Magnetic moments of Δ baryons in light cone QCD sum rules

Nuclear Physics A 678 (2000) 443–454 www.elsevier.nl/locate/npe

Magnetic moments of 1 baryons in light cone QCD sum rules T.M. Aliev, A. Özpineci ∗ , M. Savcı Physics Department, Middle East Technical University, 06531 Ankara, Turkey

Abstract We calculate the magnetic moments of 1 baryons within the framework of QCD sum rules. A comparison of our results on the magnetic moments of the 1 baryons with the predictions of different approaches is presented.  2000 Elsevier Science B.V. All rights reserved. PACS: 13.40.Em; 14.20-c Keywords: Magnetic moments; Delta baryons; Light cone QCD sum

1. Introduction The extraction of the fundamental parameters of hadrons from experimental data requires some information about physics at large distances and they can not be calculated directly from fundamental QCD Lagrangian because at large distance strong coupling constant, αs , becomes large and perturbation theory is invalid. For this reason for determination of hadron parameters, a reliable nonperturbative approach is needed. Among other nonperturbative approaches, QCD sum rules [1] is an especially powerful method in studying the properties of low-lying hadrons. In this method, deep connection between the hadron parameters and the QCD vacuum structure is established via a few condensate parameters. This method is adopted and extended in many works (see, for example, Refs. [2–4] and references therein). One of characteristic parameters of the hadrons is their magnetic moments. Calculation of the nucleon magnetic moments in the framework of QCD sum rules method using external fields technique, first suggested in [5], was carried out in [6,7]. They were later refined and extended to the entire baryon octet in [8,9]. Magnetic moments of the decuplet baryons are calculated in [10,11] within the framework of QCD sum rules using external field. Note that in [10], from the decuplet baryons, only the magnetic moments of 1++ and − were calculated. At present, the ∗ Corresponding author: [email protected]

0375-9474/00/$ – see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 0 ) 0 0 3 2 9 - 8

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magnetic moments of 1++ [12], 10 [13] and − [14] are known from experiments. The experimental information provides new incentives for theoretical scrutiny of these physical quantities. In this letter, we present an independent calculation of the magnetic moments of 1++ , + 1 , 10 , and 1− within the framework of an alternative approach to the traditional sum rules, i.e. the light-cone QCD sum rules (LCQSR). Comparison of the predictions of this approach on magnetic moments with the results of other methods existing in the literature, and the experimental results is also presented. The LCQSR is based on the operator product expansion on the light cone, which is an expansion over the twists of the operators rather than dimensions as in the traditional QCD sum rules. The main contribution comes from the lower twist operator. The matrix elements of the nonlocal operators between the vacuum and hadronic state defines the hadronic wave functions. (More about this method and its applications can be found in [15, 16] and references therein.) Note that magnetic moments of the nucleon using LCQSR approach was studied in [17]. The paper is organized as follows. In Section 2, the light cone QCD sum rules for the magnetic moments of 1++ , 1+ , 10 , and 1− are derived. In Section 3, we present our numerical analysis and conclusion.

2. Sum rules for the magnetic moments of 1 baryons According to the QCD sum rules philosophy, a quantitative estimate for the 1 magnetic moment can be obtained by equating two different representations of the corresponding correlator, written in terms of hadrons and quark-gluons. For this aim, we consider the following correlation function Z (1) 5µν = i dx eipx h 0|T ηµB (x)η¯ νB (0) |0iγ , where T is the time ordering operator, γ means external electromagnetic field. In this expression the ηµB ’s are the interpolating currents for the baryon B. This correlation function can be calculated on one side phenomenologically, in terms of the hadron properties and on the other side by the operator product expansion (OPE) in the deep euclidean region of the correlator momentum p2 → −∞ using QCD degrees of freedom. By equating both expressions, we construct the corresponding sum rules. On the phenomenological side, by inserting a complete set of one hadron states into the correlation function, Eq. (1), one obtains: X h0|ηµB |B1 (p1 )i

 hB2 (p2 )|ηνB |0i
, B1 (p1 ) B2 (p2 ) γ 5µν p12 , p22 = 2 2 p1 − M1 p22 − M22 B ,B 1

(2)

2

where p2 = p1 + q, q is the photon momentum, Bi form a complete set of baryons having the same quantum numbers as B, with masses Mi . The matrix elements of the interpolating currents between the ground state and the state containing a single baryon B with momentum p and having spin s is defined as:

T.M. Aliev et al. / Nuclear Physics A 678 (2000) 443–454



 0|ηµ B(p, s) = λB uµ (p, s),

445

(3)

where λB is a phenomenological constant parameterizing the coupling strength of the baryon to the current, and uµ is the Rarita–Schwinger spin vector satisfying (/ p − MB )uµ = 0, γµ uµ = pµ uµ = 0. (For a discussion of the properties of the Rarita– Schwinger spin vector see, e.g., [18,19].) In order to write down the phenomenological part of the sum rules, one also needs an expression for the matrix element hB(p1 )|B(p2 )iγ . In the general case, the electromagnetic vertex of spin-3/2 baryons can be written as 

(4) B(p1 ) B(p2 ) γ = ερ u¯ µ (p1 )Oµρν (p1 , p2 )uν (p2 ), where ερ is the polarization vector of the photon and the Lorentz tensor Oµρν is given by:   (p1 + p2 )ρ µρν µν f2 + qρ f3 O (p1 , p2 ) = −g γρ (f1 + f2 ) + 2MB   (p1 + p2 )ρ qµ qν g2 + qρ g3 , γρ (g1 + g2 ) + (5) − 2MB (2MB )2 where the form factors fi and gi are (in the general case) functions of q 2 = (p1 − p2 )2 . In our problem, the values of the form-factors only at one point, q 2 = 0, are needed. In our calculation, we also performed summation over spins of the Rarita–Schwinger spin vector:   X (/ p + MB ) 1 2pσ pτ pσ γτ − pτ γσ uσ (p, s)u¯ τ (p, s) = − + gσ τ − γσ γτ − . (6) 2MB 3 3MB 3MB2 s Using Eqs. (3)–(6), the correlation function can be expressed as the sum of various structures, not all of them independent. To remove the dependencies, an ordering of the / 1 ε/p / 2 γν is chosen. gamma matrices should be chosen. For this purpose the structure γµp With this ordering, the correlation function becomes:  1 gM (q 2 ) 2 g p / ε / p / 5µν = λB 2 µν 1 2 3 (p1 − MB2 )(p22 − MB2 )  (7) + other structures with γµ at the beginning and γν at the end where gM (q 2 ) is the magnetic form factor, gM /3 = f1 + f2 . From Eq. (7), it follows that / 1 /εp / 2 contains magnetic transitions only. gM (q 2 ) evaluated at q 2 = 0 the structure gµνp gives the magnetic moment of the baryon in units of its natural magneton, eh¯ /2mB c. The appearance of the factor 3 can be explained as follows. Spin dependent part of the interaction Lagrangian of the spin 32 , which is proportional to the magnetic moment, can be written as (see, for example, [20]): h i gµν + 2gµρ gνλ Ψν Fρλ , (8) Lint = bΨ µ −iσρλ 2 where Ψµ is the Rarita–Schwinger spin-vector field, Fρλ = ∂ρ Aλ −∂λ Aρ is the electromagnetic field strength tensor. To express this interaction Lagrangian via magnetic moment (at q 2 = 0) let us consider the case where A0 = 0, P0 = m, Fρλ = δρi δλj εij k Hk , where Hk is the magnetic field. Then we get

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  i ibΨ m − gmn σij εij k + 2εmnk Ψn Hk = bΨ m [gmn Σk + 2iεmnk ]Ψn Hk 2

(9)

where Σk = diag(σk , σk ). The operator gmn Σk + 2iεmnk is equal to 2Sk , where Sk is the spin operator for the Rarita–Schwinger field. So, the maximal energy of the particle in the magnetic field is equal to Eint = 3bH.

(10)

On the other hand, this interaction energy can be written in terms of the magnetic moment as Eint = µH.

(11)

Equating the two expressions, one sees that b = µ/3. In general the interpolation current might also have a nonzero overlap with spin- 21 baryons, but spin- 21 baryons do not contribute to the structure gµνp / 1 ε/p / 2 since their overlap is given by: h0|ηµ |J = 1/2i = (Apµ + Bγµ )u(p)

(12)

where (/ p − m)u(p) = 0 and (Am + 4B) = 0 [20,21]. Hence it is not possible to form the / 1 /εp / 2 using this matrix element. structure gµνp In order to calculate the correlator (1) from the QCD side, first, appropriate currents should be chosen. For the case of the 1 baryons, they can be chosen as (see, for example, [3,11]):
++ ηµ1 = εabc uaT Cγµ ub uc , 

 1 + ηµ1 = √ εabc 2 uaT Cγµ d b uc + uaT Cγµ ub d c , 3

 1 abc  aT 10 2 d Cγµ ub d c + d aT Cγµ d b uc , ηµ = √ ε 3
1− abc aT (13) u Cγµ ub uc , ηµ = ε where C is the charge conjugation operator, a, b, c are color indices. On the QCD side, for the same correlation functions we obtain: Z

1 abc def ++ 01++ ε = 5 + ε d4 x eipx γ (q) u¯ f Ai ua 51 µν µν 2  × 2Sucd γν Su0be γµ Ai + 2Sucd γν A0i γµ Sube

+

+

01 51 µν = 5µν


be + 2Ai γν Su0cd γµ Sube + Sucd Tr γν Su0 γµ Ai

cd + Sucd Tr γν A0i γµ Sube + Ai Tr γν Su0 γµ Sube |0i, Z

1 abc def − ε ε d4 x eipx γ (q) 6  cf cf be × u¯ d Ai ua 2Ai γν Sd0 γµ Su + 2Ai γν Su0 γµ Sdbe
cf cf + 2Sdbe γν A0i γµ Su + 2Ai Tr γν Su0 γµ Sdbe

(14)

T.M. Aliev et al. / Nuclear Physics A 678 (2000) 443–454

447

cf


+ Sdbe Tr γν A0i γµ Su

+ 2Su γν Sd0 γµ Ai + 2Su γν A0i γµ Sdbe be

cf

cf

+ 2Sdbe γν Su0 γµ Ai + 2Su Tr γν A0i γµ Sdbe
cf + Sdbe Tr γν Su0 γµ Ai  cf cf + d¯ e Ai d b 2Suad γν A0i γµ Su + 2Suad γν Su0 γµ Ai cf

cf




+ 2Ai γν Su0 γµ Su + 2Suad Tr γν Su0 γµ Ai
ad + Ai Tr γν Su0 γµ Suad |0i, ad

cf

cf




(15) √ where Ai = 1, γα , σαβ / 2, iγα γ5 , γ5 , a sum over Ai implied, S 0 ≡ CS T C, A0i = CATi C, with T denoting the transpose of the matrix, and Sq is the full light-quark propagator with both perturbative and nonperturbative contributions: ¯ Sq = h0|T q(x)q(0)|0i =

x2 2 hqqi ¯ i/ x − m hqqi − ¯ 2 4 12 192 0 2π x  Z1  x/ i Gµν (vx)σµν − vxµ Gµν (vx)γν 2 2 . − igs dv 16π 2 x 2 4π x

(16)

0

The 501 µν ’s in Eqs. (14) and (15) describe the emission of the photon from the freely propagating quark. Their explicit expressions can be obtained from the remaining terms by substituting all occurrences of Z ba a b 4 free free d y Fµν yν Sq (x − y)γµ Sq (y) (17) q¯ (x)Ai q Ai αβ → 2 αβ

where the Fock–Schwinger gauge, x µ Aµ (x) = 0, is used, and Sqfree is the free quark propagator, i.e. the first term in Eq. (16). The corresponding expressions for the correlation functions for the 1− and 10 baryons can be obtained from Eqs. (14) and (15), if one exchanges u-quarks by d-quarks and vice versa, respectively. In order to be able to calculate the QCD part of the sum rules, one needs to know the matrix elements hγ (q)|qA ¯ i q|0i. Upto twist-4, the nonzero matrix elements given in terms of the photon wave functions are [22–24]:

f ¯ α γ5 q|0i = eq εαβρσ εβ q ρ x σ γ (q) qγ 4

Z1 du eiuqx ψ(u), 0



¯ ¯ αβ q|0i = ieq hqqi γ (q) qσ

Z1

    du eiuqx (εα qβ − εβ qα ) χφ(u) + x 2 g1 (u) − g2 (u)

0

  + qx(εα xβ − εβ xα ) + εx(xα qβ − xβ qα ) g2 (u)

(18)

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where χ is the magnetic susceptibility of the quark condensate and eq is the quark charge. The functions φ(u) and ψ(u) are the leading twist-2 photon wave functions, while g1 (u) and g2 (u) are the twist-4 functions. Note that, since we have assumed massless quarks, mu = md = 0, and exact SU(2) ¯ the u and d quark propagators are identical, flavor symmetry, which implies huui ¯ = hddi, Su = Sd , whereas for the wave functions, the only difference is due to the different charges of the two quarks. The general expressions given by Eqs. (14) and (15), under exact SU(2) flavor symmetry lead to the following results: ++

+

1 51 µν = − 2 eu C,

1 51 µν = − 6 (2eu + ed )C, −

0

1 51 µν = − 6 (2ed + eu )C,

1 51 µν = − 2 ed C,

(19)

where C is a factor independent of the quark charges. From Eq. (19), the following exact relations between theoretical parts of the correlator functions immediately follows: +

−

++

1 1 1 51 µν = −5µν = 2 5µν ,

0

51 µν = 0.

(20)

Hence, from now on, only the results for 1++ will be reported and for the other 1’s, the corresponding results can be obtained from Eqs. (20). Using Eqs. (16) and (18), from Eq. (14) and after some algebra and after performing Fourier transformation, for / 1 /εp / 2 , we get: the coefficient of the structure gµνp Z1 5 = eu



 
hg 2 G2 i f ψ(u) 2 du 4 ln −P + 48π 2 12P 4

0

+

   χφ(u)huui  ¯ 2 m20 8 2 h uui ¯ (u) − g (u) + + 4 g 1 2 3P 4 6P 2 P2

 hg 2 G2 i 3P 2 ln(−P 2 ) 2huui ¯ 2 − − + , 3P 4 768π 4P 2 64π 4

(21)

where P = p + qu. In Eq. (21), polynomials in P 2 are omitted since they do not contribute after the Borel transformation. As stated earlier, in order to obtain the sum rules, one equates the phenomenological and theoretical expressions obtained within QCD. For this purpose, one first evaluates the spectral densities corresponding to the correlators both on the phenomonological side, and on the theoretical side. In terms of the spectral densities, the correlators can be written as: 5

p12 , p22




Z∞ =

Z∞ ds1

0

ds2 0

ρ(s1 , s2 ) + subtraction terms, (s1 − p12 )(s2 − p22 )

(22)

where the subtraction terms serve to eliminate infinities from the integral and they are usually polynomials in p12 and p22 which vanish under the double Borel transformations. The spectral density can be obtained by applying successive Borel transformations as [25]:

T.M. Aliev et al. / Nuclear Physics A 678 (2000) 443–454

 ρ

1

, 2

1



449

   


1 1 = B τ22 , 2 B τ12 , 2 B M22 , −p22 B M12 , −p12 5 p12 , p22 . (23) M2 M1

τ1 τ22

The spectral density on the phenomonological side can be written as phen

ρ phen (s1 , s2 ) = ρ1

(s1 , s2 ) + ρ cont (s1 , s2 ),

(24)

where ρ cont is the contribution of higher states and the continuum. At this point, to model ρ cont , one imposes the quark-hadron duality and approximates the unknown contribution of the continuum by the OPE spectral density above continuum thresholds s0 and s00 . Then
phen (25) ρ phen = ρ1 (s1 , s2 ) + θ (s1 − s0 )θ s2 − s00 ρ OPE (s1 , s2 ), where θ (x) is the step function. Then the sum rules, after double Borel transformation is obtained from: Z∞

Z∞ ds1

0

0

   s1 s2  ds2 exp − 2 − 2 ρ phen (s1 , s2 ) − ρ OPE (s1 , s2 ) = 0. M1 M2

(26)

After carrying out the calculation for the correlator (the details can be found in the appendix), the following sum rules is obtained for the magnetic moment of 1++ :    2    3eu M1 f ψ(u0 ) hg 2 G2 i s0 8 4 − M ¯ 2 g1 (u0 ) − g2 (u0 ) f + huui gM = 2 e M 2 1 2 2 48 3 12π M λ1    ¯ 2 2 s0 χφ(u0 )huui m0 − 4M 2 f0 + 6 M2     s0 3M 6 s0 2huui ¯ 2 hg 2 G2 iM 2 + f f + , (27) + 0 2 3 768π 4 M2 64π 4 M2 where fn (x) = 1 − e−x

n X xk k=0

u0 =

M12 M12

+ M22

,

k!

,

(28) 1 1 1 = + . M 2 M12 M22

As we are dealing with just a single baryon, the Borel parameters M12 and M22 should be taken to be equal, i.e. M12 = M22 , from which it follows that u0 = 1/2.

3. Numerical analysis From Eq. (27), one sees that, in order to calculate the numerical value of the magnetic moment of the 1++ , besides several numerical constants, one requires expressions for the photon wave functions. It was shown in [22,23] that the leading photon wave functions receive only small corrections from the higher conformal spin, so they do not deviate

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T.M. Aliev et al. / Nuclear Physics A 678 (2000) 443–454

much from the asymptotic form. Following [23,24], we shall use the following photon wave functions: φ(u) = 6uu, ¯

ψ(u) = 1,

¯ − u), g1 (u) = − 18 u(3

g2 (u) = − 14 u¯ 2 ,

where u¯ = 1 − u. The values of the other constants that are used in the calculation are: f = 0.028 GeV2 , χ = −4.4 GeV−2 [26] (in [27], χ is estimated to be χ = −3.3 GeV−2 ), ¯ = −(0.243)3 GeV3 , m20 = (0.8 ± 0.2) GeV2 [28], λ1 = hg 2 G2 i = 0.474 GeV4 , huui 3 0.038 GeV [29]. Having fixed the input parameters, our next task is to find a region of Borel parameter, M 2 , where dependence of the magnetic moments on M 2 and the continuum threshold s0 is rather weak and at the same time the higher-dimension operators, higher states and continuum contributions remain under control. We demand that these

Fig. 1. The dependence of the magnetic moment of 1++ on the Borel parameter M 2 (in nuclear magneton units) for three different values of the continuum threshold s0 . Table 1 Comparisons of 1-baryon magnetic moments from various calculations: this work (LCQSR), QCDSR [11], lattice QCD (Latt) [21], chiral perturbation theory (χ PT) [30], light-cone relativistic quark model (RQM) [31], nonrelativistic quark model (NQM) [32], chiral quark–soliton model (χ QSM) [33], chiral bag-model (χ B) [34]. All results are in units of nuclear magnetons Baryon

1++

1+

10

1−

Exp. LCQSR QCDSR Latt. χPT RQM NQM χQSM χB

4.5 ± 1.0 4.4 ± 0.8 4.13 ± 1.30 4.91 ± 0.61 4.0 ± 0.4 4.76 5.56 4.73 3.59

2.2 ± 0.4 2.07 ± 0.65 2.46 ± 0.31 2.1 ± 0.2 2.38 2.73 2.19 0.75

∼0 0.00 0.00 0.00 −0.17 ± 0.04 0.00 −0.09 −0.35 −2.09

−2.2 ± 0.4 −2.07 ± 0.65 −2.46 ± 0.31 −2.25 ± 0.25 −2.38 −2.92 −2.90 −1.93

T.M. Aliev et al. / Nuclear Physics A 678 (2000) 443–454

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contributions are less then 35%. Under this requirement, the working region for the Borel parameter M 2 is found to be 1.1 6 M 2 6 1.4 GeV2 . Finally, in this range of the Borel parameter, the magnetic moment of 1++ is found to be (4.55 ± 0.03)µN . This prediction on the magnetic moment is obtained at s0 = 4.0 GeV2 . Choosing s0 = 3.8 GeV2 or s0 = 4.2 GeV2 changes the result at most by 6% (see Fig. 1). The calculated magnetic moment is in good agreement with the experimental result (4.52 ± 0.50 ± 0.45)µN [14]. Our results on the magnetic moments for 1+ , 10 and 1− are presented in Table 1. For completeness, in this table, we have also presented the predictions of other approaches. Comparing the values presented in Table 1, it is seen that our predictions on magnetic moments is larger than the QCDSR predictions. Finally, for the calculation of the magnetic moments of other members of the decuplet (which contain at least one s-quark), the SU (3) breaking effects due to the strange quark should be taken into account. Their calculations would be presented in a future work.

Appendix In this appendix, we will present details about how to carry out the double Borel transformations with respect to the variables −p12 and −p22 and to subtract the higher states and the continuum contribution. For this purpose, consider a generic term from the correlator I

−p12 , −p22




Z1 =

du 0

h(u) (−p12 u¯ − p22 u)n

,

(29)

where h(u) is an arbitrary function of u (it can be a constant or one of the photon wave functions φ(u) or ψ(u)). First, the spectral density corresponding to Eq. (29) has to be calculated using Eq. (23). In order to take the double Borel transform of Eq. (29), let us write the denominator in exponential representation: I

−p12 , −p22




Z1 = 0

1 du h(u) 0(n)

Z∞

¯ p˜ 2 u) dt t n−1 e−t (p˜ 1 u+ , 2

2

(30)

0

where quantities with a tilde denote euclidean four momentum. Using the identity  
1 2 − α , B M 2 , p˜ 2 e−p˜ α = δ M2 the double Borel transform of Eq. (30) can be obtained as:



I M12 , M22 = B M12 , −p12 B M22 , −p22 I −p12 , −p22     M12 1 1 n−2 1 + h . = 0(n) M12 M22 M12 + M22

(31)

(32)

Introducing new variables σi = 1/Mi2 and expressing the function f (u) as a power series in u,

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T.M. Aliev et al. / Nuclear Physics A 678 (2000) 443–454

h(u) =

∞ X

a k uk ,

(33)

k=0

Eq. (32) can be written in the form ∞

I (σ1 , σ2 ) =

σ2k 1 X ak . 0(n) (σ1 + σ2 )k+2−n

(34)

k=0

As can be seen from Eq. (23), in order to obtain the spectral density, one needs to apply double Borel transformations once more, over the variables σi on Eq. (34). For this purpose, again write the denominator in Eq. (34) in exponential representation. Then ∞

1 X ak σk I (σ1 , σ2 ) = 0(n) 0(k + 2 − n) 2 k=0

Z∞

dt t k+1−n e−t (σ1 +σ2 β) ,

(35)

0

which should be evaluated at β = 1. The factor σ2k under the summation sign in Eq. (35) can be expressed as derivatives with respect to the parameter β: ∞

∂k 1 X (−1)k ak I (σ1 , σ2 ) = 0(n) 0(k + 2 − n) ∂β k k=0

Z∞

dt t 1−n e−t (σ1 +σ2 β) .

(36)

0

Finally, the spectral density is obtained by performing double Borel transformations on Eq. (36) on the variables σ1 and σ2 using Eq. (31): ∞

ρ(s1 , s2 ) =

1 X ak s k+1−n δ (k) (s1 − s2 ), 0(n) 0(k + 2 − n) 1

(37)

k=0

where si = 1/τi2 . From Eqs. (25), (26) and (37), one sees that in order to obtain the sum rules, one needs to calculate the integral Zs0

Zs0 ds1

0

ds2 e−s1 /M1 −s2 /M2 ρ(s1 , s2 ), 2

2

(38)

0

where we have used the fact that ρ(s1 , s2 ) = 0 if s1 6= s2 , and that the contribution of higher states and the continuum is equal to OPE spectral density above some thresholds s0 and s00 . Introducing new variables s, v through s1 = vs and s = s1 + s2 and integrating over v, one obtains: Zs0

Zs0 ds1

0

ds2 e−s1 /M1 −s2 /M2 ρ(s1 , s2 ) 2

2

0

    Z2s0 ∂k s(1 − v) k+1−n vs ak ds k exp − 2 − v = 1 ∂v M1 M22 v= 2 k=0 ∞ X

0

T.M. Aliev et al. / Nuclear Physics A 678 (2000) 443–454

×

s 1−n 1 . k+1 0(n)0(k + 2 − n) 2

453

(39)

When M12 = M22 = 2M 2 , which is the case in this study, the v dependence of the exponential in Eq. (39) drops and using 0(k + 2 − n) 1 ∂ k k+1−n v (40) 1 = 0(2 − n) 21−n , ∂v k v= 2

one immediately obtains: Zs0

Zs0 ds2 ρ(s1 , s2 ) e

ds1 0

−(s1 +s2 )/2M 2

0

For terms containing

  Zs0 1 1 2 =h ds e−s/M s 1−n . (41) 2 0(n)0(2 − n) 0

log(−p12 u¯ − p22 u),

d 1 log(x) = − dn x n n=0

one can use the identity (42)

and the Eq. (41) to take the Borel transformation and to subtract the contribution of the continuum. Using these equations we obtain the following results which we have used in the main text: Z1 0

    1 h(u) s0 2 →h du M f0 , 2 M2 (−p12 u¯ − p22 u)

Z1 du h(u) log

−p12 u¯ − p22 u




    1 s0 4 → −h M f1 , 2 M2

(43)

(44)

0

Z1

   

1 s0 M 6 f2 du h(u) −p12 u¯ − p22 u log −p12 u¯ − p22 u → h , 2 M2

0

where the functions fi (x) are defined in Eq. (28). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

M.A. Shifman, A.I. Vainstein, V.I. Zakharov, Nucl. Phys. B 147 (1979) 385. L.I. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. C 127 (1985) 1. B.L. Ioffe, Nucl. Phys. B 188 (1981) 317; Nucl. Phys. B 191 (1981) (E)591. V. Chung, H.G. Dosch, M. Kremer, D. Scholl, Nucl. Phys. B 197 (1982) 55. B.L. Ioffe, A.V. Smilga, JETP Lett. 37 (1983) 250. B.L. Ioffe, A.V. Smilga, Nucl. Phys. B 232 (1984) 109. I.I. Balitsky, A.V. Yung, Phys. Lett. 1290 (1983) 328. C.B. Chiu, J. Pasupathy, S.L. Wilson, C.B. Chiu, Phys. Rev. D 33 (1986) 1961. J. Pasupathy, J.P. Singh, S.L. Wilson, C.B. Chiu, Phys. Rev. D 36 (1987) 1457. V.M. Belyaev, Preprint ITEP-118, 1984, unpublished.

(45)

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