Rare radiative B
c! D
s1ð2460Þ transition in QCD
K. Azizi,1,*N. Ghahramani,2,†and A. R. Olamaei2,‡
1Physics Department, Faculty of Arts and Sciences, Dog˘us¸ University, Acbadem-Kadko¨y, 34722 Istanbul, Turkey 2Physics Department, College of Sciences, Shiraz University, Shiraz 71454, Iran
(Received 6 July 2012; revised manuscript received 9 November 2012; published 22 January 2013) We investigate the radiative Bc! Ds1 transition in the framework of QCD sum rules. In particular, we
calculate the transition form factors responsible for this decay in both weak annihilation and electro-magnetic penguin channels using the quark condensate, mixed, and two-gluon condensate diagrams, as well as propagation of the soft quark in the electromagnetic field, as nonperturbative corrections. These form factors are then used to estimate the branching ratios of the channels under consideration. The total branching ratio of the Bc! Ds1 transition is obtained to be of the order of105, and the dominant
contribution comes from the weak annihilation channel.
DOI:10.1103/PhysRevD.87.016013 PACS numbers: 11.55.Hx, 13.20.v, 13.20.He
I. INTRODUCTION
The Bcis the only heavy meson consisting of two heavy quarks with different flavors; hence the decay properties of this meson are of special interest. The difference in heavy quark flavors forbids annihilation of this meson into glu-ons, so the excited Bc states undergo pionic or radiative transition to the pseudoscalar ground state when these states lie below the threshold of decay into the pair of heavy B and D mesons. The resulting pseudoscalar ground state is more stable compared to the corresponding quar-konia and decays mostly weakly. Because of this phenome-non, it is expected that experimental study of the Bcmeson and its decay properties will constitute an important part of the physics program at LHCb. The study of the heavy mesons will not only provide a window into extracting the most accurate values of the Cabbibo-Kobayashi-Maskawa matrix elements as the sources of the CP viola-tion in the Standard Model—it will also help us better understand the perturbative and nonperturbative aspects of QCD.
In the present study, we work out the rare radiative Bc ! Ds1ð2460Þ transition in the framework of the QCD sum rules [1,2]. Here Ds1ð2460Þ is the axial vector charmed-strange meson with quantum numbers JP¼ 1þ and interpolating current ¼ s5c. This transition proceeds via both weak annihilation (WA) and electromag-netic penguin (EP) modes based on b ! s at the quark level. We calculate the transition form factors responsible for this decay in both WA and EP modes using the quark condensate, mixed, and two-gluon condensate diagrams, as well as propagation of the soft quark in the electromagnetic field, as nonperturbative corrections. We then use these form factors to estimate the branching ratios in both modes, as well as the total branching fraction of the
Bc! Ds1ð2460Þ transition. As expected, the dominant contribution comes from the weak annihilation channel. Note that similar decays like the Bc! Ds transition have been studied in the same framework [3]. Some other radiative channels of the Bc meson like Bc ! l and Bc! Bu have also been previously studied using the QCD sum rules technique [4,5]. For analysis of other decay channels of the Bcmeson, see, for instance, Refs. [6–9].
The outline of the paper is as follows: In Sec. II, we consider the radiation of the photon from both Bcand Ds1 mesons to construct the transition amplitude for the WA channel in terms of four relevant form factors. Two of the form factors [FðBcÞ
V and F ðBcÞ
A ], responsible for the emission of the photon from the initial state, are calculated in Ref. [4], and the remaining two form factors [FðDs1Þ
V and
FðDAs1Þ], representing the emission of the photon from the Ds1 meson, are calculated in Sec. III. In Sec. IV, we consider the two gluon condensate contributions to calcu-late the transition form factors responsible for the EP mode. Finally, Sec.Vis devoted to the numerical analysis of the form factors and the calculation of the decay rates and branching ratios for the modes under consideration. We also present results for the total decay rate and branch-ing ratio of the Bc ! Ds1ð2460Þ transition. This section also contains our concluding remarks.
II. WEAK ANNIHILATION AMPLITUDE In this section, we construct the WA amplitude for the radiative Bc ! Ds1 transition. Considering the quark contents of the initial and final mesonic states, the possible diagrams are shown in Fig. 1. Taking into account these diagrams, the transition amplitude for the radiative decay under consideration is written as
MWAðBc ! Ds1Þ ¼ GF ffiffiffi 2 p VcbVcshDs1ðpÞðqÞjðscÞ ðcbÞjB cðp þ qÞi; (1) *[email protected] †[email protected] ‡[email protected]
where GF is the Fermi weak coupling constant; Vij are elements of the Cabbibo-Kobayashi-Maskawa matrix; ¼ ð1 5Þ; and p, q, and p þ q are the momenta of the Ds1 meson, photon, and Bc meson, respectively. To proceed further, we use the factorization hypothesis and write the transition matrix element in Eq. (1) as
hDs1ðpÞðqÞjðscÞðcbÞjBcðp þ qÞi ¼ e""ðDs1Þf Ds1mDs1T ðBcÞ ie"ðp þ qÞfBcT ðDs1Þ ; (2) where we have divided the matrix element into two sepa-rate parts: the emission of the photon from the Bc meson, represented by the covariant tensor TðBcÞ
[diagrams (i) and (ii) in Fig.1]; and the emission of the photon from the Ds1 meson, denoted by the tensor TðDs1Þ[diagrams (iii) and (iv) in Fig.1]. In Eq. (2), fBc[fDs1] is the decay constant of the Bc [Ds1] meson, and " ["ðDs1Þ] is the polarization vector of the photon [Ds1meson]. The covariant tensors TðBcÞ
and TðDs1Þare defined as TðBcÞ ðp;qÞi Z d4xeiqxh0jTfjem ðxÞcð0Þbð0ÞgjBcðpþqÞi; (3) TðDs1Þ ðp; qÞ i Z d4xeiqxhD s1ðpÞjTfjem ðxÞsð0Þcð0Þgj0i; (4) where jem is the electromagnetic current and T is the time-ordering operator. By applying the Ward identity for the electromagnetic current and using q2¼ 0 for the real photon, ":q ¼0, and "ðDs1Þ:p ¼0—similar to what is done in Refs. [3,10,11]—we get the following results, corresponding to the emission of the photon from the initial and final mesonic states in terms of form factors:
e""ðDs1Þf Ds1mDs1T ðBcÞ ¼ efDs1mDs1f½ð":"ðDs1 ÞÞðp:qÞ ð":pÞð"ðDs1Þ:qÞiFðBcÞ A þ ifBcð":" ðDs1ÞÞ þ " "ðDs1Þ"pqFðBc Þ V g; (5) ie"ðp þ qÞf BcT ðDs1Þ ¼ iefBcf½ð":"ðDs1ÞÞðp:qÞ ð":pÞð"ðDs1Þ:qÞiFðDs1Þ A þ fDs1mDs1ð":"ðDs1ÞÞ þ " "ðDs1Þ"pqF ðDs1Þ V g; (6) where FðBcÞ VðAÞ and F ðDs1Þ
VðAÞ are the transition form factors. Using Eqs. (5), (6), and (2), we find the WA transition amplitude to be MWAðBc ! Ds1Þ ¼ eGFffiffiffi 2 p VcbVcsðfDs1mDs1f½ð":"ðDs1ÞÞðp:qÞ ð":pÞð"ðDs1Þ:qÞiFðBcÞ A þ ifBcð":" ðDs1ÞÞ þ ""ðDs1Þ"pqFðBc Þ V g ifBcf½ð":" ðDs1ÞÞðp:qÞ ð":pÞð"ðDs1Þ:qÞiFðDs1Þ A þ fDs1mDs1ð":"ðDs1ÞÞ þ ""ðDs1Þ"pqFðDs1 Þ V gÞ: (7)
As mentioned in Sec.I, the form factors FðBcÞ V and F
ðBcÞ A are calculated in Ref. [4], so what remain to be calculated are the form factors FðDs1Þ
V and F ðDs1Þ
A , which we discuss in the next section.
III. QCD SUM RULES FOR THE FORM FACTORS FðDV s1ÞAND FðDAs1Þ To calculate the transition form factors FðDs1Þ
V and F ðDs1Þ A via QCD sum rules formalism, we start considering the following correlation function:
ðp; qÞ ¼ iZ d4xeiQxhðqÞjTfcðxÞð1 5Þ
sðxÞsð0Þ5cð0Þgj0i; (8) where Q ¼ p þ q. The basic idea in this method is to calculate this correlation function, first in hadronic lan-guage (called the phenomenological or physical side), and second in terms of the QCD degrees of freedom, using the operator product expansion in deep Euclidean space (called the theoretical or QCD side). The two representa-tions are then matched in order to get the QCD sum rules for the form factors. To suppress the contributions coming from the higher energy states and continuum, we apply a Borel transformation as well as continuum subtraction, which bring two auxiliary parameters: the Borel mass parameter and the continuum threshold. We shall find their working regions, requiring that the physical observables be independent of these parameters.
First, we focus on calculation of the phenomenological side. For this aim, we insert a full set of hadronic Ds1states into Eq. (8) and perform the four-integral over x to get ðp;qÞ
¼hðqÞjcð15ÞsjDs1ðpÞihDs1ðpÞjs5cj0i
m2Ds1p2 : (9)
The matrix element hDs1ðpÞjs5cj0i is defined in terms of the decay constant and the polarization vector of the Ds1 meson as
hDs1ðpÞjs5cj0i ¼ fDs1mDs1"ðDs1 Þ
; (10)
while the transition matrix element is parametrized in terms of form factors:
hðqÞjcð1 5ÞsjDs1ðpÞi ¼ ei"""ðDs1Þq FðDs1Þ V ðQ2Þ m2Ds1 þ ½"ð"ðDs1Þ:qÞ ð":"ðDs1ÞÞq FðDs1Þ A ðQ2Þ m2ðD s1Þ : (11)
By substituting Eqs. (10) and (11) into Eq. (9) and sum-ming over the polarization vector of the Ds1meson, we find the following result for the phenomenological part of the correlation function: ðp; qÞ ¼ efDs1mDs1 m2Ds1 p2 i""q FðDs1Þ V ðQ2Þ m2Ds1 þ ½q" "qF ðDs1Þ A ðQ2Þ m2D s1 : (12)
We now compute the QCD side of the correlation func-tion within the deep Euclidean region in terms of the QCD parameters. We start by writing the correlation function in terms of the two selected structures as
ðp;qÞ ¼ i""q
1þ½q""q2; (13) where each function i(i ¼1 or 2) has perturbative and nonperturbative parts; i.e.,
i¼ perti þ nonpert
i : (14)
To calculate the perturbative parts, we consider Figs.2(a)
and2(b), where the photon can be radiated from both the charm and strange quarks. For the nonperturbative parts, we take into account the quark condensate and mixed diagrams [Figs. 2(c)–2(e)], as well as Fig. 2(f ), for the interaction of the photon with the soft quark.
The perturbative part in each case can be written via the dispersion relation as perti ¼ Z ds iðs; Q 2Þ s p2 þ subtraction terms; (15) where i are the spectral densities. Our main task is to calculate these spectral densities using the diagrams in Figs.2(a) and2(b). Here we use a method based on both Feynman and Schwinger parameterizations with several Borel transformations (see also Ref. [12]). The Feynman amplitude for the diagram in Fig.2(a)can be written as
;ðaÞ¼ eNcQsZ d 4k ð2Þ4 Trið6k þ mcÞ k2þ m2c 5 ið6p þ 6k þ msÞ ðp þ kÞ2 m2 s 6 ið 6Q þ 6k þ msÞ ðQ þ kÞ2 m2 s ð1 5Þ ; (16) where Nc¼ 3 is the number of colors and Qsis the charge of the strange quark. Using the Feynman parameterization,
FIG. 2. Diagrams for bare-loop [(a), (b)], quark and mixed condensates [(c)–(e)] and propagation of the soft quark in the electromagnetic field (f ).
we perform the four-integral over k, and then use the Schwinger parameterization 1 n¼ 1ðnÞ Z1 0 d n1e (17)
to write the denominators in exponential forms. As a result, we get pert1;a ¼eNcQs 42 Z1 0 dxx Z1 0 dy½msðmcþ msxyÞ þ p2xxð1 xyÞ þ 2p:qxx2y2Z1 0 de ; (18)
pert2;a ¼eNcQs 42 Z1 0 dxx Z1 0 dyx½msðmc msxyÞ p2xxð1 xyÞ 2p:qxx2y2Z1 0 de ; (19)
where xðyÞ ¼ 1 xðyÞ and ¼ m2cx þ m2sx p2xx y Q2xxy.
By applying a double Borel transformation ^BðM21Þ ^BðM22Þ on perti that transforms Q2! M21 and p2 ! M22, we obtain ^pert 1;a ¼ eNcQs 42 12 1þ 2 Z1 0 dx 1 xe ðm2c xþm2s xÞð1þ2Þ xx mcmsþ m2sx 1 1þ 2þ 2xð1 x 2Þ 1 ð1þ 2Þ2 ; (20) ^pert 2;a ¼ eNcQs 42 12 1þ 2 Z1 0 dx 1 xe ðm2c xþm2s xÞð1þ2Þ xx mcms m2sx 1 1þ 2 2xð1 x 2Þ 1 ð1þ 2Þ2 ; (21)
where 1;2¼ 1=M21;2, and we have used ^Bp2ðM2Þep2 ¼ ð1 M2Þ; ^Bp2ðM2Þp2ep2 ¼ d d ^Bp2ðM2Þep2 ¼ d d ð1 M 2Þ: (22)
Now, we perform a second double Borel transformation on ^perti in order to transform 1 and 2 to the new variables w and s using
%iðw; sÞ ¼ 1 ws ^B 1 w; 1 ^B1 s; 2 ^pert i 12: (23)
In our calculations, we also use the relations
^B1 w; 1 ^B1 s; 2 eð1þ2Þ¼ 1 w 1 s (24) and ne¼ d d n e: (25)
The final expressions for the spectral densities are then calculated via the following formula:
iðs; Q2Þ ¼Z dw%iðw; sÞ
w Q2: (26)
After lengthy calculations, we get the following spectral densities corresponding to Fig.2(a):
1aðs; Q2Þ ¼eNcQs 162 1 ðs Q2Þ2 Zx1 x0 dx 1 xx2fm 4 cx2ðx 5Þ þ m4 sx2ðx 6Þ m2cm2sxxð2x 11Þ þ 4mcmsxxðs Q2Þ þ m2cxx2½ðx 8ÞQ2 þxðs Q2Þ m2 sx2x½ðx 10ÞQ2 þ ðx 2Þðs Q2Þ 4x2x2Q2ðs Q2Þ 4x2x2Q4g; (27) 2aðs; Q2Þ ¼eNcQs 162 ðs Q1 2Þ2 Zx1 x0 dx 1 xx2fm 4 cx2ðx 5Þ m4 sx2ð6 þ xÞ þ m2cm2sxxð2x 11Þ þ 4mcmsxxðs Q2Þ m2cxx2½ðx 8ÞQ2 þxðs Q2Þ þ m2 sx2x½ðx 10ÞQ2 þ ðx 2Þðs Q2Þ þ 4x2x2Q2ðs Q2Þ þ 4x2x2Q4g; (28)
where the integral boundaries x0 and x1 satisfy the following inequality:
sxx ðm2cx þ m2sxÞ 0; (29) which comes from the Heaviside theta function arising in these calculations. Similarly, we calculate the contribution of Fig.2(b). The final expressions for the spectral densities corresponding to the two selected structures are
1ðs; Q2Þ ¼ eNc 322 1 ðs Q2Þ2 Qs ð4ð5 5 1ÞsQ2þ s2½3ð 3Þ ð6 þ 1Þ þ 32Þ þ 2ð4mcmsðs Q2Þ þ 8sQ2þ ð1 4 þ 9Þs2Þ ln1 þ 1 þ þ þ Qc ð4ð5 5 1ÞsQ2þ s2½3ð 3Þ ð6 þ 1Þ þ 32Þ þ 2ð4mcm sðs Q2Þ þ 8sQ2þ ð1 4 þ 9Þs2Þ ln 1 þ 1 þ þ ; (30) 2ðs; Q2Þ ¼ eNc 322 1 ðs Q2Þ2 Qs ð4ð5 þ 5 þ 1ÞsQ2þ s2½3ð 3Þ þ ð6 þ 1Þ 32Þ þ 2ð4mcmsðs Q2Þ 8sQ2 ð1 4 þ 9Þs2Þ ln1þ 1 þ þ þ Qc ð4ð5 þ 5 þ 1ÞsQ2þ s2½3ð 3Þ þ ð6 þ 1Þ 32Þ þ 2ð4mcm sðs Q2Þ 8sQ2 ð1 4 þ 9Þs2Þ ln1þ1 þ þ ; (31) where ¼m2s s , ¼ m2c s , and ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2þ 2 2 2 2 p .
For the nonperturbative parts, we first consider the quark condensate and mixed diagrams [Figs.2(c)–2(e)]. For their contributions, we get nonpert1ðc;d;eÞ ¼ mc r2R2hssi þ ms 2 hssi 2r2R2þ m2c r4R2 7 m2c r2R4 4 m4c r4R4 þm2s 2 hssi2m 3 c r2R6 8 m5c r6R4þ 2 m3c r4R4 3 mc r2R4þ 2 m3c r6R2 mc r4R2 þm20 12hssi 6m3 c r2R6 þ 24 m5c r6R4 6 m3c r4R4þ 8 mc r2R4 6 m3c r6R2þ 3 mc r4R2 4m3c r2R4hssi; (32) nonpert2ðc;d;eÞ¼ mc r2R2hssi þ ms 2 hssi 1r2R2þ 1R4 4 m4c r4R4 3 m2c r2R4þ m2c 2r4R2 þm2s 2 hssi 2m3c r2R6þ 8 m5c r6R4 2 m3c r4R4 þ 3mc r2R4 2 m3c r6R2þ mc r4R2 þm20 4 hssi 2m3 c r2R6 8 m5c r6R4þ 2 m3c r4R4 4 mc r2R4þ 2 m3c r6R2 mc r4R2 þ 4m3c r2R4hssi; (33) where r2¼ p2 m2c and R2 ¼ Q2 m2c.
The final contribution to the WA mode is that of Fig.2(f). This diagram corresponds to the propagation of the soft quark in the external electromagnetic field. Here we need to make use of the light cone version of the QCD sum rules and photon distribution amplitudes (DAs). The relevant correlation function is of the form
;ðfÞðp; qÞ ¼ iZ d4xeiQxhðqÞjTfsð0Þ5cð0ÞcðxÞ ð1 5ÞsðxÞgj0i: (34) By contracting the c-quark lines in Eq. (34) and using the propagator of the heavy quark in momentum space, we obtain ;ðfÞðp; qÞ ¼ i2Z d4x d4k ð2Þ4 eiðQkÞx m2c k2 hðqÞjs5ð6k þ mcÞð1 5Þsj0i: (35)
To relate the matrix element in the above equation to the photon DAs, we use the identities
¼ gþ i; 5¼ g5i
2";
¼ gþ g gþ i"5: (36) The relevant photon DAs of twist 2, 3, and 4 (Refs. [13,14]) are hðqÞjssj0i ¼ Qs 2 f3 Z1 0 du cðuÞx F ðuxÞ; hðqÞjs5sj0i ¼ iQs 4 f3 Z1 0 du c ðAÞðuÞx F~ ðuxÞ; hðqÞjssj0i ¼ QshssiZ1 0 duðuÞFðuxÞ þQshssi 16 Z1 0 dux 2AðuÞFðuxÞ þQshssi 8 Z1 0 duBðuÞ x ðx F ðuxÞ xF Þ; (37) where F is the field strength tensor of the electromag-netic field, defined by
FðxÞ ¼ ið"q "qÞeiqx (38) and
~
FðxÞ ¼ 1
2"FðxÞ: (39)
The wave function ðuÞ is defined in terms of the mag-netic susceptibility ðÞ at a renormalization scale ( ¼1 GeV2) in the following manner:
ðuÞ ¼ ðÞuð1 uÞ: (40)
The remaining functions, cðVÞðuÞ, cðAÞðuÞ, AðuÞ, and BðuÞ, are also defined as [13,14]
cðVÞðuÞ ¼ 20uð1 uÞð2u 1Þ þ15
16ð!A 3!VÞuð1 uÞð2u 1Þð7ð2u 1Þ2 3Þ;
cðAÞðuÞ ¼ ð1 ð2u 1Þ2Þð5ð2u 1Þ2 1Þ5 2 1 þ19 16!V 316!A ; AðuÞ ¼ 40uð1 uÞð3k kþþ 1Þ þ 8ðþ
2 32Þ½uð1 uÞð2 þ 13uð1 uÞÞ þ 2u3ð10 15u þ 16u2Þ lnu þ 2ð1 uÞ3ð10 15ð1 uÞ þ 6ð1 u2ÞÞ lnð1 uÞ;
BðuÞ ¼ 40Zu 0 dð4 Þð1 þ 3k þÞ 1 2þ 32ð2 1Þ2 ; (41)
where k, kþ, 2, þ2, and f3 are constants (see Refs. [13,14]). After combining the above equations, we perform the four-integrals over x and k. The coeffi-cients of the corresponding structures i""q and ½q" "q are obtained as follows:
nonpert1f ðp; qÞ ¼ Qs 2ðm2 c p2Þ3 Z1 0 dufm 3 chssiAðuÞ þ ðm2 c p2Þ½mchssiBðuÞ 2ð5m2c p2Þ ðmchssiðuÞ f3cðVÞðuÞÞg; (42) nonpert2f ðp; qÞ ¼ mcQs 2ðm2 c p2Þ3 Z1 0 dufAðuÞm 2 chssi þ 2ð5m2 cþ p2ÞhssiðuÞ þ ðm2c p2Þ ½BðuÞhssi þ f3mccðAÞðuÞg: (43) Now, to find the QCD sum rules for the form factors, we match the coefficients of the selected structures from both the phenomenological and QCD sides and perform the Borel transformation with respect to the momentum of the Ds1 meson ðp2 ! M2BÞ. To further suppress the contri-butions of the higher energy states and continuum, we also perform the continuum subtraction and use the quark-hadron duality assumption. As a result, we find
FðDs1Þ V;A ðQ2Þ ¼ mDs1 fDs1 e mDs1=M2B ^BZs0 ðmsþmcÞ2 ds 1;2ðs; Q 2Þ s p2 þ nonpert1;2ðcþdþeþfÞ; (44) where s0is the continuum threshold, and the V (A) on the left-hand side corresponds to the 1 (2) on the right-hand side. To obtain the expressions for the above sum rules in the Borel scheme, we perform the Borel transformation using the standard rule
^B 1 ðp2 sÞn ¼ ð1Þ n e s=M2 B ðnÞðM2 BÞn1 : (45)
IV. QCD SUM RULES FOR THE FORM FACTORS RESPONSIBLE FOR THE ELECTROMAGNETIC PENGUIN MODE At the quark level, the EP transition of the Bc ! Ds1 proceeds via b ! s, whose effective Hamiltonian is written as Heff ¼ GFe 42p Vffiffiffi2 tbVtsC7ðÞs mb 1 þ 5 2 þ ms 1 5 2 bF: (46)
The amplitude of this mode is obtained from
MEP¼ hDs1ðpÞjHeffjBcðQÞi; (47) hence to proceed further, we need to calculate the following matrix elements:
hDs1jsð1 5ÞqbjB
ci; (48)
which can be parametrized in terms of two gauge-invariant form factors, T1ðq2Þ and T2ðq2Þ, in the case of a real photon, i.e., hDs1ðp; "ðDs1ÞÞjsq 5bjBcðQÞi ¼ i""ðDs1ÞpQT1ð0Þ; hDs1ðp; "ðDs1ÞÞjsqbjB cðQÞi ¼ ½ðm2 Bc m 2 Ds1Þ" ðDs1Þ ð"ðDs1Þ:qÞðp þ QÞT2ð0Þ; (49) where these two form factors are not independent from each other. Using the relation 5¼ 2i",
we see T1ð0Þ ¼12T2ð0Þ. Therefore, we need to calculate just one of them, and here we choose to calculate the form factor T2ð0Þ. The corresponding correlation function is chosen as
ðp2; Q2Þ ¼ i2ZZ d4xd4yeiðQxpyÞh0jTfcðyÞ 5sðyÞ sð0Þqbð0Þ bðxÞ
5cðxÞgj0i; (50) where b5c and c5s are the interpolating currents of the initial and final mesonic states, respectively. Here also, sqb is the transition current. Using the general phi-losophy of the QCD sum rules, we calculate this correla-tion funccorrela-tion again in two different languages: the hadronic language and the quark-gluon language. For the hadronic or phenomenological side, we get
ðp2; Q2Þ ¼ ifDs1fBcmDs1m2Bc ðm2 Bc Q 2Þðm2 Ds1 p2Þðmbþ mcÞ ðm2 Bc m 2 Ds1ÞgT2ð0Þ mB2c m 2 Ds1 m2Ds1 ppT2ð0Þ þ ðp þ QÞ p:q m2Ds1p q T2ð0Þ þ ; (51) where the ellipsis denotes contributions of the higher energy states and continuum which will be suppressed by applying the Borel transformation as well as the continuum subtraction. In deriving the above equation, we have used the following definition of the decay constant of the Bc meson:
hBcj b5cj0i ¼ i fBcm 2 Bc
ðmbþ mcÞ: (52) Note that to calculate the form factor T2ð0Þ, we choose the structure g.
On the QCD side, the correlation function is written in terms of the selected structure as
¼ gðp2; Q2Þ; (53) where
ðp2; Q2Þ ¼ pertðp2; Q2Þ þ nonpertðp2; Q2Þ: (54) Here the perturbative part is related to the spectral density pertðs0; tÞ by a double dispersion integral,
pertðp2; Q2Þ ¼ 1 ð2Þ2 ZZ ds0dt pertðs0; tÞ ðs0 Q2Þðt p2Þ þ subtraction terms; (55)
and for the nonperturbative contributions we will calculate the two-gluon condensate diagrams.
Now, we focus our attention on calculating the spectral density. Using the Cutkosky method [15], we get
perðs0; tÞ ¼2NcfI0½½ðmb mcÞðmcþ msÞ þ t 0½ðm
b mcÞðmcþ msÞ þ s0 þ 2mc½ðmcþ msÞs0þ m
bt mct
mcðmbþ msÞu þ 2A1ð2s0þ uÞg; (56) where
I0 ¼ 1
4pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0ðs0; t; q2Þ;
0ða; b; cÞ ¼ a2þ b2þ c2 2ab 2ac 2bc; A1 ¼ I0 ð4s0t þ u2Þ2½2t þ02s0 0u þ m2cð4s0t þ u2Þ; ¼ s0þ m2 c m2b; 0¼ t þ m2 c m2s; u ¼ t þ s0 q2: (57)
Note that, to obtain the above spectral density, we have performed the integrals over the delta functions, which restricts the boundaries of the integrals over s0 and t:
m2c t t0; t tm 2 b m2c t s 0 s0 0; (58)
where s00 and t0are the continuum thresholds in the initial and final channels in the case of EP mode. There are several sources for nonperturbative contributions, such as quark-quark, quark-gluon and gluon-gluon condensates. However, the quark-quark and quark-gluon condensates give zero contributions after applying the double Borel transformation with respect to Q2 (Q2! M12) and p2 (p2 ! M22). Therefore, the remaining source of the non-perturbative contributions would be the gluon condensates (see Fig. 3). The calculation of such contributions is lengthy but standard. For the nonperturbative part in the Borel scheme, we get
nonpert¼ M2 1M22 s G 2 C G2; (59)
where CG2 is the Wilson coefficient of the gluon conden-sates, defined as CG2 ¼ Ca G2þ C b G2þ C c G2þ C d G2þ C e G2þ C f G2: (60) The explicit expressions of CiG2are given in the Appendix. Using a similar procedure to that presented in the previous section, we find the sum rule for the form factor T2ð0Þ to be
T2ð0Þ ¼ ðmbþ mcÞe m2Bc=M21 em2Ds1=M22 ifDs1fBcmDs1m 2 Bcðm 2 Bc m 2 Ds1Þ 1 ð2Þ2 ZZ ds0dtes0=M12et=M22 pertðs0; tÞ þ M2 1M22 s G 2 C G2 : (61) V. NUMERICAL ANALYSIS
This section is devoted to the numerical analysis of the form factors, estimating the branching ratio in the WA and EP channels and the total branching fraction of the Bc ! Ds1ð2460Þ transition. For this aim, we use the following quark and meson masses: mc¼ ð1:275 0:015Þ GeV, ms’ 142 MeV [16], mb ¼ ð4:7 0:1Þ GeV [17], mDs1 ¼ ð2459:6 0:6Þ MeV, and mBc¼ ð6:277 0:006Þ GeV [18]. For the values of the decay constants, we use fDs1 ¼ ð225 25Þ MeV and fBc ¼ ð350 25Þ MeV [19–21]. The values of the condensates are as follows [17]: h c cj¼1 GeVi¼ð24010MeVÞ3, hssi ¼ ð0:8 0:2Þ h c ci, m20 ¼ ð0:8 0:2Þ GeV2, and hs
G2i ¼ ð0:012 0:004Þ GeV4. The parameters entered the photon DAs are also taken as ¼ ð3:15 0:30Þ GeV2, k ¼0:2, kþ¼ 0, 1 ¼ 0:4, 1þ¼ 0, 2 ¼ 0:3, 2þ¼ 0, f3¼ ð4 2Þ 103 GeV2, !A
¼ 2:1 1:0, and !V ¼ 3:8 1:8 [13,14,22]. The remaining parameters are chosen as jVcsj ¼ 0:957 0:017, jVcbj ¼ 0:0416 0:0006, jVtbj ¼ 0:77þ0:18
0:24, jVtsj ¼ ð40:6 2:7Þ 103 [18], C7ð ¼ mcÞ ¼ 0:0068 0:02i [23], and Bc ¼ 0:52 1012 s.
The sum rules for the form factors also contain the continuum thresholds and the Borel mass parameters as auxiliary objects. We find the working regions for these parameters such that the physical observables are practi-cally independent of them. The continuum thresholds are not completely arbitrary but are correlated with the energy of the first excited states in the initial and final mesonic channels. Our numerical results show that the results depend weakly on the thresholds in the intervals s0 ¼ t0 ¼ ð6–8Þ GeV2and s0
0 ¼ ð45–50Þ GeV2. The working regions for the Borel parameters are obtained by demanding not only that the contributions of the higher states and contin-uum be effectively suppressed, but also that the contribu-tions of the higher-order operators and higher-twist DAs remain small; i.e., the series of sum rules must converge. These conditions lead to the intervals 6 GeV2 M2B 12 GeV2, 10 GeV2 M2
1 30 GeV2, and 5 GeV2 M22 12 GeV2 for the Borel mass parameters.
Now, we proceed to find the fit functions of the form factors using the aforesaid working regions for the auxil-iary parameters. Here we would like to mention that to calculate the decay rates, we need only the values of the form factors FðDs1Þ V and F ðDs1Þ A at Q2 ¼ m2Bc, F ðBcÞ V and F ðBcÞ A at p2 ¼ m2Ds1, and T2 at q2¼ 0. However, we determine their fit functions in general and give their values at these fixed points. The fit functions for the form factors FðDs1Þ
V and FðDs1Þ A are fðQ2Þ ¼ fð0Þ 1 þ a Q2 m2 Ds1 þ bð Q2 m2 Ds1 Þ2; (62)
where fð0Þ, a, and b are the fit parameters whose values are given in TableI.
The values of these form factors at Q2 ¼ m2B care FðDs1Þ V ðQ2 ¼ m2BcÞ ¼ 0:055 0:016; FðDs1Þ A ðQ2 ¼ m2BcÞ ¼ 0:102 0:030; (63)
where the errors on the values are due to the uncertainties in the determination of the working regions for the auxil-iary parameters, as well as those coming from the DAs and other input parameters.
FIG. 3. Feynman diagrams for gluon condensate corrections.
TABLE I. Fit parameters for the form factors FðDs1V Þand FðDs1A Þ.
Form factors fð0Þ a b
FðDs1V ÞðQ2Þ 0.098 0.171 0:008
The fit functions for the form factors FðBcÞ A;V are [4] FðBcÞ V ðp2Þ ¼ FVð0Þ 1 p2=m2 1 ; FðBcÞ A ðp2Þ ¼ FAð0Þ 1 p2=m2 2 ; (64) where the fit parameters are
FVð0Þ ¼ 0:44 GeV; m21¼ 43:10 GeV2; FAð0Þ ¼ 0:21 GeV; m22¼ 48:00 GeV2: The values of the form factors FðBcÞ
V;A calculated at p2¼ m2Ds1 are FðBcÞ V ðp2 ¼ m2Ds1Þ ¼ ð0:51 0:14Þ GeV; FðBcÞ A ðp2 ¼ m2Ds1Þ ¼ ð0:24 0:07Þ GeV: (65)
For the form factor induced by the EP at q2 ¼ 0, we obtain
T2ð0Þ ¼ 0:298 0:085: (66) At the end of this section, we would like to calculate the decay widths and branching ratios. Using the amplitudes of each decay mode, we find the following expressions for
the decay rates at fixed points in WA and EP channels, as well as for the total decay rate of the transition under consideration: ðWAÞðB c ! Ds1Þ ¼ G2FjVcbVcsj2 16 m2 Bc m 2 Ds1 mBc 3 f2B c½ðF ðDs1Þ A Þ2þ ðF ðDs1Þ V Þ2 þ 2fBcfDs1F ðBcÞ V F ðDs1Þ V mDs1 m2B c þ f2 Ds1m2Ds1 ðFðBcÞ A Þ2 m4B c þðF ðBcÞ V Þ2 m4B c ; (67) ðEPÞðB c! Ds1Þ ¼ G2FjC7j2jVtbVtsj2 10244 m2 Bc m 2 Ds1 mBc 3 ð16ðmbþ msÞ2þ ðmb msÞ2Þ ½T2ð0Þ2; (68) ðtotalÞðB c ! Ds1Þ ¼ G2F 10244 m2 Bc m 2 Ds1 mBc 3 644jV cbVcsj2 fB2 cfðF ðDs1Þ A Þ2þ ðF ðDs1Þ V Þ2g þ 2fBcfDs1F ðBcÞ V F ðDs1Þ V mDs1 m2B c þ f2 Ds1m2Ds1 ðFðBcÞ A Þ2 m4B c þðF ðBcÞ V Þ2 m4B c þ jC7j2jV tbVtsj2ð16ðmb msÞ2þ ðmbþ msÞ2Þ½T2ð0Þ2 þ 162T 2ð0ÞjVcbVcsjjVtbVtsj fDs1mDs1X 4F ðBcÞ A m2B c ðmb msÞ þFðBcÞV m2B c ðmbþ msÞ þ fBcfFðDs1Þ V ðmbþ msÞX 4F ðDs1Þ A ðmb msÞYg ; (69)
where X and Y are the real and imaginary parts of the Wilson coefficient C7, respectively. In these formulas, as we previously mentioned, the fixed-point values of the form factors are used.
Finally, the numerical values of the corresponding branching ratios for the radiative decay under considera-tion are obtained as follows:
BðEPÞðB c! Ds1Þ ¼ ð1:769 0:582Þ 108; BðWAÞðB c! Ds1Þ ¼ ð2:243 0:736Þ 105; BðtotalÞðB c! Ds1Þ ¼ ð2:351 0:795Þ 105; (70)
where the dominant contribution to each channel comes from the perturbative part. From these values, we also see
that the Bc! Ds1ð2460Þ transition proceeds mostly via the WA mode. The order of the total branching ratio indicates that this decay channel can be detected at LHCb in the near future. Any measurement of this decay and comparison of the obtained data with our predictions in the present work can give valuable information about the nature and internal structure of the participating particles, especially the Ds1 meson.
ACKNOWLEDGMENTS
A. R. O. would like to thank R. Khosravi and S. Zarepour for useful discussions. Also, the partial support of the Shiraz University research council is appreciated.
APPENDIX The explicit expressions for Ci
G2 are given as follows: Ca
G2 ¼ mbf2mbm 2
c½I0½1; 3; 1 3m2bI0½1; 4; 1 þ 2m2bðmb msÞðmbþ msÞI0½1; 5; 1 þ 2mcðI0½1; 2; 1 4m2 bI0½1; 3; 1 mbmsI0½1; 3; 1 þ 5m4bI0½1; 4; 1 þ 3m3bmsI0½1; 4; 1 3m2bm2sI0½1; 4; 1 þ 2m3 bðmbþ msÞðmbþ msÞ2I0½1; 5; 1Þ þ 2ms½I0½1; 2; 1 2m6bI0½1; 5; 1 þ m4bð5I0½1; 4; 1 þ 2I½0;10 ½1; 5; 1Þ þ m2 bð4I0½1; 3; 1 3I ½0;1 0 ½1; 4; 1 þ 3I½1;00 ½1; 3; 1Þ þ mbð3I0½0;1½1; 3; 1 þ I½0;20 ½1; 4; 1 þ I0½1;0½1; 3; 1 þ 2m4bð3I½0;1 0 ½1; 5; 1 þ I½1;00 ½1; 5; 1Þ m2bð9I0½0;1½1; 4; 1 þ 2I½0;2 0 ½1; 5; 1 þ 3I½1;00 ½1; 4; 1 2I0½2;0½1; 4; 1Þ 4I3½0;1½1; 5; 1 þ 4I½1;03 ½1; 3; 1Þg; (A1) Cb G2 ¼ 7m2bm2cm2sI0½1; 1; 3 þ m3bI0½1; 1; 4 þ mcI0½1; 1; 4 þ mbmcI0½1; 1; 4 m2bmcI0½1; 1; 4 m3 bmsI0½1; 1; 4 mcmsI0½1; 1; 4 þ 2mbmcmsI0½1; 1; 4 þ m2bmcmsI0½1; 1; 4 mcI0½1; 1; 5 mbmcI0½1; 1; 5 þ m3bmcI0½1; 1; 5 m2bm2cI0½1; 1; 5 þ mcmsI0½1; 1; 5 2mbmcmsI0½1; 1; 5 þ 2m3bmcmsI0½1; 1; 5 2m2bm2cmsI0½1; 1; 5 mbI0½0;1½1; 1; 4 þ msI0½0;1½1; 1; 4 3=2I0½0;1½1; 1; 5 þ 3=2m2bI ½0;1 0 ½1; 1; 5 þ mbmsI½0;10 ½1; 1; 5 msI0½0;2½1; 1; 2 þ 3=2I0½1;0½1; 1; 3 þ mbI½1;00 ½1; 1; 3 þ 3msI0½1;0½1; 1; 3 þ 1=2m2 bI0½1;0½1; 1; 4 mbmsI0½1;0½1; 1; 4 þ 1=2I½2;00 ½1; 1; 4 þ msI½2;00 ½1; 1; 4; (A2) Cc G2 ¼ 1=6f2m5b½msðI0½3; 1; 1 þ m2cI0½3; 1; 2 þ I0½0;1½3; 1; 1Þ þ mcðm2cI0½3; 2; 2 þ I½0;10 ½3; 1; 2Þ 3I½0;1 0 ½3; 2; 1 þ I0½0;1½3; 2; 2 þ 3I0½0;2½3; 1; 2 2I½0;20 ½3; 2; 1 þ I0½0;3½3; 2; 2 þ 3I0½1;0½3; 1; 1 I½1;00 ½3; 1; 2 þ 2m3 cmsfI0½3; 2; 1 I0½3; 2; 2 4I½0;10 ½3; 1; 2 þ 3I0½0;1½3; 2; 1 þ 3I0½1;0½3; 1; 12I0½1;0½3; 2; 1g I½1;10 ½3; 1; 1 þ I0½1;1½3; 1; 2 2m3 b½mcð2I ½0;1 0 ½3; 2; 2 þ I0½1;0½3; 1; 2 þ m2cðI0½3; 1; 2 2I0½3; 2; 2 3ðI0½0;1½3; 2; 2 þ I½1;00 ½3; 2; 2ÞÞ þ I0½1;1½3; 2; 1Þ þ msðð1 þ 2m2 cÞI0½3; 2; 2 þ 2I½0;10 ½3; 1; 1 2I½0;10 ½3; 2; 1 þ 4m2cI0½0;1½3; 2; 2 þ I½0;20 ½3; 1; 2 þ I0½2;1½3; 1; 2ÞÞ m2bð2m6cI0½3; 2; 2 þ I½0;10 ½3; 1; 2 6I½0;10 ½3; 2; 1 2I0½0;2½3; 1; 2 þ 6I0½0;2½3; 2; 2 þ I½0;30 ½3; 2; 2 þ 2m3 cmsðI0½3; 1; 2 5I0½0;1½3; 2; 2 I½1;00 ½3; 2; 1Þ þ 3I0½1;0½3; 2; 1 2I½1;00 ½3; 2; 2 7m4 cI0½1;0½3; 2; 2; þ2mcmsðI0½3; 1; 2 I0½3; 2; 1 þ 2I½0;20 ½3; 1; 2 I0½1;1½3; 2; 1Þ þ 2I½1;10 ½3; 2; 1 3I½1;20 ½3; 2; 2 þ 2I0½2;0½3; 2; 1 2I0½2;1½3; 2; 2Þ þ I½2;10 ½3; 2; 2 þ m2 cð2I½0;10 ½3; 2; 2 þ 10I½0;20 ½3; 1; 2 2I0½0;2½3; 2; 1 þ I0½1;0½3; 2; 1 þ I½1;00 ½3; 2; 2 14I0½1;1½3; 1; 2 þ 14I0½1;1½3; 2; 2 þ 3I0½1;2½3; 2; 2 þ 2I0½2;0½3; 1; 2 10I½2;00 ½3; 2; 1 3I½2;10 ½3; 2; 2 3I0½3;0½3; 2; 2Þ I½3;00 ½3; 2; 2g; (A3) CdG2 ¼ 1=12f2m4cI0½3; 2; 1 þ 2m3cmsI0½3; 1; 1 þ 24mb7ðmcþ msÞI0½3; 2; 2 þ I0½0;1½3; 2; 2 þ 2m6bð8mcmsI0½3; 1; 1 þ 8m2 cI0½3; 2; 2 þ 3I½0;10 ½3; 1; 2 þ I½1;00 ½3; 1; 1Þ þ 6m5bð2m3cI0½3; 1; 1 þ 2m2cmsI0½3; 2; 2 2msð2I0½3; 1; 2 þ 2m2sI0½3; 2; 1 þ 2I0½0;1½3; 2; 2Þ þ mc 9I0½0;1½3; 2; 1 þ I½1;00 ½3; 1; 2ÞÞ þ 3I0½1;0½3; 2; 1 I0½1;0½3; 2; 2 2mcmsð2I0½0;1½3; 1; 2 I½1;00 ½3; 1; 1 I0½1;0½3; 1; 2Þ þ 2I½1;10 ½3; 2; 2 þ I½1;2 0 ½3; 1; 2 m2cð2I0½3; 1; 2 þ 2I0½3; 2; 2 þ 2I0½0;1½3; 2; 2 þ I½0;20 ½3; 2; 2 5I½1;00 ½3; 2; 1 þ 3I0½1;0½3; 2; 2 I0½2;0½3; 2; 2Þ I0½2;0½3; 2; 2 þ m4 bð4m4cI0½3; 2; 1 þ 4mc3msI0½3; 2; 2 þ 2I0½0;1½3; 2; 1 15I½0;10 ½3; 2; 2 2I0½0;2½3; 1; 1 þ 5I0½0;1½3; 2; 2 4I½1;00 ½3; 1; 2Þ þ 6I½1;00 ½3; 2; 2
þ 3ðI½0;10 ½3; 2; 2 þ I0½1;0½3; 2; 2Þ þ 6I½1;10 ½3; 1; 2 þ 4I½2;00 ½3; 2; 2Þ þ 3mbð2m2 cmsðI0½3; 2; 1 2I0½0;1½3; 2; 2 þ 2I½1;00 ½3; 2; 2Þ 4I0½3; 2; 2; þ3I½0;10 ½3; 2; 1 9I0½0;1½3; 2; 2 3I0½0;2½3; 1; 2 þ 5I½1;00 ½3; 1; 2 þ I½1;00 ½3; 2; 1 þ 3I½2;00 ½3; 1; 2Þ 4msðI0½3; 1; 2 þ I0½0;1½3; 2; 2 þ I½1;00 ½3; 1; 2 2I0½1;0½3; 2; 1 I0½1;1½3; 2; 1 þ I½2;00 ½3; 2; 2Þ 3m3 bð4m3cI0½3; 2; 2 þ 2m2cms 4I0½0;1½3; 2; 2 þ 4I½1;00 ½3; 1; 2Þ 4msð3I0½3; 2; 1 þ 4I0½3; 2; 2 þ 3I½0;10 ½3; 2; 2 4I½1;00 ½3; 2; 2 2I0½1;1½3; 2; 2 þ 2I0½2;0½3; 2; 2Þ þ mcð16I0½3; 1; 2 þ 12I0½3; 2; 2 27I0½0;1½3; 1; 2 þ 6I½0;10 ½3; 2; 1 6I½0;20 ½3; 2; 1 þ 3I0½1;0½3; 1; 1 þ 10I0½1;0½3; 2; 2 þ 6I½2;0 0 ½3; 2; 2Þ þ m2bðm4cð6I0½3; 1; 2 þ 4I0½3; 2; 2Þ þ m3cð4msI0½3; 1; 2 6msI0½3; 2; 1Þ 3I½0;10 ½3; 1; 1 þ 12I0½0;1½3; 1; 2 9I½1;00 ½3; 1; 2 þ 4I½1;00 ½3; 2; 1 þ 2mcmsð7I0½3; 1; 1 þ 9I0½0;1½3; 2; 2 2I0½1;0½3; 2; 1 6I0½1;0½3; 2; 2Þ 9I½1;10 ½3; 1; 1 2I0½1;2½3; 2; 1 þ m2cð14I 0½3; 1; 2 12I0½3; 2; 2 þ 9I½0;10 ½3; 1; 2 2I½0;1 0 ½3; 2; 2 þ 2I½0;20 ½3; 1; 2 10I½1;00 ½3; 1; 2 þ 9I½1;00 ½3; 2; 1 2I0½2;0½3; 1; 2Þ 6I½2;00 ½3; 1; 2 þ 6I0½2;0½3; 2; 2 þ 2I½3;00 ½3; 1; 0Þ I0½3;0½3; 2; 2g; (A4) Ce G2 ¼ 1=6f2mcmsðI0½1; 3; 2 þ I0½1; 3; 3Þ þ 4m 5 bðmcI0½1; 2; 3 2msI0½1; 3; 3Þ 2m2c½I0½1; 2; 2 I0½1; 3; 3 3I½0;10 ½1; 1; 3 þ I0½0;1½1; 2; 2 I0½0;2½1; 2; 3 I½1;00 ½1; 2; 2 þ 3I0½1;0½1; 3; 1 2m4 bð2mcmsI0½1; 2; 1 2m2 cI0½1; 3; 3Þ þ 3I0½0;1½1; 3; 3 þ I½1;00 ½1; 2; 2Þ 2m3b½mcð3I0½1; 2; 3 þ 2I0½1; 3; 3Þ þ 2msðI0½1; 3; 3 2I½0;10 ½1; 2; 2 þ 2I0½1;0½1; 3; 2Þ þ 2mbfmc½I0½1; 2; 2 þ I0½1; 3; 2 þ 2ms½I0½1; 2; 3 þ I0½1; 3; 2 I½0;10 ½1; 1; 3 þ I0½1;0½1; 2; 3 3I0½1; 3; 1 þ 2I½0;10 ½1; 2; 3 2I½1;00 ½1; 3; 3Þg þ I½2;00 ½1; 3; 2 þ m2
bð2m2cð2I0½1; 2; 3 3I0½1; 3; 2Þ þ 2mcmsð2I0½1; 1; 2 3I0½1; 3; 3Þ þ 9I ½0;1 0 ½1; 2; 3 2I½0;10 ½1; 3; 2 þ 2I½0;20 ½1; 3; 3 6I0½1;0½1; 1; 3 þ 3I0½1;0½1; 3; 2 2I0½2;0½1; 3; 3Þg; (A5) Cf G2 ¼ 2=3msfm 3 bðm2cI0½2; 1; 4Þ þ m2bmcðm2cI0½2; 1; 4 2I½0;10 ½2; 1; 4Þ þ mbðI0½0;1½2; 1; 3 2I½0;10 ½2; 1; 4 þ I0½0;2½2; 1; 2 þ I½1;00 ½2; 1; 3 þ m2 cI0½1;0½2; 1; 3 I½1;10 ½2; 1; 4Þ þ mcð2I½0;1 0 ½2; 1; 3 þ I½0;20 ½2; 1; 3 3I½1;00 ½2; 1; 4Þg; (A6) where we have ignored terms with higher powers of the strange quark mass. The functions In½a; b; c and I½i;jn ½a; b; c are defined as
I0½a; b; c ¼ ð1Þ aþbþc
162ðaÞðbÞðcÞðM21Þ2abðM22Þ2acU0ða þ b þ c 4; 1 c bÞ; I1½a; b; c ¼ ð1Þ
aþbþcþ1
162ðaÞðbÞðcÞðM21Þ2abðM22Þ3acU0ða þ b þ c 5; 1 c bÞ; I2½a; b; c ¼ ð1Þ
aþbþcþ1
162ðaÞðbÞðcÞðM21Þ3abðM22Þ2acU0ða þ b þ c 5; 1 c bÞ; In½i;j½a; b; c ¼ ½M21i½M22j di dðM21Þi dj dðM22Þj½M12 i½M2 2jIn½a; b; c: (A7)
whereU0ða; bÞ is given by
U0ða; bÞ ¼Z1 0 dyðy þ M 2 1þ M22Þaybexp B1 y B0 B1y ; (A8) and B1¼m 2 b M12½M 2 1þ M22; B0 ¼ 1 M21M22½M 2 1m2cþ M22ðm2cþ m2bÞ; B1 ¼ m2c M21M22: (A9) 1 PHYSICAL REVIEW D 87, 016013 (2013)
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![FIG. 2. Diagrams for bare-loop [(a), (b)], quark and mixed condensates [(c)–(e)] and propagation of the soft quark in the electromagnetic field (f ).](https://thumb-eu.123doks.com/thumbv2/9libnet/4066163.57871/3.918.477.840.689.1012/diagrams-quark-mixed-condensates-propagation-quark-electromagnetic-field.webp)
