Analytical Forms for Cross Sections of Di-lepton Production from e^+e^- Collisions around the J/ψ Resonance
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Analytic Forms for Cross Sections of Di-lepton Productionfrom e + e − Collisions around the
J/ψ
Resonance *Xing-Yu Zhou ( 周 兴 玉 ) Ya-Di Wang ( 王 雅 迪 ) Li-Gang Xia ( 夏 力 钢 ) Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Helmholtz Institute Mainz, Mainz 55128, Germany Department of Physics, Tsinghua University, Beijing 100084, China
Abstract:
A detailed theoretical derivation of the cross sections of e + e − → e + e − and e + e − → µ + µ − around the J/ψ resonance is reported. The resonance and interference parts of the cross sections, related to
J/ψ resonanceparameters, are calculated. Higher-order corrections for vacuum polarization and initial-state radiation are consid-ered. An arbitrary upper limit of radiative correction integration is involved. Full and simplified versions of analyticformulae are given with precision at the level of 0.1% and 0.2%, respectively. Moreover, the results obtained in thepaper can be applied to the case of the ψ (3686) resonance. Key words: initial-state radiation, vacuum polarization, e + e − collision, di-lepton production, the J/ψ resonance
PACS:
The
J/ψ resonance is frequently referred to as a hy-drogen atom for QCD, and its resonance parameters(mass M , total width Γ tot , leptonic widths Γ ee and Γ µµ ,and so on) describe the fundamental properties of thestrong and electromagnetic interactions. In theory, thedecay widths can be predicted by different potentialmodels [1, 2] and lattice QCD calculations [3]. In ex-periment, with results from BABAR [4], CLEO [5] andKEDR [6], determinations of these decay widths haveentered a period of precision measurement.In 2012, data samples were taken at 15 center-of-massenergy points around the J/ψ resonance with the BESIIIdetector [7] operated at the BEPCII collider [7]. In thisenergy region, BEPCII provides high luminosity and BE-SIII shows excellent performance, which helps us accu-rately measure the cross sections of e + e − → e + e − and e + e − → µ + µ − . To measure J/ψ decay widths, ac-curate theoretical formulae taking into account higher-order corrections are also needed. If one wishes to havea high-efficiency optimization procedure, it is better tohave analytic expressions for the theoretical cross sec-tions. Because the continuum parts of these cross sec-tions do not involve
J/ψ decay widths and can be eval-uated precisely by Monte-Carlo generators such as the Babayaga generator [8], only the analytic forms for theresonance and interference parts are derived in this pa-per.We will start with theoretical fundamentals on thestructure function method, its applications to the casesof e + e − → e + e − and e + e − → µ + µ − , Born cross sec-tions and the vacuum polarization function in Section .Then, we will give the definitions and resulting formu-lae for the resonance and interference parts of the crosssections of e + e − → e + e − and e + e − → µ + µ − in Section .Most of the purely mathematical derivation is given inAppendix A to make the text easier to read. Generally, initial-state radiation (ISR), final-state ra-diation (FSR) and their interference (ISR-FSR relation)must be considered when one makes higher-order correc-tions to cross sections. Here, the ISR-FSR relation in-cludes interference of diagrams with emission of real andvirtual photons between initial- and final-state particles.The suppression level of the ISR-FSR relation betweenthe production and decay stages of heavy unstable parti-cles is discussed in Ref. [9]. According to the conclusionin Ref. [9], there is no need to take into account the ISR- ∗ Supported by National Natural Science Foundation of China (11275211) and Istituto Nazionale di Fisica Nucleare, Italy1) E-mail: [email protected]) E-mail: [email protected]) E-mail: [email protected] xxxxxx-1ccepted by Chinese Physics C
FSR relation in the case of
J/ψ , because it is suppressedby Γ tot /M (about 3 × − ). As for FSR, a universal cal-culation is impossible if one has no explicit knowledgeof selection criteria, so it needs to be handled separatelywith a numerical method, which is outside the scope ofthis paper. Thus, in this paper the calculation with ISRonly is presented.The structure function method [10] is adopted hereto deal with ISR. Its fundamental formula is σ ( s ) = Z Z X d ¯ σd Ω ( s (1 − x ) , cos θ ) F ( s, x ) dxd Ω . (1)Here, σ stands for the cross section after correction, d ¯ σd Ω for the differential cross section before correction, F forthe radiator, s for the square of the center-of-mass energyand θ for the polar angle of the positively charged finalparticle in the center-of-mass frame. The upper limit X of the integration variable x is usually set as 1 − s ′ min /s ,where s ′ min is the minimum of the invariant mass squaredof the final-state particle system excluding the emittedphotons.The radiator F adopted in this paper was first de-rived in Ref. [11] and slightly revised in Ref. [12]. Bothdocuments are in Chinese, although the former has anEnglish-language preprint (Ref. [13]). It is different frombut a very good approximation of the classical one in Ref.[10]. Its expression is F ( s, x ) = x v − v (1 + δ )+ x v (cid:18) − v − v (cid:19) + x v +1 (cid:18) v − v (cid:19) , (2)where δ ( v ) = απ ( π −
12 ) + 34 v + (cid:18) − π (cid:19) v (3)and v ( s ) = 2 απ (cid:18) ln sm e − (cid:19) . (4)Here, α stands for the fine structure constant and m e denotes the electron mass. e + e − → e + e − and e + e − → µ + µ − Applying the structure function method to the casesof e + e − → e + e − and e + e − → µ + µ − , one can get (cid:18) dσd Ω (cid:19) ee | µµ ( s, cos θ )= Z X (cid:18) d ¯ σd Ω (cid:19) ee | µµ ( s (1 − x ) , cos θ ) F ( s, x ) dx, (5) where the symbol | stands for “or”, (cid:18) d ¯ σd Ω (cid:19) ee = (cid:18) dσ d Ω (cid:19) S ee (cid:12)(cid:12)(cid:12)(cid:12) − Π( s ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) dσ d Ω (cid:19) T ee (cid:12)(cid:12)(cid:12)(cid:12) − Π( t ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) dσ d Ω (cid:19) STI ee Re (cid:18) − Π( s ) 11 − Π( t ) (cid:19) (6)and (cid:18) d ¯ σd Ω (cid:19) µµ = (cid:18) dσ d Ω (cid:19) S µµ (cid:12)(cid:12)(cid:12)(cid:12) − Π( s ) (cid:12)(cid:12)(cid:12)(cid:12) . (7)Here, t denotes the square of the 4-momentum trans-ferred in the t channel. As for e + e − → e + e − , the relationbetween t and s is t ≈ − s − cos θ ) . (8)In addition, (cid:0) dσ d Ω (cid:1) S ee , (cid:0) dσ d Ω (cid:1) T ee , (cid:0) dσ d Ω (cid:1) STI ee and (cid:0) dσ d Ω (cid:1) S µµ areBorn cross sections, and − Π is the vacuum polarizationfunction. They will be discussed in the following twosubsections. The quantities (cid:0) dσ d Ω (cid:1) S ee , (cid:0) dσ d Ω (cid:1) T ee and (cid:0) dσ d Ω (cid:1) STI ee are the schannel part, the t channel part and the s-t interferencepart of the Born cross section of e + e − → e + e − (cid:0)(cid:0) dσ d Ω (cid:1) ee (cid:1) ,respectively, that is (cid:18) dσ d Ω (cid:19) ee = (cid:18) dσ d Ω (cid:19) S ee + (cid:18) dσ d Ω (cid:19) T ee + (cid:18) dσ d Ω (cid:19) STI ee , (9)where (cid:18) dσ d Ω (cid:19) S ee = α s (1 + cos θ ) , (10a) (cid:18) dσ d Ω (cid:19) T ee = α s (1 + cos θ ) + 4(1 − cos θ ) , (10b) (cid:18) dσ d Ω (cid:19) STI ee = − α s (1 + cos θ ) − cos θ . (10c)The Born cross section of e + e − → µ + µ − (cid:16)(cid:0) dσ d Ω (cid:1) µµ (cid:17) hasonly an s channel part (cid:0) dσ d Ω (cid:1) S µµ , which equals exactly (cid:0) dσ d Ω (cid:1) S ee given by Eq. (10a). In Section 4 of Ref. [14], the distinction and relation-ship between the “bare” and “dressed” parameters of J PC = 1 −− resonances (for example J/ψ ) are discussedin detail. In the discussion there, the vacuum polariza-tion function is written as11 − Π( q ) = 11 − Π ( q ) + Π R ( q ) , (11) xxxxxx-2ccepted by Chinese Physics C where Π R is expressed with the “dressed” parameters M ,Γ tot and Γ ee asΠ R ( q ) = 3Γ ee α q M q − M + iM Γ tot . (12)Here, Π R stands for the contribution from the resonanceitself (in our case, it is J/ψ ), while Π denotes contribu-tions from other sources. Based on the lepton universal-ity assumption, Γ ee in Eq. (12) can be substituted by p Γ ee Γ µµ in the case of e + e − → µ + µ − .According to Eq. (11), − Π( s ) and − Π( t ) in Eq. (6)and (7) can be expressed as11 − Π( s ) = 11 − Π ( s ) + Π R ( s ) (13)and 11 − Π( t ) = 11 − Π ( t ) . (14) No Π R ( t ) term appears in Eq. (14) because it can besafely ignored in the spacelike region. Besides, the imag-inary parts of − Π ( s ) and − Π ( t ) can be safely ignored aswell. Consequently, − Π ( s ) and − Π ( t ) will be regardedas real in the following section. Considering (cid:0) d ¯ σd Ω (cid:1) ee and (cid:0) d ¯ σd Ω (cid:1) µµ given by Eq. (6) and(7) as well as − Π( s ) and − Π( t ) given by Eq. (13) and(14), one can expand (cid:0) dσd Ω (cid:1) ee and (cid:0) dσd Ω (cid:1) µµ via Eq. (5) intomany small terms. With these small terms regrouped,the resonance and interference parts of (cid:0) dσd Ω (cid:1) ee and (cid:0) dσd Ω (cid:1) µµ , namely (cid:0) dσd Ω (cid:1) R ee , (cid:0) dσd Ω (cid:1) CRI ee , (cid:0) dσd Ω (cid:1) R µµ and (cid:0) dσd Ω (cid:1) CRI µµ ,can be defined as (cid:18) dσd Ω (cid:19) R ee = Z X (cid:18) dσ d Ω (cid:19) S ee ( s (1 − x ) , cos θ ) | Π R ( s (1 − x )) | F ( s, x ) dx, (15a) (cid:18) dσd Ω (cid:19) CRI ee = Z X (cid:18) dσ d Ω (cid:19) S ee ( s (1 − x ) , cos θ )2 Re (cid:18) − Π ( s (1 − x )) Π R ( s (1 − x )) (cid:19) + (cid:18) dσ d Ω (cid:19) STI ee ( s (1 − x ) , cos θ ) Re (cid:18) Π R ( s (1 − x )) 11 − Π ( t (1 − x )) (cid:19) ! F ( s, x ) dx, (15b) (cid:18) dσd Ω (cid:19) R µµ = Z X (cid:18) dσ d Ω (cid:19) S µµ ( s (1 − x ) , cos θ ) | Π R ( s (1 − x )) | F ( s, x ) dx, (15c) (cid:18) dσd Ω (cid:19) CRI µµ = Z X (cid:18) dσ d Ω (cid:19) S µµ ( s (1 − x ) , cos θ )2 Re (cid:18) − Π ( s (1 − x )) Π R ( s (1 − x )) (cid:19) F ( s, x ) dx. (15d)With (cid:0) dσ d Ω (cid:1) S ee | µµ and (cid:0) dσ d Ω (cid:1) STI ee expressed in Eq. (10a)and (10c) as well as Π R expressed in Eq. (12) further employed, one can rewrite (cid:0) dσd Ω (cid:1) R ee , (cid:0) dσd Ω (cid:1) CRI ee , (cid:0) dσd Ω (cid:1) R µµ and (cid:0) dσd Ω (cid:1) CRI µµ more explicitly as (cid:18) dσd Ω (cid:19) R ee = 9Γ ee M · I R · (1 + cos θ ) , (16a) (cid:18) dσd Ω (cid:19) CRI ee = 3Γ ee α M · I CRI · (cid:18) (1 + cos θ ) 11 − Π ( s ) − (1 + cos θ ) − cos θ − Π ( t ) (cid:19) , (16b) (cid:18) dσd Ω (cid:19) R µµ = 9Γ ee Γ µµ M · I R · (1 + cos θ ) , (16c) xxxxxx-3ccepted by Chinese Physics C (cid:18) dσd Ω (cid:19) CRI µµ = 3 p Γ ee Γ µµ α M · I CRI · (1 + cos θ ) 11 − Π ( s ) , (16d)where I R = Z X s (1 − x )( s (1 − x ) − M ) + M Γ F ( s, x ) dx, (17a) I CRI = Z X s (1 − x ) − M ( s (1 − x ) − M ) + M Γ F ( s, x ) dx. (17b)Here, in the cases of (cid:0) dσd Ω (cid:1) CRI ee and (cid:0) dσd Ω (cid:1) CRI µµ , − Π ( s ) and − Π ( t ) are used as very good approximations to theequivalents of − Π ( s (1 − x )) and − Π ( t (1 − x )) after integra-tion in Eq. (15). Numerical calculation indicates thatthe resulting deviations are less than 0.01%.As can be seen from Eq. (16), to evaluate further,only I R and I CRI have to be calculated. Detailed calcu-lations of the two integrals are put in Appendix A, whichincludes three parts: A.1, A.2, A.3. Their analytic for-mulae are fully derived in part A.1. Due to complexity,simplified versions of the analytic formulae are furtherobtained in part A.2. Finally, both versions of the ana- lytic formulae are compared with numerical computingresults in part A.3.Based on those of I R and I CRI , we will list directly thefull and simplified version of analytic results of (cid:0) dσd Ω (cid:1) R ee , (cid:0) dσd Ω (cid:1) CRI ee , (cid:0) dσd Ω (cid:1) R µµ and (cid:0) dσd Ω (cid:1) CRI µµ and discuss briefly theircomparisons with numerical computing results in the fol-lowing three subsections. With I R and I CRI expressed in Eq. (A14) and (A15)adopted, the full versions of the analytic formulae for (cid:0) dσd Ω (cid:1) R ee , (cid:0) dσd Ω (cid:1) CRI ee , (cid:0) dσd Ω (cid:1) R µµ and (cid:0) dσd Ω (cid:1) CRI µµ can be written as (cid:18) dσd Ω (cid:19) R ee = 9Γ ee M · s ( P − Q ) · (1 + cos θ ) , (18a) (cid:18) dσd Ω (cid:19) CRI ee = 3Γ ee α M · (( s − M ) P − sQ ) · (cid:18) (1 + cos θ ) 11 − Π ( s ) − (1 + cos θ ) − cos θ − Π ( t ) (cid:19) , (18b) (cid:18) dσd Ω (cid:19) R µµ = 9Γ ee Γ µµ M · s ( P − Q ) · (1 + cos θ ) , (18c) (cid:18) dσd Ω (cid:19) CRI µµ = 3 p Γ ee Γ µµ α M · (( s − M ) P − sQ ) · (1 + cos θ ) 11 − Π ( s ) , (18d)where P = 1 s ( A G ( a, β, v, X ) + B G ( a, β, v + 1 , X ) + C H ( a, β, v, X )) , (19a) Q = 1 s ( D G ( a, β, v + 1 , X ) + E H ( a, β, v, X ) + C H ( a, β, v + 1 , X )) (19b)with a = s(cid:18) M s − (cid:19) + M Γ s , (20a) β = cos − (cid:16) M s − (cid:17)q(cid:0) M s − (cid:1) + M Γ s , (20b) xxxxxx-4ccepted by Chinese Physics C A = 1 + δ, (20c) B = 1 v + 1 (cid:18) − v − v (cid:19) , (20d) C = v − v , (20e) D = Avv + 1 , (20f) E = B ( v + 1) (20g)and G ( a, β, v, X )= a v − (cid:16) πv sin πv (cid:17) (cid:18) sin[(1 − v ) β ]sin β (cid:19) + vX v − (cid:18) X v −
2+ 2 a (cos β ) Xv − − a (4 cos β − v − (cid:19) (0 < v < , (21a) H ( a, β, v, X )= h ( a sin β, a cos β, v + 1 , X + a cos β ) − h ( a sin β, a cos β, v + 1 , a cos β ) , (21b) h ( a, b, c, x ) = − i ac · (cid:18) − ia + x (cid:19) − c F (cid:18) − c, − c, − c, a + iba + ix (cid:19) − (cid:18) ia + x (cid:19) − c F (cid:18) − c, − c, − c, ia + bia + x (cid:19) ! . (21c)Here, F is the Gauss hypergeometric function. With I R and I CRI given by Eq. (A23) and (A24), thesimplified versions of the analytic formulae for (cid:0) dσd Ω (cid:1) R ee , (cid:0) dσd Ω (cid:1) CRI ee , (cid:0) dσd Ω (cid:1) R µµ and (cid:0) dσd Ω (cid:1) CRI µµ can be written as (cid:18) dσd Ω (cid:19) R ee = 9Γ ee M Γ tot (1 + δ ) Im F · (1 + cos θ ) , (22a) (cid:18) dσd Ω (cid:19) CRI ee = − ee α M s (1 + δ ) Re F · (cid:18) (1 + cos θ ) 11 − Π ( s ) − (1 + cos θ ) − cos θ − Π ( t ) (cid:19) , (22b) (cid:18) dσd Ω (cid:19) R µµ = 9Γ ee Γ µµ M Γ tot (1 + δ ) Im F · (1 + cos θ ) , (22c) (cid:18) dσd Ω (cid:19) CRI µµ = − p Γ ee Γ µµ α M s (1 + δ ) Re F · (1 + cos θ ) 11 − Π ( s ) , (22d)where F = (cid:16) πv sin πv (cid:17) (cid:18) sM − s − iM Γ tot (cid:19) − v . (23) As one can see from Eq. (16) and (17), (cid:18) ∆ σσ (cid:19) R | CRI ee | µµ (F | S , N) = σ R | CRI ee | µµ (F | S) − σ R | CRI ee | µµ (N) σ R | CRI ee | µµ (N)= I R | CRI (F | S) − I R | CRI (N) I R | CRI (N) = (cid:18) ∆ II (cid:19) R | CRI (F | S , N) . Here, the symbols F, S and N stand for the full ver-sion of the analytic results, the simplified version of theanalytic results and the numerical computing results, re-spectively. According to part A.3 (the last part of Appendix A),from √ s = M − tot to √ s = M + 10Γ tot with X set at1 as well as M and Γ tot at their PDG values [15]: (cid:18) ∆ σσ (cid:19) R | CRI ee | µµ (F , N) = (cid:18) ∆ II (cid:19) R | CRI (F , N) < . (cid:18) ∆ σσ (cid:19) R | CRI ee | µµ (S , N) = (cid:18) ∆ II (cid:19) R | CRI (S , N) < . . Taking into account the precision of the structure func-tion method itself is 0.1% [10], we regard 0.1% and 0.2%as the precision of the full and simplified versions ofthe analytic formulae for (cid:0) dσd Ω (cid:1) R ee , (cid:0) dσd Ω (cid:1) CRI ee , (cid:0) dσd Ω (cid:1) R µµ and (cid:0) dσd Ω (cid:1) CRI µµ , respectively. xxxxxx-5ccepted by Chinese Physics C We have derived the detailed formulae for the res-onance and interference parts of the cross sections of e + e − → e + e − and e + e − → µ + µ − around the J/ψ reso-nance with higher-order corrections for vacuum polariza-tion and initial-state radiation considered. In the deriva-tion, the arbitrary upper limit of radiative correction in-tegration X has been involved. Two (full and simplified)versions of the analytic formulae are given with precisionat the levels of 0.1% and 0.2%, which are accurate enoughfor the measurement of J/ψ decay widths at present.In our derivation, only a very few steps rely on the values of
J/ψ resonance parameters and they can be eas-ily verified to be workable for the case of the ψ (3686) res-onance. In the coming round of data-taking at BESIII,there is a plan for an energy scan around the ψ (3686) res-onance for the measurement of the resonance parameters.By that time, the results obtained in this paper will begood references. The authors would like to thank Prof. Wei-Guo Lifor his suggestion on the contributing as well as Prof.Ping Wang, Prof. Hai-Ming Hu and Prof. Chang-ZhengYuan for their kind help and beneficial discussions.
Appendix ACalculations of I R and I CRI
A.1 Full versions of analytic formulae
In the appendix, we evaluate the two integrals I R and I CRI required in Section . For the convenience of furthercalculations, it is necessary to make some simple transforma-tions by introducing some new variables. The first transfor-mation is 1( s (1 − x ) − M ) + M Γ = 1 s x + 2 a (cos β ) x + a , (A1)where a = s(cid:18) M s − (cid:19) + M Γ s , (A2a) β = cos − (cid:16) M s − (cid:17)r(cid:16) M s − (cid:17) + M Γ s . (A2b)The second transformation is F ( s, x ) = x v − v (1 + δ )+ x v (cid:18) − v − v (cid:19) + x v +1 (cid:18) v − v (cid:19) = Avx v − + B ( v + 1) x v + Cx v +1 , (A3)where A = 1 + δ, (A4a) B = 1 v + 1 (cid:18) − v − v (cid:19) , (A4b) C = v − v . (A4c) The third transformation is xF ( s, x ) = x v v (1 + δ )+ x v +1 (cid:18) − v − v (cid:19) + x v +2 (cid:18) v − v (cid:19) = D ( v + 1) x v + Ex v +1 + Cx v +2 , (A5)where D = Avv + 1 , (A6a) E = B ( v + 1) . (A6b)In addition, some integral formulae are crucial for furthercalculations. From the following two integral formulae Z ∞ vx v − x + 2 a (cos β ) x + a dx = a v − (cid:16) πv sin πv (cid:17)(cid:18) sin[(1 − v ) β ]sin β (cid:19) (0 < v <
2) (A7)and Z ∞ X vx v − x + 2 a (cos β ) x + a dx ≃ vX v − − X v − − a (cos β ) Xv − a (4 cos β − v − ! ( v < , (A8)one obtains for the first integral formula G ( a, β, v, X ) = Z X vx v − x + 2 a (cos β ) x + a dx ≃ a v − (cid:16) πv sin πv (cid:17)(cid:18) sin[(1 − v ) β ]sin β (cid:19) + vX v − (cid:18) X v −
2+ 2 a (cos β ) Xv − − a (4 cos β − v − (cid:19) (0 < v < . (A9)xxxxxx-6ccepted by Chinese Physics CThe second integral formula is H ( a, β, v, X ) = Z X x v +1 x + 2 a (cos β ) x + a dx = Z X x v +1 ( x + a cos β ) + ( a sin β ) dx = Z X + a cos βa cos β ( y − a cos β ) v +1 y + ( a sin β ) dy = h ( a sin β, a cos β, v + 1 , X + a cos β ) − h ( a sin β, a cos β, v + 1 , a cos β ) , (A10) where h ( a, b, c, x ) = Z x ( y − b ) c y + a dy = − i ac · (cid:18) − ia + x (cid:19) − c F (cid:18) − c, − c, − c, a + iba + ix (cid:19) − (cid:18) ia + x (cid:19) − c F (cid:18) − c, − c, − c, ia + bia + x (cid:19) ! . (A11)Here, F is the Gauss hypergeometric function.Using the newly introduced variables and the importantintegral formulae, we get P = Z X s (1 − x ) − M ) + M Γ F ( s, x ) dx = 1 s Z X x + 2 a (cos β ) x + a ( Avx v − + B ( v + 1) x v + Cx v +1 ) dx = 1 s (cid:18) A Z X vx v − x + 2 a (cos β ) x + a dx + B Z X ( v + 1) x v x + 2 a (cos β ) x + a dx + C Z X x v +1 x + 2 a (cos β ) x + a dx (cid:19) = 1 s ( A G ( a, β, v, X ) + B G ( a, β, v + 1 , X ) + C H ( a, β, v, X )) (A12)and Q = Z X x ( s (1 − x ) − M ) + M Γ F ( s, x ) dx = 1 s Z X xx + 2 a (cos β ) x + a ( Avx v − + B ( v + 1) x v + Cx v +1 ) dx = 1 s Z X x + 2 a (cos β ) x + a ( D ( v + 1) x v + Ex v +1 + Cx v +2 ) dx = 1 s (cid:18) D Z X ( v + 1) x v x + 2 a (cos β ) x + a dx + E Z X x v +1 x + 2 a (cos β ) x + a dx + C Z X x v +2 x + 2 a (cos β ) x + a dx (cid:19) = 1 s ( D G ( a, β, v + 1 , X ) + E H ( a, β, v, X ) + C H ( a, β, v + 1 , X )) , (A13)and then get I R = Z X s (1 − x )( s (1 − x ) − M ) + M Γ F ( s, x ) dx = s Z X − x ( s (1 − x ) − M ) + M Γ F ( s, x ) dx = s (cid:18)Z X s (1 − x ) − M ) + M Γ F ( s, x ) dx − Z X x ( s (1 − x ) − M ) + M Γ F ( s, x ) dx (cid:19) = s ( P − Q ) (A14)and I CRI = Z X s (1 − x ) − M ( s (1 − x ) − M ) + M Γ F ( s, x ) dx = Z X ( s − M ) − sx ( s (1 − x ) − M ) + M Γ F ( s, x ) dx = ( s − M ) Z X s (1 − x ) − M ) + M Γ F ( s, x ) dx − s Z X x ( s (1 − x ) − M ) + M Γ F ( s, x ) dx = ( s − M ) P − sQ. (A15)Equations (A14) and (A15) give the analytic formulae for I R and I CRI . Since there are no approximations made in thederivation, we refer to the formulae as the full versions of theanalytic formulae. Considering all the quantities involved in P and Q ( A , B , C and so on), the results are actually verycomplicated. For ease of use, simplified versions of the ana-lytic formulae are needed. A.2 Simplified versions of analytic formulae
In this part, we will make some approximations to obtainsimplified versions of the analytic formulae. The first step isto reduce F ( s, x ) to x v − v (1+ δ ). Since 0 ≤ x ≤ v ≈ . J/ψ region, the parts discarded are negligible. Thisreduction leads to B = 0, C = 0, E = 0.xxxxxx-7ccepted by Chinese Physics CThe second step is to reduce G ( a, β, v, X ) to a v − (cid:0) πv sin πv (cid:1) (cid:16) sin[(1 − v ) β ]sin β (cid:17) . This reduction means that X → + ∞ , which is unreasonable from the physical point of view.However, since v ≈ .
08 and a ∈ (3 × − , × − ), the reduc-tion itself is a reasonable mathematical approximation when X is large enough. In addition, in the cases of (cid:0) dσd Ω (cid:1) R ee and (cid:0) dσd Ω (cid:1) R µµ , a reasonable reduction of sin[(1 − v ) β ] − a sin[( − v ) β ]to sin[(1 − v ) β ] is also carried out at this step. With the twosteps of approximation applied, one can get I R ≈ sa sin β (1 + δ ) a v − (cid:16) πv sin πv (cid:17) sin[(1 − v ) β ] (A16)and I CRI ≈ − s (1 + δ ) a v − (cid:16) πv sin πv (cid:17) cos[(1 − v ) β ] . (A17)At this point, if one introduces a complex variable F = (cid:16) πv sin πv (cid:17) ( a cos β − ia sin β ) v − , (A18)then a v − (cid:16) πv sin πv (cid:17) sin[(1 − v ) β ] = Im F , (A19) a v − (cid:16) πv sin πv (cid:17) cos[(1 − v ) β ] = Re F . (A20)Getting a and β back to r(cid:16) M s − (cid:17) + M Γ s and cos − (cid:18) M s − (cid:19)r(cid:16) M s − (cid:17) + M s , respectively, one has a sin β = M Γ tot s (A21)and F = (cid:16) πv sin πv (cid:17) (cid:18) sM − s − iM Γ tot (cid:19) − v . (A22)With Eq. (A19), (A20) and (A21), I R and I CRI can beexpressed further as I R ≈ M Γ tot (1 + δ ) Im F (A23)and I CRI ≈ − s (1 + δ ) Re F . (A24)These are the simplified versions of the analytic formulae weneed. A.3 Comparisons of analytic formulae with numericalcomputing results
To check the validity of these analytic formulae, we com-pare them with numerical computing results. In the com-parisons, the two integrals I R and I CRI are compared from √ s = M − tot to √ s = M +10Γ tot with X set at 1 as well as M and Γ tot at their PDG values [15]. The results are shownin Fig. 1. (GeV)s3.096 3.0965 3.097 3.097500.050.1(%) (F,N) R I I ∆ (S,N) R I I ∆ (GeV)s3.096 3.0965 3.097 3.0975-0.500.5(%) (F,N) CRI
I I ∆ (S,N) CRI
I I ∆ Fig. 1. Comparisons of analytic formulae with numerical computing results. In the middle of the right-hand plot,the dotted line has a similar structure to the solid one. It does not show clearly in the plot because of its smallscale.The variables in the legends are defined as (cid:18) ∆ II (cid:19) R | CRI (F | S , N) = I R | CRI (F | S) − I R | CRI (N) I R | CRI (N) . Here, the symbols | , F, S and N are same as those used atthe beginning of Subsections and .As can be seen from the dotted lines, the full versions ofthe analytic formulae agree very well with the numerical com-puting results. In fact, detailed numbers show that their rel-ative differences are less than 0.01%. Similarly, from the solid lines, one can see that except for I CRI at energies very closeto the
J/ψ peak, the simplified versions of the analytic formu-lae agree with the numerical computing results to better than0.1%. The upward and downward peaks of (cid:0) ∆ II (cid:1) CRI (S , N) atenergies near the
J/ψ peak is caused by the smallness of theabsolute values (very close to 0) of I CRI , which makes σ CRI values negligible when compared with their corresponding σ R values. Because in the end, only the sum of σ R and σ CRI willbe used in our data analysis, the peaks of (cid:0) ∆ II (cid:1) CRI (S , N) arenot worrying for us.xxxxxx-8ccepted by Chinese Physics C
References : 1206(2009)2 O. Lakhina and E.S. Swanson, Phys. Rev. D, : 014012 (2006)3 J.J. Dudek, R.G. Edwards, and D.G. Richards, Phys. Rev. D, : 074507 (2006)4 B. Aubert et al (BABAR Collaboration), Phys. Rev. D, :011103 (2004)5 G.S. Adams et al (CLEO Collaboration), Phys. Rev. D, :051103 (2006)6 V.V. Anashin et al (KEDR Collaboration), Phys. Lett. B, :134 (2010)7 M. Ablikim et al (BESIII Collaboration), Nucl. Instrum. Meth-ods A, : 345 (2010) 8 C.M. Carloni Calame, G. Montagna, O. Nicrosini et al, Nucl.Phys. B (Proc. Suppl.), : 48 (2004)9 V.S. Fadin, V.A. Khoze and A.D. Martin, Phys. Lett. B, :141 (1994)10 E.A. Kuraev and V.S. Fadin, Sov. J. Nucl. Phys., : 466(1985)11 F.Z. Chen, P. Wang, J.M. Wu et al, HEP & NP, (7): 585(1990) (in Chinese)12 X.H. Mo, Measurement of ψ (2 S ) Resonance Parameters, Ph.D.Thesis (Beijing: Institute of High Energy Physics, CAS, 2001)(in Chinese)13 F.Z. Chen, P. Wang, C.M. Wu et al, BIHEP-EP-90-0114 V.V. Anashin et al (KEDR Collaboration), Phys. Lett. B, :280 (2012)15 C. Patrignani et al (Particle Data Group), Chin. Phys. C, (10): 1 (2016)(10): 1 (2016)