Published for SISSA by Springer
Received: October 10, 2012 Revised: March 21, 2013
Accepted: April 6, 2013
Published: April 24, 2013
Standard Model prediction and new physics tests for D0 ! h+1h2+(h = , K; = e, )
Luigi Cappiello,a,b Oscar Catc and Giancarlo DAmbrosiob
aDipartimento di Scienze Fisiche, Universit di Napoli Federico II, Via Cintia, 80126 Napoli, Italia
bINFN-Sezione di Napoli,
Via Cintia, 80126 Napoli, Italia
cArnold Sommerfeld Center for Theoretical Physics, Fakultat fr Physik, Ludwig-Maximilians-Universitat Mnchen,D80333 Mnchen, Germany
E-mail: mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected]
Abstract: Motivated by the recent evidence for direct CP-violation in D0 h+h decays, we provide an exhaustive study of both Cabibbo-favored and Cabibbo-suppressed (singly and doubly) D0 h+1h2+ decays. In particular, we study the Dalitz plot for the
long-distance contributions in the (m2ll, m2hh) parameter space. We nd that near-resonant e ects, i.e., D0 V (h+1h2)+ with V = , K, , are sizeable and even dominant (over
Bremsstrahlung) for the + decay modes, bringing the branching ratios close to the LHCb reach. We also provide a detailed study of the angular asymmetries for such decays and identify signatures for new physics detection. In particular, new physics signals can be neatly isolated in asymmetries involving the semileptonic operator Q10, where for typical new physics scenarios the e ects can be as sizeable as O(1%) for the doubly Cabibbo-
suppressed modes.
Keywords: Rare Decays, Standard Model, Beyond Standard Model
c
JHEP04(2013)135
SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP04(2013)135
Web End =10.1007/JHEP04(2013)135
Contents
1 Introduction 1
2 Long-distance hadronic contributions 32.1 Bremsstrahlung 52.2 Resonant contributions 62.3 Results and discussion 112.3.1 D0 K+l+l 122.3.2 D0 +l+l 122.3.3 D0 K+Kl+l 13
3 Short-distance e ective Hamiltonian 133.1 Standard model prediction 133.2 New physics scenarios 14
4 Angular asymmetries 154.1 T-odd asymmetry 164.2 Forward-backward asymmetry 17
5 Conclusions 20
A Kinematics 22
B Hadronic parameterization 24
C Di erential decay widths 25
1 Introduction
Processes involving avor changing neutral currents (FCNC) are loop-suppressed in the standard model and therefore are especially suited as probes of new physics. In D decays the suppression is even more accentuated that in B or kaon decays: in charm physics FCNC involve down-type quarks and as a result the GIM mechanism is more e cient. Since new physics does not have to be subject to the same GIM suppression, charm decays are in principle an ideal arena to test physics beyond the standard model. The situation is of course not so simple: light quarks carry the bulk contribution to the decays, which means that they are long-distance dominated. The resulting hadronic uncertainties, due to their nonperturbative nature, are very di cult to estimate and overshadow short-distance e ects, making their detection rather challenging. The situation gets worse because charm
1
JHEP04(2013)135
physics does not seem to accept an e ective eld theory description and one has to resort, for instance, to lattice computations or hadronic models.
Quite recently the LHCb [1] and CDF [2] collaborations reported convincing evidence for direct CP violation in D0 +, K+K decays. Specically, they found that the
CP asymmetry aCP aKKCP aCP gives a nonzero value, that averaged with the previous
results of B factories [3, 4] gives [5]
aCP = (0.68 0.15)% (1.1)
A standard model interpretation of the previous results is not ruled out [68] but seems hard to accomodate (see, e.g. [9, 10]), even allowing for generous uncertainties in the estimation of hadronic matrix elements. (This picture could change drastically, however, if the results of the latest LHCb analysis [11, 12] are conrmed, where no signicant deviation from the standard model is observed.)
If new physics is the explanation behind eq. (1.1), then one should scrutinize other decay modes in search of similar large e ects. For instance, large CP-violating e ects for radiative D0 V decays would naturally point out at sizeable electromagnetic pen
guins [13], which is a rather common feature of many extensions of the standard model.
In this paper we will study the semileptonic 4-body decays D0 h+1h2+, ( = e, ) in all its variants: Cabibbo allowed (K+), singly Cabibbo-suppressed (+; K+K)
and doubly Cabibbo-suppressed (K+). Such a study is rather timely: LHCb has recently reported the potential for reaching branching ratios of 106 in the dimuon decays at the 3 level [14, 15], improving previous upper bounds [16, 17] by one order of magnitude. BESIII [18] should also be able to study 4-body semileptonic D0 decays in the near future.
On the theoretical side, there are a number of motivations to study these rare decays:
The angular structure of a 4-body decay allows to dene a variety of di erential
distributions. The associated Dalitz plots become essential tools for detailed tests of both the standard model and new physics. Of special importance are the di erent angular asymmetries that one can construct, which allow for a clean separation of short and long-distance e ects.
Access to short-distance physics is not limited to the charge asymmetry. In particular,
some observables o er the possibility of disentangling new physics contributions in observables with tiny standard model backgrounds, like forward-backward asymme-tries. This is especially interesting for the doubly Cabibbo-suppressed mode, which is particularly sensitive to new physics.
In a nutshell, the penalty of small branching fractions one naturally pays in 4-body decays as opposed to 2 or 3-body decays is overly compensated by the diversity (and the size) of the asymmetries one can build. This implies that the semileptonic D0 2h2l decays have large potential to single out exceptionally clean experimental signatures.
The only reference for D0 h+1h2+ in the literature is the recent work in [19],
where generic estimates are given for the branching ratios. In this work we will rene these estimates for the eight channels under study, detailing the di erential distributions
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of the di erent long-distance contributions (bremsstrahlung and both magnetic and electric hadronic components) in the (m2, m2hh) plane. Our results for the branching ratios are substantially larger than previously estimated in [19]: D0 K+K+ hovers around 107, while we expect D0 ++ to be within the reach of LHCb. The Cabibbo-
allowed mode D0 K++ is predicted in the high 106 while the doubly Cabibbo-
suppressed D0 K++ is estimated at 108.
We will then proceed to study angular asymmetries that isolate short-distance e ects. We will concentrate on two asymmetries that, due to tiny standard model backgrounds, are clean tests of new physics. First we will consider a T-odd asymmetry A, resulting from the interference between the electric and magnetic hadronic pieces, where is the angle between the dihadron and dilepton planes. This asymmetry involves four-quark operators and can be parametrized in terms of a weak phase W . Next we will consider the forward-backward asymmetry for the dilepton pair, AFB, which involves the semileptonic penguin operator Q10. While no generic prediction of their magnitude can be given (without resorting to models of new physics), we show that in both cases the signal is concentrated around the resonant region, i.e., along the line dened by m2 (0.5 1) GeV2 and m2hh = m2H, where
H = , K, for hh = +, K+, K+K, respectively. Taking some reference values for W and C10 we show that A (1 8)%, where the D0 ++ modes are
the most favored, while AFB can reach O(1%) for the doubly Cabibbo-suppressed decay
D0 K++.
We will organize this paper as follows: in section 2 we describe long distances, both Bremsstrahlung and hadronic contributions, and discuss their properties for the di erent decay modes in the (m2ll, m2hh) Dalitz plot. In section 3 we rst review the short-distance e ects to D0 2h2 within the standard model and then discuss new physics scenarios
that can enhance semileptonic operators while complying with aCP and current bounds from avor physics. Section 4 is devoted to angular asymmetries, where we study A and
AFB as two examples of clean tests of new physics. Conclusions are given in section 5, while technical details are collected in three appendices.
2 Long-distance hadronic contributions
As mentioned in the Introduction, D0 h+1h2+ are largely dominated by long-distance
e ects. The bulk of the decay width thus comes from light down-type quarks running into one-loop diagrams. At low energies, when s 1 and dynamics become nonperturbative,
the picture that applies is shown in gure 1, where the blob collects hadronized strange and down quarks. At the same time, the dilepton pair creation is dominated by photon exchange. The amplitude for D0 h+1h2 h+1h2+ can be parametrized as
MLD L(k+, k)H(p1, p2, q) (2.1)
where L is the leptonic current
L(k+, k) =
3
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e q2
(k)v(k+)
[bracketrightbig]
(2.2)
h+1(p1)
h2(p2)
D0(P)
(q)
+(k+)
(k)
Figure 1. Photon-mediated D0 h+1h2+ decay with our kinematic conventions. The blob
represents the hadronic tensor H.
and H is the hadronic vector, which can be written in terms of three form factors Fi:
H(p1, p2, q) = F1p1 + F2p2 + F3p1p2q (2.3)
The previous hadronic vector is also present in radiative D0 h+1h2 decays. There
it is common to perform a multipole expansion to distinguish the electric and magnetic components [20], depending on whether the dihadron pair is in an intrinsic parity-even (electric) or parity-odd (magnetic) state. In terms of the form factors in eq. (2.3), F1,2 are
electric contributions, while F3 is the magnetic one. Here we will import this language to our decay.
In terms of the momenta p1,2, q = k+ + k and Q = k+ k, the squared amplitude
can be cast as
Xspins|MLD|2 =2e2 q4
JHEP04(2013)135
[bracketleftBigg]
3 |Fi|2Tii + 2Re
3
Xi<j(F iFj)Tij
[bracketrightBigg]
(2.4)
where (i, j = 1, 2)
Tij = q piq pj Q piQ pj q2pi pj;
Ti3 = (Q pi)p1p2qQ;
T33 = 4m2
(m2h1m2h2 (p1 p2)2)q2 m2h1(q p2)2
m2h2(q p1)2 + 2p1 p2q p1q p2
+ (Q p1)2
(q p2)2 q2m2h2
+ (Q p2)2
(q p1)2 q2m2h1
+ 2Q p1Q p2(q2p1 p2 q p1q p2) (2.5)
Quite generally, there are three dynamically distinct long-distance contributions: (i) internal Bremsstrahlung, i.e., QED radiation of photons away from the weak D0 h+1h2 vertex; (ii) direct emission, where strong and weak e ects get combined and resonance exchange is typically assumed to be the main contribution; and (iii) the charge radius contribution, where the photon gets radiated as a result of D0 D0 mixing.
In the following subsections we will discuss, in turn, the Bremsstrahlung and direct emission contributions, giving expressions for the angular-integrated decay rates and branching ratios. The results will be shown in Dalitz plots for d2 /dq2dp2.
4
In this paper we will not discuss the charge radius contribution since it bears no e ect on the di erent angular asymmetries. Its contribution is limited to the decay width and subject to large uncertainties [19]. With this in mind, our results for the decay width should thus be considered as a lower bound. However, its hard to imagine that the charge radius can increase the decay width dramatically, so for the purposes of this paper omitting its contribution should be a reasonable approximation.
2.1 Bremsstrahlung
The radiation of photons away from the weak D0 h1h2 vertex is a genuine infrared e ect
that can be computed with Lows theorem [21]:
Mb(D0 h+1h2) = 2eM(D0 h+1h2) [bracketleftbigg]
p1 2p1 q + q2
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p2 2p2 q + q2
[bracketrightbigg]
(2.6)
Since the photon emission is factored out, the only ingredient that one needs is |M(D0 h+1h2)|, which can be extracted from experiment [22] for the di erent decay modes. Hence
forth we will assume that there are no CP-violating phases in the D0 h+1h2 amplitudes.
In terms of the form factors entering H, eq. (2.6) implies that
F (b)j = (1)j1
2ie(1 j3)
2q pj + q2 M(Dh1h2)
(2.7)
Inserting the previous expression in eq. (2.4), one can easily build the di erential decay rate. The kinematics of 4-body decays requires ve variables. In this paper we will use the Cabibbo-Maksymowicz set of variables, Xi = (q2, p2, h, , ), whose concrete denitions
are given in appendix A. In terms of these variables, one can express the di erential decay rate quite generically as [23, 24]
d5
dxdy = A1(x) + A2(x)s2 + A3(x)s2c2 + A4(x)s2c
+ A5(x)sc + A6(x)c + A7(x)ss + A8(x)s2s + A9(x)s2s2 (2.8)
where the vectors x = (q2, p2, cos h) and y = (cos , ) split the dynamical variables (entering the form factors Ai(x)) from the pure kinematical angular distribution described
by y. In the following we will integrate the full angular dependence above and concentrate on the quantity d2 /dq2dp2. Details thereof, including the analytical formulae, can be found in appendix C.
In gure 3 we show the Dalitz plot in the (q2, p2) plane for the di erent decay modes. Since Bremsstrahlung is an infrared e ect driven by photon emission, one expects a signicant contribution only in the low-q2 region, with a sharp increase close to the dilepton threshold, whose maximum is reached in the region of large hadron recoil (p2 (mD
2m)2). In table 1 one can read o the resulting branching ratios for the di erent decay modes. For dimuon decays, branching ratios span between 107 1010, where the di er
ences are solely due to the hierarchy between the hadronic branching ratios Br(D0 h1h2). For electron-positron decays there is an increase of two orders of magnitude per channel,
5
Decay mode Bremsstrahlung Direct emission (E) Direct emission (M) D0 K+e+e 9.9 106 6.2 106 4.8 107
D0 +e+e 5.3 107 1.3 106 1.3 107
D0 K+Ke+e 5.4 107 1.1 107 5.0 109
D0 K+e+e 3.7 108 1.7 108 1.3 109
D0 K++ 8.6 108 6.2 106 4.8 107
D0 ++ 5.6 109 1.3 106 1.3 107
D0 K+K+ 3.3 109 1.1 107 5.0 109
D0 K++ 3.3 1010 1.7 108 1.3 109 Table 1. Long-distance contributions to the branching ratio for the di erent decay modes.
with branching ratios in the window 105 108. This increase precisely illustrates the
strength of Bremsstrahlung at low-q2: electron-positron decays have a lower q2-threshold and therefore probe infrared physics deeper. Notice that our results for the Bremsstrahlung are substantially larger than the rough estimates given in [19] by 23 orders of magnitude.
Finally, we want to emphasize that the previous results for the Bremsstrahlung contribution only rely on Lows theorem (i.e., QED) and experimental input for D0 h1h2.
Therefore, the results in the rst column of table 1 should be seen as a solid lower estimate of the total branching ratios for the di erent decay channels.
2.2 Resonant contributions
Besides Bremsstrahlung, there are also long-distance contributions associated with hadronic e ects, the most important of which are the near-resonant regions in both the dihadron and dilepton sectors. Depending on the strength of the resonant e ect, these contributions have the potential of overcoming the Bremsstrahlung contribution discussed in the previous section. Similar contributions have been studied in the B K(K)l+l decay [25]. The
situation is in sharp contrast with what one encounters in kaon 4-body decays [2628] at least in two aspects: (i) in Ke4 decays the energy range is always far below resonance thresholds. As a result, Bremsstrahlung is overwhelmingly dominant; and (ii) additionally, the kaon direct emission contribution can be estimated in chiral perturbation theory.
In the present case, the phase space is larger and decays like D0 V 0+ have nonnegligible branching ratios [29]. As a result, D0 V 0(h+1h2)+ are expected to play
a signicant role. However, unlike kaon decays, such contributions cannot be estimated with e ective eld theories and one has to resort to hadronic models. Therefore, one has to exercise some caution when interpreting the results: hadronic e ects are genuinely nonperturbative and as such uncertainties are di cult to estimate. One should bear in mind that the results of this section are not on the same solid ground as the Bremsstrahlung contributions discussed above.
Quite generally, we will be considering underlying processes of the form depicted in
6
JHEP04(2013)135
h1(p1)
h2(p2)
T (p, q)
B(p1, p2)
D0(pD)
V (p)
(q)
Figure 2. Diagrammatic representation of the hadronic model adopted, with the weak T (p, q)
and strong B(p1, p2) correlators.
gure 2 where
T V(p, q) = i3 [integraldisplay]
d4xe(ipx+iqy)h0|Jc(0)JV (x)J(y)|0i= tV1 qq + tV2 qp + tV3 g + tV4 pq + tV5 pp + tV6 pq (2.9)
and
BV (p1, p2) = i3
[integraldisplay]
d4xe(ip1x+ip2y)h0|JV (0)J1(x)J2(y)|0i= bV1 p1 bV2 p2 (2.10)
parameterize the most general tensorial decomposition of the D0 V (p)(q) and V (p)
h1(p1)h2(p2) vertices. tVi and bVi are functions of the kinematical invariants of the problem, namely tVi (q2, p2) and bVj (p2), where the upper index denotes their dependence on the exchanged resonance.
The hadronic vector H can be expressed as
H(p1, p2, q) =
XV
hh1h2|H|V i
()(q)
JHEP04(2013)135
PV (p2)hV |H|D0i (2.11)
where we will describe particle widths with a Breit-Wigner propagator, i.e., Pj(q2) =
q2 m2j + i jmj, and
hh1h2|H|V i = BV (p1, p2)(V )(p)
hV |H|D0i = T V(p, q)(V )(p)()(q) (2.12)
In order to evaluate the matrix elements above we need to determine the functions bi and ti.
The former can be easily determined from the experimental V h1h2 decays. Notice that
for the equal mass case, charge conjugation invariance (in our case V is neutral) imposes that bV1 = bV2 . When mh1 6= mh2 the form factors di er by a term proportional to the mass
di erence (see appendix B). In general, it is a good approximation to neglect this e ect and henceforth we will use that bV1 = bV2 bV for all decay modes. We will also assume that the
momentum dependence is soft and that to a good approximation bV (p2) = bV (m2V ) bV .
7
The V h1h2 decay width is
V h1h2 =
148 b2V m5V 3/2(m2V , m2h1, m2h2) (2.13)
and comparison with the experimental determinations [22] gives
b = 5.92 GeV; bK = 5.46 GeV; b = 4.41 GeV (2.14)
The contribution of the (782) to D0 ++ is extremely suppressed (b/b 3%)
and will be neglected.
The evaluation of the weak vertex requires the c = 1 e ective hamiltonian. For the decay channels that we are considering, the relevant operators for the Cabibbo-allowed (C), singly Cabibbo-suppressed (SCS) and doubly Cabibbo-suppressed (DCS) transitions are
HC c=1 =
GF
2 [bracketleftBig]
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sdC(sd)2Qsd
[bracketrightBig]
HSCS c=1 =
GF
2 [bracketleftBig]
dC(d)2Qd + sC(s)2Qs
[bracketrightBig]
HDCS c=1 =
GF
2 [bracketleftBig]
dsC(ds)2Qds
[bracketrightBig]
(2.15)
where j = V cjVuj, sd = V csVud, ds = V cdVus and
Qsd = (c)L(sd)L
Qd = (c)L( dd)LQs = (c)L(ss)LQds = (c)L( ds)L (2.16)
In the previous equations we have neglected the penguin operators. To proceed further one needs to make some approximations. In this work we will assume that the factorizable contributions are the dominant ones, which means that
hV |JijJc|D0i = hV |Jij|0ih|Jc|D0i + h|Jij|0ihV |Jc|D0i
+ hV |Jij|0ih0|Jc|D0i (2.17) where ij = dd, ss, sd, ds. We will further assume that
(i) The weak annihilation contribution (second line) is negligible compared to the spectator ones (rst line) for all the processes to be considered (see, e.g., [30] for a detailed discussion).
(ii) The Zweig rule is at work, i.e., avor annihilation is suppressed and a possible enhancement due to nal state interactions is excluded.
(iii) The photon is created mainly through vector meson exchange and we will neglect a direct photon coupling. We will consider the exchange of the lowest-lying neutral states (, , ) to be dominant.
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Figure 3. Long-distance contributions to the di erential decay width (in arbitrary units) in the (m2ll, m2hh) plane for the di erent decay modes (from top to bottom: K, + and K+K, respectively). mll and mhh are given in GeV. Above we show the e+e modes. For the dimuon case the only di erence happens for the Bremsstrahlung, which is strongly suppressed due to lepton threshold e ects.
9
The last point implies that
hV |JijJc|D0i =
XV
h|HV |V i
1PV (q2)hV V |JijJc|D0i (2.18)
and requires the electromagnetic couplings of vector mesons, which can be inferred from
HV =
14hVV i +
fV e
2mV FhQV i (2.19)
where fV is dened by
hV (k, )|J|0i = fV mV (k) (2.20)
For phenomenological purposes we will break the SU(3) symmetry and take the experimental values for the decay couplings. The interaction term in the previous Hamiltonian will thus be replaced by
HV = e
fm +
f3m
2f
3m [parenrightBigg]
A (2.21)
To complete the picture, we need the hV (p)|Jc|D0(P )i matrix elements, which can
be parameterized as (p+ P + p, k P p)
hV (p, )|Jc|D0(P )i = D1(k2)p+ + D2(k2)k + D3(k2) + iD4(k2)p+k (2.22)
A determination of the di erent form factors Di(k2) is given in ref. [31]. We refer to appendix B for details. Matching eqs. (2.17) and (2.12) the result for the ti(q2, p2) form
factors is
tV2 (q2, p2) =
2iV mD + m
JHEP04(2013)135
hJV (q2)2(p2) + V W (q2)2(q2)
[bracketrightBig]
tV3 (q2, p2) = iV (mD + m)
hJV (q2)1(p2) + V W (q2)1(q2)
[bracketrightBig]
tV4 (q2, p2) = 2V
mD + m
hJV (q2) V (p2) + V W (q2) V (q2)
[bracketrightBig]
(2.23)
where j = Cj2j eGF
2 , V = 0, 1 depending on the channel and
JV (q2) = q2
[parenleftbigg]
f mP(q2) +
f 3mP(q2)
fV mV
W (q2) = q2 f2
P(q2) +
f2 3P(q2)
2f2
3P(q2)
[parenrightBigg]
2(k2) = hA2m2A2 m2A2 k2
1(k2) = hA1m2A1 m2A1 k2
V (k2) = hV 1m2V 1
m2V 1 k2
(2.24)
10
As side remarks, we note that: (i) the rst and second terms in eq. (2.23) correspond to the rst and second lines of eq. (2.17), respectively. The absence of -exchange in JV (q2) is
thus a consequence of the Zweig rule; and (ii) the global prefactor (mD +m) in eqs. (2.23) results from considering the (770) and (782) nearly degenerate.
Bringing all the pieces together and matching to eq. (2.3) the form factors take the form:
F (V )1 = iaV21q p1 + aV22q p2 + aV31
PV (p2)
F (V )2 = iaV21q p1 + aV22q p2 + aV32
PV (p2)
F (V )3 = aV4
PV (p2) (2.25)
where
aV21(q2, p2) = aV22(q2, p2) = ibV tV2 (q2, p2) aV31(q2, p2) = aV32(q2, p2) = ibV tV3 (q2, p2)
aV4 (q2, p2) = 2bV tV4 (q2, p2) (2.26)
2.3 Results and discussion
In order to proceed to a numerical analysis for the di erent decay channels we need to estimate the parameters entering the di erent form factors. For the time being, and given the theoretical uncertainty in our results (which, while di cult to estimate, can easily amount to 30 50%), we will content ourselves with reference values for the di erent
input parameters.In addition to the couplings in eq. (2.14) for the V h+1h2 couplings, we will take [32] f = 216 MeV, fK = 220 MeV,f = 187 MeV, f = 215 MeV (2.27)
for the vector meson coupling constants. Regarding the2,1, V form factors in eq. (2.24), their residues have been determined experimentally for the D0 K transition [22, 33, 34].
In this work we will consider rV = hV 1/hA1 1.7 and r2 = hA2/hA1 1. Additionally, we
will assume that hA1 = 0.55 [35], which leads to hV 1 = 0.94 and hAi = 0.55. We will consider the previous numbers to be avor-blind. Regarding the poles, we will adopt the usual values, namely mV 1 = 2110 MeV and mA1 = mA2 = 2530 MeV [31]. We will also assume that the Wilson coe cient C2 0.55 for all the channels [30]. The remaining parameters
(CKM matrix elements, decay widths and vector meson masses) are taken from [22].
The main results are collected in table 1 and gure 3. In table 1 we have listed the contributions of Bremsstrahlung, electric and magnetic emission to the branching ratios for the di erent channels (the electric-Bremsstrahlung interference is only at the 1 2% level
and has been omitted). The amplitudes turn out to be largely dominated by the electric pieces (Bremsstrahlung and electric emission for the e+e decays, or only electric emission for the + decays) with weights that depend on the channel under study. Figure 3 shows
11
JHEP04(2013)135
the di erential decay widths for the di erent channels in the (m2ll, m2hh) plane. Analytical expressions thereof can be found in appendix C. Notice that, in agreement with Lows theorem, the low-q2 region is dominated by the Bremsstrahlung. The resonance contributions (or any other contributions) should have a smooth low-q2 behavior. Therefore, we do not quite understand the low-q2 enhancement found in ref. [36] for D0 V +, but it
denitely cannot be ascribed to factorization, as suggested in [38].1
In the following we will discuss each separate channel in turn, focussing on their specic features.
2.3.1 D0 K+l+l
The Bremsstrahlung contribution is listed in appendix C. The dominant resonance contribution to this Cabibbo-allowed decay comes from D0 KV , with subsequent decays
K K+ and V l+l. One can easily get convinced that strangeness conser
vation forbids the second line in the factorization formula of eq. (2.17). The form factors get therefore reduced to
tK2(q2, p2) =
tK3(q2, p2) = iK (mD + m)JK(q2)1(p2)
tK4(q2, p2) = 2K
mD + m JK(q2)
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2iK
mD + m JK(q2)
2(p2)
V (p2) (2.28)
The combination of long-distance contributions is shown in the top panel of gure 3, where one can see the hadronic emission sitting (mostly) at (m2, m2K). The (782) contributions is also present but too small to be noticed by the naked eye. As expected, the
Bremsstrahlung and resonance regions lie rather far apart, which explains why their interference is negligible. For the e+e case, the total branching ratio adds up to 1.6 105,
with a slight dominance of the Bremsstrahlung, while for + we nd 6.7 106, coming
largely from the electric emission.
The same qualitative features apply for the doubly Cabibbo-suppressed modes. In this case, branching ratios add to 5.5 108 for electrons and 1.8 108 for muons.
2.3.2 D0 +l+l
The direct emission contribution for the D0 + decay comes mainly from the near-
resonant decays D0 0 and D0 . As we argued in the previous subsection, for
phenomenological purposes it is a good approximation to neglect the contribution due to its tiny branching ratio into two pions.
Figure 3 shows both Bremsstrahlung and direct emission contributions. The resonance region now shows distinct peaks at (m2, m2) and (m2, m2) of comparable size. Contrary to the Cabibbo-allowed mode, in this case both spectator pieces contribute and eq. (2.23)
assumes its more general form for the ti.
The branching ratios for the e+e and + are respectively 2.0 106 and 1.5 106.
We note that our results for the direct emission are in very good agreement with ref. [38],
1Apparently, the low-q2 enhancement was corrected in [37] (S. Fajfer, private communication).
12
where the 3-body decays D0 l+l were considered. Our results are especially interesting
in view of the prospects of LHCb to be able to detect signals at 106 in Cabibbo-suppressed D0 h+1h2+ decays [14, 15].
2.3.3 D0 K+Kl+l
The direct emission contribution comes in this case mainly from a near-resonant D0 .
Similarly to the Cabibbo-allowed case, only the rst line survives in factorization and therefore the form factors reduce to eq. (2.28), with the obvious substitution K .
The resonance region is peaked at (m2, m2), such that Bremsstrahlung and direct emission populate opposite sides of the phase space in the Dalitz plot. The total branching ratios for the e+e and + are respectively 6.5 107 and 1.1 107, with a slight
dominance of the Bremsstrahlung for the electron case.
3 Short-distance e ective Hamiltonian
In the previous sections we have studied the (dominant) long-distance contributions to the D0 h+1h2+ decays. In this section we turn our attention to short distances. Our main
aim is to single out an observable without long-distance background. In charm decays, such an object is automatically a new physics probe, since the standard model contribution is not only loop-suppressed, but additionally shrinked by the GIM mechanism. In the following, we will rst assess the standard model prediction for D0 h+1h2+ and then discuss new physics scenarios with enhanced semileptonic operators and their connection with aCP.
3.1 Standard model prediction
The c ul+l transition inside the standard model is mediated by electromagnetic and Z
penguin diagrams and by W box diagrams at one-loop order. At energies right below the charm threshold it is described by the following e ective lagrangian
H(cul
+l)e = H(cu)e
where i = V ciVui, i = d, s run through the down-type quarks and
Q9 = (PLc)(
)
5) (3.2)
The full list of operators in H(cu)e can be found in [29]. The leading contribution in
c u comes from the electromagnetic penguin operator Q7L
Q7L = i e2
42 mc
JHEP04(2013)135
GF
2
Xii
hC(i)9Q9 + C(i)10Q10
[bracketrightBig]
(3.1)
Q10 = (PLc)(
qq2 (PRc)(
) (3.3)
The remaining operators are O(s)-suppressed [39] and will be neglected. We have also
omitted Q7R and Q9,10, which are weighted by mu and can be safely neglected.
13
The RG running of the previous operators was rst computed in [40] to the leading logarithm approximation. QCD corrections to c u were shown to lead to a signicant
enhancement of C7L [29], where the strongest e ect comes from the two-loop contribution: mixing with mainly Q2 softens the GIM suppression from power-like to logarithmic [41]. This phenomenon was rst pointed out in [42] and later applied to B decays [43] and kaon decays [44, 45]. Despite this enhancement, when it comes to c ul+l, Q7L is overshadowed by Q9 at least by one order of magnitude, since its mixing with Q2 happens already at tree level [39]. Although one-loop QCD corrections are signicant [46] and e ectively reduce the value of C9, it still constitutes the dominant contribution to the short-distance standard model estimate of c ul+l [48]. In contrast, Q10 does not mix
with Q2 and is invariant under the renormalization ow, which makes its value extremely suppressed in the standard model:
C10(mc) = C10(mW )
m2s m2W
JHEP04(2013)135
(3.4)
For inclusive c ul+l transitions, the branching ratio has recently been estimated at
3.7 109 for electron-positron dilepton pair [48]. For the dimuon case, it is reasonable to
expect a factor 5 suppression. Taking into account our results in table 1, this entails that short-distance e ects have a negligible impact on the decay width, as expected. Even if one is extremely conservative and makes the comparison only with the Bremsstrahlung, short distances might be competitive only for the dimuon Cabibbo-suppressed modes. However, disentangling short from long distances in those cases is extremely challenging: since the bulk of the short distances comes from C9, the short-distance piece has the same distribution in the Dalitz plot as the long-distance one, i.e., the short-distance contribution will simply pile up on top of the long-distance contribution.
3.2 New physics scenarios
In this section we will discuss two di erent new physics scenarios, namely SUSY in the single-insertion approximation and generic Z-enhanced models, that can generate aCP at
the experimentally observed level through enhanced magnetic penguins. Our main goal will be to determine the typical sizes they induce for semileptonic operators.
In SUSY models within the single-insertion approximation, the s-quark gluino loop can easily account for aCP if [13]
|ImC7,8(mc)| 4 103 (3.5)
Quite generally one can show that, for the hadronic part of the D0 2h2l decay, qh+|PRc|D0i = iT q2h+|PLc|D0i (3.6)
where T parametrizes the relative strength of the vector and tensor matrix elements. Its magnitude can be estimated with di erent hadronic models, but one naturally expects that T O(1). Using the denitions of Q7L and Q9 given in the previous section, one
concludes that
h2h2l|Q7L|D0i =
mcT h2h2l|Q9|D0i (3.7)
14
While the previous relation holds at the operator level, in practice one expects that C9 will be suppressed for chirality reasons.
Larger values for both real and imaginary parts of C9 and C10 can be generated with other mechanisms, for instance with double-insertion diagrams correcting the Z vertex (e.g. the s-quark gluino loop). In order to be more general, one can parametrize these new physics e ects correcting Z vertices in an e ective eld theory language. These Z-enhanced scenarios are governed by the Lagrangian:
LNP =
g2cW qi [bracketleftBig]
qjZ (3.8)
Semileptonic operators receive contributions from tree-level Z-exchange diagrams, with the result:
CNP9 =
CNP10 = gucL
D0 mixing puts stringent bounds on the up-type couplings, |gucL| < 2 104 [47].
Assuming gucL to be real, this translates into CNP9 < 0.1 and CNP10 < 1.25. In [47] it was observed that with guc couplings alone magnetic penguins cannot be enhanced to t aCP. However, one can account for aCP if one-loop diagrams with top exchange are sizeable. This is feasible because the top sector is only loosely constrained (|gutL| < 2 102). For the semileptonic operators, such one-loop contributions will typically be of the form
C1loop9,10
C9,10
gutL(gctL)gucL O
gijLPL + gijRPR
gucL
b (1 4s2W )
[bracketrightBig]
JHEP04(2013)135
b (3.9)
D0
(1) (3.10)
As we will discuss in the following section, new physics in 4-body D0 decays is not restricted to CP-violating observables. In particular, one can dene clean observables for new physics sensitive to the real parts of C9,10, hence not constrained by aCP. In the following, we will assume that the CP-violating phases entering aCP come mainly from the right-handed Z couplings while the left-handed couplings are mainly real and generate C9,10 O(1).
Incidentally, we note that this complies with the values adopted in Little Higgs Model scenarios [37, 48]. In particular, ref. [37] used C9,10 = 4. In the following we will adopt a more conservative C9,10 = 1.
4 Angular asymmetries
The most general angular distribution for a 4-body decay can be parametrized, using the Cabibbo-Maksymowicz set of variables, as in eq. (2.8), which we repeat here for convenience:
d5
dxdy = A1(x) + A2(x)s2 + A3(x)s2c2 + A4(x)s2c
+ A5(x)sc + A6(x)c + A7(x)ss + A8(x)s2s + A9(x)s2s2 (4.1)
15
The pieces contributing to the decay width were studied in section 2 and correspond to the rst line above. The remaining angular structures can be generated by interference e ects, either at the hadronic or at the leptonic vertex, and can be probed with di erent angular asymmetries. It is important to emphasize that, due to the rich angular structure of eq. (4.1), angular asymmetries are not restricted to charge asymmetries. Moreover, there are asymmetries that, due to tiny standard model backgrounds, turn out to be clean probes of new physics. In this section we will concentrate on two such asymmetries. Far from being exhaustive, we just want to provide some exploratory indications of what are their expected signals in the Dalitz plot and parametrize its size in terms of a few short-distance parameters.
4.1 T-odd asymmetry
It was long noted, in the context of KL +e+e, that the interference between
the magnetic and Bremsstrahlung contributions can lead to large CP violation [49]. This interference is genuine of 4-body decays and contributes to A8,9 in eq. (4.1). Therefore, it
can be singled out by an angular asymmetry in the diplane angle :
A = hsgn(sc)i =
2
3/2
[integraldisplay]
ddd (4.2)
where we have dened the piece-wise angular integration
[integraldisplay]
2
1
2
0
JHEP04(2013)135
0 d [bracketleftBigg][integraldisplay]
/2
[integraldisplay]
[integraldisplay]
[bracketrightBigg]
/2 + [integraldisplay]
3/2
d (4.3)
This asymmetry actually collects not only the Bremsstrahlung vs. magnetic interference, but also the electric vs. magnetic one. In kaon physics the former dominates and, since the Bremsstrahlung and magnetic contributions have di erent strong phases, A becomes a probe of long-distance physics (see, e.g., [28]).
In D decays, in contrast, the electric vs magnetic interference is expected to be dominant, because both pieces have the same structure in the Dalitz plot. In comparison, the Bremsstrahlung vs magnetic interference here is severely suppressed, even allowing for large strong phases. Interestingly, the electric vs magnetic interference can only be nonzero in the presence of weak phases. In section 2 we evaluated the hadronic matrix elements for the dominant Q2 four-quark operator. Weak phases emerge when other four-quark operators are also considered, such as Q1 or the QCD penguins, so that their presence is naturally expected. In the standard model, however, they are extremely suppressed and can only become sizeable in the presence of new physics. Therefore, as opposed to kaon physics, in D physics A is a probe of new physics, with signals mostly concentrated on the resonance region of the Dalitz plot.
In gure 4 we show the di erential electric-magnetic interference as a function of m2ll for the di erent decay modes. For comparison, we have included the long-distance background (dashed line). For the sake of illustration we have picked W /4 as a
reference value for the weak phases. As expected, the contributions are sizeable close to
16
0
Mode (e+e) (+)
K+ 5.4 107 ( 3%) 4.8 107 ( 7%)
+ 1.3 107 ( 6%) 1.1 107 ( 8%)
K+K 7.9 109 ( 1%) 6.9 109 ( 6%)
K+ 1.5 109 ( 3%) 1.3 109 ( 7%)
Table 2. Branching ratios for the interference between the electric and magnetic terms assuming W /4 for the weak phase. In parenthesis we show the value for the T-odd asymmetry A.
the exchanged-resonance peaks, which vary depending on the nal dihadron state. The interference between magnetic and Bremsstrahlung (not shown in the plot) amounts to roughly 1% of the integrated asymmetry and can be thus safely neglected. In table 2 we have listed the resulting values for A for W /4. Notice that in that case the
asymmetry hovers in the O(110%) window, depending on the channel. Quite generically,
dimuon modes give bigger signals than e+e modes, which should be taken as an additional motivation to study the dimuon decays by LHCb, specically D0 ++.
4.2 Forward-backward asymmetry
In the previous subsection P violation was induced in the hadronic vertex. It is also interesting to consider P violation in the leptonic vertex.
Consider the matrix element stemming from the semileptonic operators Q9 and Q10, namely
M(9)SD 9L(k+, k)H(p1, p2, q)
M(10)SD 10L5(k+, k)H(p1, p2, q) (4.4)
where
L(k+, k) =(k)v(k+)
L5(k+, k) =(k)5v(k+) (4.5)
The short-distance hadronic tensor is dened as
H(p1, p2, q) hh1h2|Jc|D0i (4.6)
and
2 bC(9,10) (4.7)
Above, the CKM unitarity relation d + s + b = 0 has been used. The short-distance hadronic vector H admits a decomposition in terms of form factors F1,2,3 akin to the
long-distance one in eq. (2.3). As in the long-distance analysis of section 2, the dominant contribution comes from the near-resonant region. Thus,
hh1h2|Jc|D0i = hh1h2|H|V i
17
JHEP04(2013)135
(9,10) = GF
1
PV (p2)hV |Jc|D0i (4.8)
2 [LBracket1] K- + e+ e- [RBracket1]
2 [LBracket1] K- + + - [RBracket1]
10
10
10
10
10
10
1GdGdm ee
10
1GdGdm
10
10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
10
0.2 0.4 0.6 0.8 1.0 1.2 1.4
mee2
m2
JHEP04(2013)135
2[LBracket1]+ - e+ e- [RBracket1]
10
2 [LBracket1] + - + - [RBracket1]
10
10
10
1GdGdm ee
10
1GdGdm
10
10 0.0 0.5 1.0 1.5 2.0 2.5
10 0.0 0.5 1.0 1.5 2.0 2.5
mee2
m2
2[LBracket1]K+ K- e+ e- [RBracket1]
10
2 [LBracket1] K+ K- + - [RBracket1]
10
10
10
1GdGdm ee
10
1GdGdm
10
10 0.0 0.2 0.4 0.6
10
0.1 0.2 0.3 0.4 0.5 0.6 0.7
mee2
m2
2 [LBracket1] K+ - e+ e- [RBracket1]
2 [LBracket1] K+ - + - [RBracket1]
10
10
10
10
10
10
1GdGdm ee
10
1GdGdm
10
10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
10
0.2 0.4 0.6 0.8 1.0 1.2 1.4
mee2
m2
Figure 4. The angular integrated (with the prescription of eq. (4.3)) di erential decay width as a function of m2ll (in GeV2) for the di erent decay modes (solid lines). The vertical axis displays
d dm2ll
[integraltext]
2 0
ddm2lld d, which corresponds to eq. (C.10) integrated over p2. For concreteness, W =
/4 is chosen as a reference value. For convenience the result is normalized to the total decay width, such that the area under the curve is A (see eq. (4.2)). The dashed lines correspond to the (angular symmetric) di erential decay width, which is included for comparison. From top to bottom: K+, +, K+K and K+ modes. Left and right panels collect, respectively, the e+e and + modes.
18
Following the steps of section 2 and the expressions given in appendix B, the form factors can be shown to be:
FV1 (q, p) = bV
2(q2)q (p1 p2) +3(q2) PV (p2)
FV2 (q, p) = bV
2(q2)q (p1 p2) 3(q2) PV (p2)
FV3 (q, p) = 2bV
4(q2)
PV (p2) (4.9)
where
2(q2) =
2i mD + m
2(q2)
3(q2) = i(mD + m)1(q2)
4(q2) = 2 mD + m
V (q2) (4.10)
and we have used that bV1 = bV2 bV .
The interference with the long-distance (photon-mediated) contribution studied in section 2 reads
Re[MLDM(10)SD]=
2e q2
JHEP04(2013)135
Xi<jGijIm[FiF jFjF i] (4.11)
where Gij are given by [28]
G12 = p1p2Qq
G13 =
p1 Q(q2p1 p2 p1 qp2 q) + p2 Q((p1 q)2 m2h1q2) G23 =
p2 Q(q2p2 p1 p2 qp1 q) + p1 Q((p2 q)2 m2h2q2)
(4.12)
The term proportional to G12 describes the interference between the electric components of long and short distances, while G13 and G23 collect the electric vs magnetic interference terms. They contribute to the second line in eq. (4.1). The forward-backward asymmetry in :
AFB = hsgn(c)i =
1
[bracketleftbigg][integraldisplay]
1
0 dy
d dy
[integraldisplay]
0
1dy
d dy
[bracketrightbigg]
(4.13)
singles out A6 and it is sensitive to the electric vs magnetic interference. Due to the tiny
value for C10 in the standard model discussed above, a nonvanishing AFB is a rather clean test for new physics.
In gure 5 we show the di erential electric-magnetic interference as a function m2ll for the di erent decay modes. The gure actually shows the absolute value of the interference: since strong phases are accounted for as Breit-Wigner widths, the contribution ips sign at each resonance pole. For comparison, we have included the long-distance background (dashed lines). Notice that the angular asymmetry stays rather constant from the dilepton
19
Mode (e+e)SDLD (+)SDLD (e+e)SDSD (+)SDSD
K+ 1.1108 ( 0.07%) 1.0108 ( 0.06%) 1.01010 ( 6104%) 7.11011 ( 4104%) + 7.1109 ( 0.4%) 6.5109 ( 0.5%) 1.31010 ( 7103%) 1.01010 ( 7103%)
K+K 7.01010 ( 0.1%) 6.11010 ( 0.5%) 3.41011 ( 5103%) 2.21011 ( 0.02%) K+ 5.91010 ( 1%) 5.31010 ( 3%) 1.01010 ( 0.2%) 7.11011 ( 0.4%)
Table 3. Branching ratios for: (i) the interference between short distances (Q10) and long-distance direct emission (denoted SD LD) and (ii) the pure short-distance interference between Q9 and
Q10 (denoted SD SD) for the di erent decay modes. In parenthesis we include the value of the
forward-backward asymmetry AFB. As reference values for the semileptonic coe cients we have chosen C9,10 = 1. Results for di erent values of C9,10 can be easily obtained by noting that the rst two columns are proportional to C10, while the last two are proportional to C9C10.
threshold up until the resonance peak, beyond which it falls o rather steeply. In order to be quantitative, and following the discussion of section 3.2, we have considered a new physics scenario in which C10 = 1. The expected size of AFB for the di erent decay modes is summarized in the rst two columns of table 3. Notice that since C10 is avor-universal,i.e. not subject to the CKM hierarchy, the signal steadily increases from the Cabibbo-allowed to the doubly Cabibbo-suppressed modes, where it can reach the 3% level for the dimuon mode.
This last observation suggests to consider also the interference between C9 and C10 in new physics scenarios where C9 C10 O(1). In this case the CKM hierarchy is absent
altogether: both C9 and C10 are avor-blind. The results are summarized in gure 5 and the second column of table 3. Analytical expressions can be found in appendix C. Notice, as compared to the previous case, that the gradual dominance as one goes from the Cabibbo-allowed to the doubly Cabibbo-suppressed is more pronounced, as expected. However, we note that even with this huge enhancement in C9 and C10, the purely short-distance interference is only appreciable for the doubly Cabibbo-supressed modes.
5 Conclusions
The recent evidence of CP violation in D0 h+h decays well above the expected standard
model prediction makes the study of D decays a priority in the search of new physics. In this paper we have provided the rst detailed analysis of the rare 4-body decays D0
h+1h2+, (h = , K; = e, ) in the standard model. We have studied the dominant long-distance contributions (Bremsstrahlung and hadronic e ects) in the (m2ll, m2hh) Dalitz plots and the total branching ratios, which turn out to be substantially larger than previously estimated. Both the Dalitz plots and the associated branching ratios should be useful tools in view of the upcoming analyses of LHCb and BESIII.
For the Cabibbo-allowed, singly Cabibbo-suppressed and doubly Cabibbo-suppressed
20
JHEP04(2013)135
2 [LBracket1] K- + e+ e- [RBracket1]
2 [LBracket1] K- + + - [RBracket1]
10
10
10
10
1GdGdm ee
10
10
10
1GdGdm
10
10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
10
0.2 0.4 0.6 0.8 1.0 1.2 1.4
mee2
m2
JHEP04(2013)135
2[LBracket1]+ - e+ e- [RBracket1]
10
2 [LBracket1] + - + - [RBracket1]
10
10
10
1GdGdm ee
10
1GdGdm
10
10 0.0 0.5 1.0 1.5 2.0 2.5
10 0.0 0.5 1.0 1.5 2.0 2.5
mee2
m2
2[LBracket1]K+ K- e+ e- [RBracket1]
10
2 [LBracket1] K+ K- + - [RBracket1]
10
10
10
1GdGdm ee
10
1GdGdm
10
10 0.0 0.2 0.4 0.6
10
0.1 0.2 0.3 0.4 0.5 0.6 0.7
mee2
m2
2 [LBracket1] K+ - e+ e- [RBracket1]
2 [LBracket1] K+ - + - [RBracket1]
10
10
10
10
1GdGdm ee
1GdGdm
10
10
10
10
10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
10
0.2 0.4 0.6 0.8 1.0 1.2 1.4
mee2
m2
Figure 5. The angular integrated (with the prescription of eq. (4.13)) di erential decay width as a function of m2ll (in GeV2) for the di erent decay modes. The vertical axis displays
d FB
dm2ll
[bracketleftBig][integraltext]
1 0
[integraltext]
0 1
[bracketrightBig]
ddm2lldy dy. The spiky and smooth solid lines correspond, respectively, to the absolute value of eq. (C.13) and eq. (C.14) integrated over p2. For concreteness, we have chosen C9 = C10 = 1 as reference values. For convenience the result is normalized to the total decay width, such that the area under the curve is AFB (see eq. (4.13)). The dashed lines correspond to the (angular symmetric) di erential decay width, which is included for comparison. From top to bottom: K+, +,
K+K and K+ modes. Left and right panels collect, respectively, the e+e and + modes.
21
modes one nds
Br[D0 K++] 105
Br[D0 ++] 106 Br[D0 K+K+] 107
Br[D0 K++] 108 (5.1)
where an important contribution comes from the near-resonant processes D0 V hV , with subsequent decays V h h1h2 and V +. Assuming factorization, lowest meson
dominance and strong phases coming mainly from the resonance widths, the di erent form factors involved can be determined from experimental input. Our results for the hadronic contribution are in agreement with ref. [38], which is quite a nontrivial agreement, given that the assumptions and methods going into both analyses are rather di erent. Finally, we want to emphasize that our results comply with Lows theorem, i.e., the main contribution in the low dilepton invariant mass region comes from the Bremsstrahlung.
After having determined the standard-model contribution to D0 h+1h2+ we have also explored signals for new physics detection. In particular, we have shown that two angular asymmetries, namely the T-odd diplane asymmetry and the forward-backward dilepton asymmetry can provide direct tests of new physics due to tiny standard model backgrounds. Motivated by new physics scenarios proposed to explain aCP (supersym-metric and Z-enhanced models), we estimate the size of the short-distance parameters C9 and C10 compatible with aCP and avor constraints. We show that new physics e ects in D0 h+1h2+ can generically reach the % level.
Acknowledgments
We want to thank Gino Isidori and Sbastien Descotes-Genon for useful discussions. O. C. wants to thank the University of Naples for very pleasant stays during the di erent stages of this work. G. DA. is grateful to the organizers of the Workshop Implications of LHCb measurements and future prospects (CERN, April 2012), where this work started. L. C. andG. DA. are supported in part by the EU under Contract MTRN-CT-2006-035482 (FLAVIAnet) and by MUIR, Italy, under Project 2005-023102. O. C. is supported in part by the DFG cluster of excellence Origin and Structure of the Universe. This work was performed in the context of the ERC Advanced Grant project FLAVOUR (267104).
A Kinematics
We dene p = p1 + p2 and q = k+ + k as the momenta of the dihadron and dilepton pairs,
respectively. Then one can obtain for the phase space
d = 1
4m2D
22
JHEP04(2013)135
(2)5
[integraldisplay]
[integraldisplay]
dp2
dq2
pD h (A.1)
where
h = 1
(2)5
1 8p2
ph
[integraldisplay]
d cos h
= 1
(2)6
1 8
s1 4m2 q2
[integraldisplay]
d
[integraldisplay]
d cos (A.2)
Above we have dened h (p2, m2h1, m2h2) and D (p2, m2D, q2) where (a, b, c) =
a2 + b2 + c2 2ab 2ac 2bc. The angular variables are dened as in ref. [23]: if p1 is
the h1 momentum in the dihadron CM system; k+ the + momentum in the dilepton CM system; n the direction of the dihadron system as seen from the D0 rest frame; and p1 and k+ the components of p1 and k+ perpendicular to n, then
cos h = n p1
|p1|
; cos =
n k+ |k+|
; cos = p1 k+
|p1||k+|
(A.3)
Intuitively, is the angle between the + momentum and the dihadron system as measured from the dilepton CM while is the angle between the dihadron and dilepton planes.
The nal result for the phase space therefore reads
d5 = 12146m2D
1 p2
s1 4m2 q2
pDh dp2dq2d cos hd cos d (A.4)
where the range of the kinematical variables is
4m2 q2 (mD (mh1 + mh2))2 (mh1 + mh2)2 p2 (mD
JHEP04(2013)135
pq2)2 0 (h, )
0 2 (A.5)
The relevant kinematic products can be expressed as:
p1 p2 =
1
2(p2 m2h1 m2h2)
q p1,2 =
14(m2D p2 q2)
1 4p2
phD cos h
Q p1,2 =
pD 4 cos m2D p2 q2 4
h
p2 cos h cos
h 2p2
pq2p2 sin h sin cos
[bracketrightbigg]
p1p2qQ =
sq2p2hD4 sin h sin sin (A.6)
where
=
s1 4m2q2 , =
1 2 p2
[parenrightbigg]
(A.7)
and 2 = m2h1 m2h2.
23
B Hadronic parameterization
The hadronic matrix element for the weak vertex can be written, using vector meson dominance and assuming factorization, as
hV (p)(q)|JijJc|D0(P )i =XV
h|HV |V i
1PV (q2)
nfV mV (V)hV |Jc|D0i + fVmV (V)hV |Jc|D0i[bracerightBig](B.1)
The most general parametrization for the remaining matrix elements is
hV (p, )|Jc|D0(P )i = D1(k2)p+ + D2(k2)k+ D3(k2) + iD4(k2)p+k (B.2)
with p+ = P +p and k = P p. A similar expression holds for hV |Jc|D0i. The divergence
of the current is proportional to the di erence of quark masses, which can be parametrized as the squared mass di erence of the hadrons. In other words,
khV (p, )|Jc|D0(P )i (m2D m2V )k (B.3) which is satised if
D1(k2) = A1(k2)k
D2(k2) = A2(k2)k
m2D m2V k2
D3(k2) = A3(k2)(m2D m2V ) (B.4)
In particular, the second equation above implies that A2(0) = 0, such that the divergence is avoided. If one now shifts A2 A2 A3 A1 one obtains
hV (p)|Jc|D0(P )i = A1(k2)k [bracketleftbigg]p+
JHEP04(2013)135
(m2D m2V )k2 k[bracketrightbigg]
+ A2(k2)k
(m2D m2V ) k2 k
+ A3(k2)(m2D m2V ) [bracketleftbigg]
k
k2 k[bracketrightbigg]
+ iD4(k2)p+k (B.5)
which can be straightforwardly compared with the parametrization of ref. [31]. The form factors are related as
A1(k2) = i(mD + mV )1
2(k2)
A3(k2) = i(mD mV )1
1(k2)
D4(k2) = i(mD + mV )1 V (k2) (B.6)
The hatted form factors are determined in ref. [31] in the nearest pole approximation (vector meson resonance) as
V (k2) = hV 1m2V 1
m2V 1 k2
;1(k2) = hA1m2A1 m2A1 k2
;
2(k2) = hA2m2A2 m2A2 k2
(B.7)
24
On the other hand, one has that
hV (p)(q)|JijJc|D0(P )i = T (p, k)(p)(q)= tV2 (q V )(p ) + tV3 (V ) + tV4 V pq (B.8)
Comparison between eqs. (B.1) and (B.8) yields the expressions appearing in eq. (2.23) in the main text.
Regarding the strong vertex, the most general parametrization is
B(p+, p) = B1(p2+)p+ + B2(p2+)p (B.9)
where p = p1p2. Since
p+B(p+, p) (m2h1 m2h2) (B.10)
this entails that
B1(p2+) = 1(p2+)m2h1 m2h2
p2+
B2(p2+) = 2(p2+) (B.11)
where 1(0) = 0 to smoothen the divergence out. Therefore,
B(p1, p2) = (2 + 12)p1 (2 12)p2
b1p1 + b2p2 (B.12)
where 2 (m2h1 m2h2)p2+. In the equal dihadron mass case, b1 = b2, in agreement with
charge conjugation invariance. In the general case one nds that b1 6= b2. However, since
bi O(GeV), the mass correction amounts at most to an O(1%) correction and can be
safely neglected.
C Di erential decay widths
In this appendix we will provide analytical expressions for the angular-integrated di erential decay widths for the generic case D0 h+1h2+, i.e., for mh1 6= mh2. The singly
Cabibbo-suppressed case, i.e., mh1 = mh2, can then be worked out as a particular case. We start with
d = 1
2mD
JHEP04(2013)135
Xspins|M|2d (C.1)
where the di erential phase space d is given in appendix A and
M= MLD + M(9)SD + M(10)SD (C.2)
whose denitions can be found in the main text. It is useful to dene the following functions:
h+(q2, p2) = (m2D + q2 p2) h(q2, p2) = (m2D q2 p2)
h+(q2, p2) = (m2D q2 + p2)h++(q2, p2) = (m2D + q2 + p2) (C.3)
25
and the mass combinations
2 = m2h1 m2h2m2 = m2h1 + m2h2 (C.4)
For the long-distance contributions with symmetric angular integration one nds
d2
dq2dp2
[vextendsingle][vextendsingle][vextendsingle][vextendsingle]
Br.
=B
[bracketleftBigg]
q2 4m2h1 (2h+p2h+)2hD
+ q2 4m2h2(2hp2h+)2hD
+ 2( m2p2) + q2 2hDp2h+
JHEP04(2013)135
log
4h2 (p2h+ hD)2 4h2 (p2h+ + hD)2 [bracketrightBigg]
(C.5)
d2
dq2dp2
[vextendsingle][vextendsingle][vextendsingle][vextendsingle]
= E
p2(p2 2 m2)2D + 44(D + 3q2p2)D[bracerightBig]|
aV21|2
El.
|PV |2
[bracketleftbigg][braceleftBig]
+ n4h(h2+ 4m2Dq2)(h + 34)[bracerightBig]
(C.6)
Re
haV21aV 31
[bracketrightBig]
+ 4
np2(p2 2 m2)(D + 12q2p2) + 44(D + 3q2p2)[bracerightBig]| aV31|2[bracketrightbigg]
d2
dq2dp2
[vextendsingle][vextendsingle][vextendsingle][vextendsingle]
= M |aV4 |2
|PV |2
(C.7)
Mag.
d2
dq2dp2
[vextendsingle][vextendsingle][vextendsingle][vextendsingle]
= EB
[bracketleftbigg] [parenleftbigg]
8h+Re
aV21PV[bracketrightbigg]+ 16Re
aV31PV[bracketrightbigg][parenrightbigg]
Br.El.
p2 hD
Xjlog (1)jp2h+ 2h hD (1)jp2h+ 2h + hD
2h+
(1)j22 h+
Re
aV21 PV
[bracketrightbigg]
+ 4
(1)j22 h++ + 8m2hj
Re
aV31 PV
[bracketrightbigg] [parenrightbigg][bracketrightbigg]
(C.8)
where
B = 2
483m3D
(2m2l + q2)
hDp2
q4 |MD|2
E =
hD
q6p6
368644m3D
(2m2l + q2)
M =
184324m3D
(2m2l + q2) (hD)3/2
q4p4
EB =
3/2
15367/2m3D
(2m2l + q2)
hD
q4p2 |MD| (C.9)
and we have dened MD M(Dh1h2). We have included for completeness the
Bremsstrahlung vs. electric interference, even though its e ect is extremely suppressed. Expressions for the mode-dependent form factors aV21, aV31 and aV4 can be found in the main text.
26
For the short-distance contribution entering A one nds that
d2
dq2dp2
[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]El.Mag.
= EM
haV31aV 4
|PV (p2)|2
sin W Re
[bracketrightBig]
(C.10)
d2
dq2dp2
[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]Br.Mag.
= BM sin SRe
[bracketleftbigg]
aV4 PV
[bracketrightbigg] [bracketleftbigg]
h+ 2
phD
+ m2h1(m2Dh p2h+) + q2(4 + m2Dp2) m2h2q2h+
log
p2h+ + 2h hD p2h+ + 2h + hD [bracketrightbigg]
+ mh1 mh2, 2 2
[parenrightbig]
(C.11)
where the asterisk indicates that the integration over is done as in eq. (4.3). W and S are, respectively, weak and strong phases and
EM =
23045m3D
3
3/2hD q2p4
q2 |MD| (C.12)
Again for completeness we have included the Bremsstrahlung vs magnetic interference term, despite being strongly suppressed.
Finally, we list the expression for the contributions entering the forward-backward asymmetry AFB, namely the terms Re[MLDM(10)SD] and Re[M(9)SDM(10)SD]:d2 FB
dq2dp2
[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]SDLD
= (1)FB
JHEP04(2013)135
BM =
3/2
969/2m3D
3
h
|PV |2
Re
haV31V 4 +V31aV 4
[bracketrightBig]
(C.13)
d2 FB
dq2dp2
= (2)FB
(C.14)
The subindex F B is a reminder that the integration over is done using the prescription of eq. (4.13), and
(1)FB =
1/210
61449/2m3D
[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]SDSD
hV31V 4
|PV |2
Re
[bracketrightBig]
2 3/2hD
p4
(2)FB =
910
61445m3D
2 3/2hDq2
p4 (C.15)
and we have used the short-hand notation
V21(q2) = ibV2(q2) V31(q2) = ibV3(q2)
V4 (q2) = 2bV4(q2) (C.16)
Expressions for the short-distance coe cients 9,10 and the short-distance form factors i(q2) can be found in the main text.
27
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JHEP04(2013)135
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SISSA, Trieste, Italy 2013
Abstract
(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)
Abstract
Motivated by the recent evidence for direct CP-violation in D ^sup 0^ [arrow right] h ^sup +^ h ^sup -^ decays, we provide an exhaustive study of both Cabibbo-favored and Cabibbo-suppressed (singly and doubly) ... decays. In particular, we study the Dalitz plot for the long-distance contributions in the ... parameter space. We find that near-resonant effects, i.e., ... with V = [rho], K*, , are sizeable and even dominant (over Bremsstrahlung) for the [mu] ^sup +^ [mu] ^sup -^ decay modes, bringing the branching ratios close to the LHCb reach. We also provide a detailed study of the angular asymmetries for such decays and identify signatures for new physics detection. In particular, new physics signals can be neatly isolated in asymmetries involving the semileptonic operator Q ^sub 10^, where for typical new physics scenarios the effects can be as sizeable as ...(1%) for the doubly Cabibbo-suppressed modes.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer