenta of lepton and antineutrino pair are given aswhere , , and is the angle between the and three-momenta. In this frame, the polarization vector takes the form

2.3. Hadronic Helicity Amplitudes

For the decay, the hadronic helicity amplitudes and are defined bywhich describe the decay of three helicity states of meson into a pseudoscalar meson and the four helicity states of virtual . It should be noted that in , (15) and (16), should always be equal to .

For transition, the matrix elements of the vector and axial-vector currents can be written in terms of form factors and aswith the sign convention . Furthermore, using the equations of motion,one can write the matrix elements of scalar and pseudoscalars currents asin which and are the running quark masses.

Then, by contracting above hadronic matrix elements with the polarization vectors in the -meson rest frame, we obtain five nonvanishing helicity amplitudesIt is obvious that only the amplitudes with survive.

2.4. Leptonic Helicity Amplitudes

Expanding the leptonic tensor in terms of a complete set of Wigner’s -functions [9, 83, 87], can be rewritten as a compact formin which and run over 1 and 0, and run over their components, and massless right-handed antineutrinos with . In (27), are the leptonic helicity amplitudes defined asIn the center-of-mass frame, taking the exact forms of the spinors and polarization vectors, we finally obtain four nonvanishing contributions

2.5. Observables of Decays

With the amplitudes obtained in above subsections, we then present the observables considered in our following evaluations. The double differential decay rate of decay is written aswhere the factor is caused by averaging over the spins of initial state . Using the standard convention for -function [88], we finally obtain the double differential decay rates with a given leptonic helicity state , which areUsing (35) and (36), one can get the explicit forms of various observables of decays as follows: (i)The differential decay rate is(ii)The dependent ratio is where denotes the light lepton.(iii)The lepton spin asymmetry is(iv)The forward-backward asymmetry is

The SM results can be obtained from above formulae by taking .

In the following evaluations, in order to fit the NP spaces, we also need the observables of decays, which have been fully calculated in the past years. In this paper, we adopt the relevant theoretical formulae given in [46].

3. Numerical Results and Discussions

3.1. Input Parameters

Before presenting our numerical results and analyses, we would like to clarify the values of input parameters used in the calculation. For the CKM matrix elements, we use [89]For the well-measured Fermi coupling constant , the masses of mesons and leptons, and the running masses of quarks at , we take their central values given by PDG [88]. The total decay widths (or lifetimes) of mesons are essential for estimating the branching fraction; however there is no available experimental data until now. According to the fact that the electromagnetic process dominates the decays of meson, we take the approximation ; the latter has been evaluated within different theoretical models [90–96]. In this paper, we adopt the most recent results [95, 96]

Then the residual inputs are the transition form factors, which are crucial for evaluating the observables of and decays. For the transitions, the scheme of Caprini, Lellouch, and Neubert (CLN) parametrization [97] is widely used, and the CLN parameters can be precisely extracted from the well-measured decays; numerically, their values read [7]However, for the transition, there is no experimental data and ready-made theoretical results to use at present. Here, we employ the Bauer-Stech-Wirbel (BSW) model [98, 99] to evaluate the form factors for both and transitions. Using the inputs , , , , and , we obtain the results at ,To be conservative, 15% uncertainties are assigned to these values in our following evaluation. Moreover, with the assumption of nearest pole dominance, the dependences of form factors on read [98, 99]where is the state of with quantum number of ( and are the quantum numbers of total angular momenta and parity, respectively).

With the theoretical formulae and inputs given above, we then proceed to present our numerical results and discussion, which are divided into two scenarios with different simplification for our attention to the types of NP couplings as follows:(i)Scenario I: taking , i.e., only considering the NP effects of couplings(ii)Scenario II: taking , i.e., only considering the NP effects of couplings

In these two scenarios, we consider all the NP parameters to be real for our analysis. In addition, we assume that only the third generation leptons get corrections from the NP in the processes and for the NP is absent. In the following discussion, the allowed spaces of NP couplings are obtained by fitting to and (1), with the data varying randomly within their error, while the theoretical uncertainties are also considered and obtained by varying the inputs randomly within their ranges specified above.

3.2. Scenario I: Effects of and Type Couplings

In this subsection, we vary couplings and while keeping all other NP couplings to zero. Under the constraints from the data of and , the allowed spaces of new physics parameters, and , are shown in Figure 1. In the fit, the form factors based on CLN parametrization and BSW model are used, respectively; it can be seen from Figure 1 that their corresponding fitting results are consistent with each other, but the constraint with the former is much stronger due to the relatively small theoretical error. Therefore, in the following evaluations and discussions, the results obtained by using CLN parametrization are used. In addition, our fitting result in Figure 1 agrees well with the ones obtained in the previous works, for instance, [26, 35].

(a)

(b)

From Figure 1, we find that (i) the allowed spaces of are bounded into four separate regions, namely, solutions A-D. (ii) Except for solution A, the other solutions are all far from the zero point and result in very large NP contributions. Taking solution C (D) as an example, the SM contribution is completely canceled out by the NP contribution related to , and the coupling presents sizable positive (negative) NP contribution to fit data. The situation of solution B is similar, but only coupling presents sizable NP contribution. Numerically, one can easily conclude that the NP contributions of solutions B-D are about two times larger than the SM, which seriously exceeds our general expectation that the amplitudes should be dominated by the SM and the NP only presents minor corrections. In this point of view, the minimal solution (solution A) is much favored than solutions B-D. So, in our following discussions, we pay attention only to solution A, which is replotted in Figure 1(b) and numerical result is

Using the values of NP couplings given by (54), we then present our theoretical predictions for and -integrated in Table 1, in which the SM results are also listed for comparison. The -dependence of differential observables , , , and for and decays is shown in Figure 2; the case of and ( and ) is similar to the one of () decay, not shown here. The following are some discussions and comments:(1)From Table 1, it can be seen that the branching fractions of induced decays are at the level of , while the induced decays are relatively rare due to the suppression caused by the CKM factor. In addition, the difference between the branching fractions of three decay modes induced by (or ) transition is mainly attributed to the relation of total decay widths, , illustrated by (42), (43), and (44).(2)Comparing with the SM results, one can easily find from Table 1 that are enhanced about 20% by the NP contributions of and . It can also be clearly seen from Figures 2(a) and 2(b). However, as shown in Figures 2(a) and 2(b), due to the large theoretical uncertainties caused by the form factors, the NP hints are hard to be totally distinguished from the SM results.(3)The theoretical uncertainties can be well controlled by using the ratio instead of decay rate due to the cancelation of nonperturbative errors; therefore is much suitable for probing the NP hints. From the last three rows of Table 1, it can be found that the NP prediction for significantly deviates from the SM result. In particular, as Figures 2(c) and 2(d) show, the NP effects can be totally distinguished from the SM at even though the theoretical errors are considered. So, future measurements on decays can make further test on the NP models which provide possible solutions to the and problems.(4)From Figures 2(e)–2(h) it can be found that the NP contribution of solution A has little effect on the observables and in the whole region, which can be understood from the following analyses. Because the NP contribution of solution A is dominated by the left-handed coupling , we can find that in the limit of . As a result, the NP contributions (solution A) to the numerator and denominator of and cancel each other out to a large extent. For , the cases of solutions B, C, and D are similar to solution A.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

3.3. Scenario II: Effects of and Type Couplings

In this subsection, we only consider the effects of scalar interactions and and take the other NP couplings to be zero. Under the constraint from the data of and , the allowed spaces of and are shown in Figure 3. Similar to scenario I, four solutions for and are found in scenario II, which can be seen from Figure 3(a); and the fitting results obtained by using form factors in CLN parametrization and BSW model are consistent with each other. Solutions B-D result in so large NP contributions; therefore, in the following discussion, we pay our attention to solution A, which are replotted in Figure 3(b). The numerical result of solution A isUsing these values, we present in Table 1 our numerical predictions of scenario II for the observables, and -integrated . Moreover, the distributions of differential observables , , , and are shown in Figure 4. The following are some discussions for these results: (1)From Table 1 and Figures 4(a) and 4(b), it can be found that the and can be enhanced about 15% compared with the SM results by the NP contributions. Similar to the situation of scenario I, the NP effect of and on is much more significant than the one on branching fraction due to the theoretical uncertainties of which can be well controlled. In particular, as Figures 4(a) and 4(b) show, the spectra of the SM and NP for can be clearly distinguished at middle region.(2)The main difference between the effects of scalar and vector couplings on the decays is that the former only contributes to the longitudinal amplitude, which can be found from (37). As a result, their effects on and are a little different, which can be seen by comparing Figures 2(a)–2(d) with Figures 4(a)–4(d).(3)Another significant difference between the scalar and vector couplings is that only the leptonic helicity amplitudes of scalar type with survive, which can be easily found from (35) and (36). Therefore, as Figures 4(e) and 4(f) show, the scalar couplings lead to significant NP effects on , which is obviously different from predictions of vector couplings in scenario I (Figures 2(e) and 2(f)). Besides, as Figures 4(e) and 4(f) show, and couplings also have large contributions to the at all region, which is another difference with the vector couplings (Figures 2(g) and 2(h)). Therefore, the future measurements on these observables will provide strict tests on the SM and various NP models.

(a)

(b)

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

4. Summary

In this paper, motivated by the observed “ and puzzles" and its implication of NP, we have studied the NP effects on the induced semileptonic () decays in a model-independent scheme. Using the allowed spaces of vector and scalar couplings obtained by fitting to the data of and , the NP effects on the decay rate, ratio , lepton spin asymmetry, and forward-backward asymmetry are studied in vector and scalar scenarios, respectively. It is found that the vector couplings present large contributions to the decay rate and , but their effects on and are very tiny. Different from the vector couplings, the scalar couplings present significant effects not only on the decay rate and but also on and . The future measurements on the decays will further test the predictions of the SM and NP and confirm or refute possible NP solutions to and .

Data Availability

The experimental data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11475055), the Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant No. 201317), and Program for Innovative Research Team in University of Henan Province (Grant No. 19IRTSTHN018).

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Copyright © 2018 Qin Chang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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