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Two-loop matrix element of the current–current operator in the decay B→Xsγ
Nuclear Physics B 611 (2001) 488–502 www.elsevier.com/locate/npe
Two-loop matrix element of the current–current operator in the decay B → Xs γ Andrzej J. Buras a , Andrzej Czarnecki b , Mikołaj Misiak c , Jörg Urban a a Physik Department, Technische Universität München, D-85748 Garching, Germany b Physics Department, University of Alberta, Edmonton, Alberta, T6G 2J1 Canada c Institute of Theoretical Physics, Warsaw University, Ho˙za 69, 00-681 Warsaw, Poland
Received 23 May 2001; accepted 3 July 2001
Abstract We evaluate important two-loop matrix element sγ |Q2 |b of the operator (cγ ¯ µ PL b)(¯s γµ PL b) contributing to the inclusive radiative decay B → Xs γ . The calculation is performed in the NDR scheme, by means of asymptotic expansions method. The result is given as a series in z ≡ m2c /m2b up to O(z6 ). We confirm the result of Greub, Hurth and Wyler obtained by a different method up to O(z3 ). Higher-order terms are found to be numerically insignificant. 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The radiative decay B → Xs γ plays an important role in the present tests of the Standard Model (SM) and of its extensions [1]. In particular, in the supersymmetric extensions of the SM, the best bounds on several new parameters come from the data on B → Xs γ [2]. The short distance QCD effects are very important for this decay. They are known to enhance the branching ratio Br(B → Xs γ ) by roughly a factor of three, as first pointed out in [3,4]. Since these first analyses, a lot of progress has been made in calculating these important QCD effects in the renormalization group improved perturbation theory, beginning with the work in [5,6]. Let us briefly summarize this progress. A peculiar feature of the renormalization group analysis in B → Xs γ is that the mixing under infinite renormalization between the four-fermion operators (Q1 , . . . , Q6 ) and the magnetic penguin operators (Q7γ , Q8G ), which govern this decay, vanishes at the oneloop level. Consequently, in order to calculate the coefficients C7γ (µb ) and C8G (µb ) at µb = O(mb ) in the leading logarithmic approximation, two-loop calculations of O(egs2 ) E-mail address: [email protected] (J. Urban). 0550-3213/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 ( 0 1 ) 0 0 3 3 6 - 4
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and O(gs3 ) are necessary. Such calculations were completed in [7,8] and confirmed in [9–11]. Earlier analyses contained either additional approximations or mistakes. It turns out that the leading order expression for the branching ratio Br(B → Xs γ ) suffers from sizable renormalization scale uncertainties [12,13] implying that a complete NLO analysis including also dominant higher order electroweak effects to this decay is mandatory. By 1998, the main ingredients of such an analysis had been calculated. It was a joint effort of many groups: • The O(αs ) corrections to C7γ (µW ) and C8G (µW ) were first calculated in [14] and subsequently confirmed by several groups [15–18]. • The two-loop mixing involving the four fermion operators Q1 , . . . , Q6 and the two-loop mixing in the sector (Q7γ , Q8G ) was calculated in [19–24] and [25], respectively. The very difficult three-loop mixing between the set (Q1 , . . . , Q6 ) and the operators (Q7γ , Q8G ) was calculated in [11]. • Leading order matrix elements sγ gluon|Qi |b were calculated in [26,27], and the challenging two-loop calculation of sγ |Q2 |b was presented in [28]. • Higher order electroweak corrections were incorporated with increasing level of sophistication in [29–33]. In addition, non-perturbative corrections were calculated in [34–39]. The most recent analysis of B → Xs γ incorporating all these calculations can be found in [40]. Now, among the perturbative ingredients listed above, three have been calculated only by one group. These are: • Two-loop mixing in the sector (Q7γ , Q8G ) [25]; • Three-loop mixing between the set (Q1 , . . . , Q6 ) and the operators (Q7γ , Q8G ) [11]; • Two-loop matrix element sγ |Q2 |b [28]. Moreover, • Two-loop matrix elements of the QCD penguin operators sγ |Qi |b with i = 3, . . . , 6 have not yet been calculated. It should be emphasized that all these four ingredients enter not only the analysis of B → Xs γ in the SM but are also necessary ingredients of any analysis of this decay in the extensions of this model. It is therefore desirable to check the first three calculations and to perform the last one. In the present paper we will make the first step in this direction by calculating the twoloop matrix element sγ |Q2 |b using the method of asymptotic expansions. This matrix element turned out [28,40] to be the most important ingredient of the NLO-analysis for Br(B → Xs γ ) enhancing this branching ratio by roughly 20%. In [28] the matrix element sγ |Q2 |b was found by applying the Mellin–Barnes representation to certain Feynmanparameter integrals, and the result was presented as an expansion in z = m2c /m2b up to and including terms O(z3 ). In order to be sure about the convergence of this expansion, we will include also the terms O(z4 ), O(z5 ) and O(z6 ). Asymptotic expansions [41–43] provide a systematic approach to computing Feynman integrals which depend on two (or more) mass or momentum scales. The general idea is to
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expand the integrand in the small parameters. The coefficients of the resulting expansion depend on a single mass scale and are simpler to evaluate than the original integral, since they are given by numbers rather than potentially complicated functions of dimensionless quantities — ratios of the various scales. Of course, it is well known that if an integral depends on a parameter, an expansion does not, in general, commute with the integration. The “error” one makes can be systematically corrected. Below, these corrections will be reflected in consideration of various momentum regions in each integral. A nice formulation of the expansion procedure can be given in the language of counterterms (and theory of distributions); see [44] for a review and further references.
2. Two-loop contribution to sγ |Q2 |b 2.1. Preface The effective Hamiltonian for the process b → sγ is given by 8 4GF Heff (b → sγ ) = − √ Vt b Vt∗s Ci (µ)Qi . 2 i=1
(2.1)
Here GF is the Fermi constant, Vij are the CKM matrix elements and Ci are the Wilson coefficients of the operators Qi evaluated at µ = O(mb ). We have dropped the negligible ∗ . The full list of operators in (2.1) can be found in [28]. contributions proportional to Vub Vus In the present paper we need only the expressions for two of them. These are µ Q2 = cγ ¯ PL b s¯ γµ PL b , e Q7 = s¯α σ µν mb (µ)PR + ms (µ)PL bα Fµν , 2 16π
(2.2) (2.3)
where PL = (1 − γ5 )/2, PR = (1 + γ5 )/2 and σ µν = 2i [γµ , γν ]. As in [28], we will set ms = 0. In this section we present the details of the calculation of the matrix element sγ |Q2 |b in the NDR scheme. This matrix element vanishes at the one-loop level. Therefore, in order to obtain a non-vanishing contribution, one has to calculate two-loop diagrams. They are shown in Figs. 1, 4, 5, and 6, where the wavy and dashed lines represent the photon and gluon, respectively. Following [28], we have divided the contributing diagrams into four sets: • In sets 1 and 2, the photon is emitted from an internal s-quark (set 1) or an internal b-quark (set 2), which also exchanges a gluon with the charm quark. • Sets 3 and 4 are obtained from sets 1 and 2, respectively, by emitting this time the photon from the charm quark propagator. • Thanks to QED gauge invariance, it is not necessary to consider diagrams with a real photon emission from external quark lines.
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Fig. 1. Diagram M1 . Momentum assignments: Q, q, and γ denote respectively four-momenta of the b and s quarks, and of the photon; p and k are loop momenta. Double lines denote a massive particle (the b quark) and single lines are light particle propagators. The gluon is denoted by a dashed line.
It is convenient to write the regularized contributions from each set of the diagrams as follows: 1 mb −4' αs CF Qd sγ |Q7 |btree , M1 = (2.4) + M1 36' κµ π 5 mb −4' αs M2 = − (2.5) + M2 CF Qd sγ |Q7 |btree , 36' κµ π 1 mb −4' αs M3 = − (2.6) CF Qu sγ |Q7 |btree , + M3 8' κµ π 1 mb −4' αs CF Qu sγ |Q7 |btree . M4 = − (2.7) + M4 4' κµ π We work in D = 4 − 2' dimensions, with κ 2 = 4πe−γE , Qu = 23 , Qd = − 13 , CF = 43 and γE is Euler’s constant. The tree level matrix element of the operator Q7 is given by e sγ |Q7 |btree = mb (µb ) 2 u(p )/'q/ PR u(p). (2.8) 8π In order to make the two-loop matrix element finite, counterterms have to be added. These counterterms can easily be calculated by using the operator renormalization constants that were needed in the context of the calculation of the leading order anomalous dimension matrix. The complete counterterm is found to be [28] Qd mb −2' 3Qu Qd κ 4' αs count − CF sγ |Q7 |btree . = + M2 (2.9) 6 µ 8 18 ' π The above expression includes two-loop counterterms as well as contributions from oneloop diagrams with one-loop counterterm insertions. Adding the contributions from (2.4)–(2.7) and (2.9), we find the two-loop matrix element sγ |Q2 |b in the NDR-scheme: αs 416 mb ln + r2 , sγ |Q2 |b = sγ |Q7 |btree (2.10) 4π 81 µ with r2 = −
16 3 + M 2 + 32 M 4 . M1 + M 9 9
(2.11)
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Fig. 2. One-loop diagram (the k propagator can have fractional power) corresponding to Eq. (2.12).
The rest of this section is devoted to a detailed presentation of the calculation of the i using the method of asymptotic expansions. All the momentum integrals contributions M are performed in the Euclidean space. 2.2. Diagram M1 Of the four sets of diagrams which we have to consider, M1 and M2 shown in Figs. 1 and 4 are relatively simple. In M1 , the photon is emitted from a massless s-quark propagator. There are four momentum regions we have to consider. Denoting the momenta in the cquark and vertex loops by p and k, we have 1. k ∼ p ∼ mb : the “hard–hard” region; 2. k ∼ mb and p ∼ mc : the “hard-soft” region (which comes in two variants, depending on the routing of k through the c-quark loop); 3. a “collinear-soft” region, in which p ∼ mc and k · γ ∼ m2b but k · q ∼ m2c . Below, we describe the treatment of those regions in some detail. 2.2.1. Hard contribution: Taylor expansion in mc Since all scalar products are now large compared with m2c , we can expand the c-quark propagators in mc . This leads to a massless loop integration, which simply modifies the power of the momentum (k) in the gluon propagator (see Fig. 2). After this first integration, the integrand has the following form: dD k (scalar products involving k) . (2.12) (k 2 )a1 +' (k 2 + 2k · Q + Q2 )a2 (k 2 + 2k · q)a3 We combine propagators 1 and 3 using the Feynman parameter x, and then combine the result (·(1 − y)) with propagator 2 (·y). We shift the variable k → K − x(1 − y)q − yQ, and the denominator becomes a power of k 2 + y(1 − y)(x − 1 − i0). Since x − 1 < 0, this diagram has an imaginary part; its sign is determined by assigning a small negative imaginary mass to the massless lines. After the momentum shift, we have additional factors x m and y n in the numerator, and denominator has the power d + '. The result is
−2'
Γ (d − 2 + 2') B(m + a3 , a1 + 2 − d − ') Γ (d + ') B(a3 , a1 + ')B(a2 , a1 + a3 + ') × B(n + a2 + 2 − d − 2', a1 + a3 + 2 − d − '),
exp(−iπ)
(−1)d
where B and Γ are the standard Euler functions Beta and Gamma.
(2.13)
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Fig. 3. Subgraph corresponding to one of the two hard–soft contributions.
2.2.2. Hard–soft contributions from k ∼ mb , p ∼ mc This region can be handled as in the “large momentum expansion” [41–43]. In Fig. 3 we see one of the two configurations to be computed. We first integrate over p, and then evaluate the one-loop vertex. An analogous procedure is executed for the other (left) cquark line. 2.2.3. Collinear region Naively, one might think that we should consider the region k ∼ p ∼ mc , in which (Q + k)2 Q2 , (q + k)2 2k · q. However, this does not lead to a consistent power counting, since we have to shift the momentum k by an amount proportional to q, which is ∼ mb . Instead, we have to consider the collinear configurations [45]. We define k · γ = Q0 k+ ,
k · q = Q0 k− , Q2 = Q20 ,
2 k 2 = k+ k− − k⊥ ,
k · Q = Q0 (k+ + k− ).
(2.14)
The relevant contribution is k+ ∼ Q0 , k− ∼ m2c /Q0 . Now (Q + k)2 Q2 + 2k · γ , while (k + q)2 = k 2 + 2k · q is already homogeneously of order m2c and cannot be expanded. First, we integrate over p; for this we combine the two c propagators (we assume here, for generality, that their powers are a1 and a2 ) using x, shift p → P − xk, and obtain (m and d are exponents arising from the presence of p in the numerator) polynomial(p) Γ (d − 2 + ') x m+a1 +1−d−' (1 − x)a2 +1−d−' → , 2 2 a 2 2 a [(p + k) + mc ] 1 [p + mc ] 2 B(a1 , a2 )Γ (d) (k 2 + m2x )d−2+' m2x ≡
m2c . x(1 − x)
For the k integration, the integrand has the form dD k polynomial(k) . 2 a 2 2 d−2+' (k ) 3 (k + mx ) (k 2 + 2k · q)a4 (Q2 + 2γ · k)a5
(2.15)
(2.16)
Now, we combine the first two propagators multiplying them by 1 − z and z, respectively. We multiply the result by y and combine it with the “a4 ” propagator multiplied by 1 − y. Finally, we use the parameter u to include the last propagator (i = a3 + a4 + d − 2 + '): [k 2
+ yzm2x
1 + 2(1 − y)k · q]i (Q2 + 2k · γ )a5
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1 = B(i, a5 )
∞ du 0
ua5 −1 . [k 2 + yzm2x + 2(1 − y)k · q + 2uk · γ + uQ2 ]i+a5
(2.17)
Now, we shift k → K − (1 − y)q − uγ and, using q · γ = Q2 /2, find that the denominator simplifies to become [K 2 +y(zm2x +uQ2 )]. Because of the presence of K in the numerator, we generate additional powers of Feynman parameters. After integrating over K, we integrate over u, ∞ du 0
un−1 B(n, m − n) = . (zm2 + uQ2 )m (Q2 )n (zm2 )m−n
(2.18)
We see that the dependence on Q separates (so the integral really depends only on a single scale). Also, only mc appears in fractional power, so that there is no imaginary part in this integral, even though Q2 < 0 (after the Wick rotation). 2.3. M2 In this diagram (see Fig. 4) we have again four momentum regions, of which the first three are similar to those discussed for M1 . The fourth one is different: it is a “soft–soft” contribution, with k ∼ p ∼ mc . For the hard–hard contribution, we first integrate over p, just like in M1 . After that, the integrand takes on the following form: dD k (scalar products involving k) (2.19) . 2 a (k ) 1 +' (k 2 + 2k · Q)a2 (k 2 + 2k · q − Q2 )a3 We combine propagators 2 and 3 using a Feynman parameter x, and then combine the result (·y) with the propagator 1 (·(1 − y)). We shift the variable k → K − y((1 − x)Q + xq), and the denominator becomes a power of k 2 + y[y + x(1 − y)]. After shifting the momentum and simplifying the numerator, additional factors x m and y n appear in the numerator, and the denominator has the power d + 2'. After the k integration, we find that the x and y integrations do not separate, and we first integrate over x using Eq. (2.30), to be discussed below. The remaining y integration is a Beta function. The hard–soft contributions are similar to those in M1 and lead to rather trivial products of one-loop integrals.
Fig. 4. Diagram M2 .
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On the other hand, the soft–soft contribution is less standard. We have k ∼ mc and the two b-quark propagators can be expanded. In the lowest order we have 1 1 → , k 2 + 2k · Q 2k · Q
1 1 →− 2. k 2 + 2k · q − Q2 Q
(2.20)
Since the second propagator becomes independent of k, the whole two-loop diagram becomes equivalent to a simpler, two-point function. The resulting integrals are similar, but not the same, as those considered in the eikonal expansion study [46]. Details of their evaluation have been recently described in a completely different context of bound-state calculations [47]. In that work, exactly such type of integrals appeared when energies of bound-states consisting of two particles with very different masses were expressed as expansions in the ratio of those masses. 2.4. M3 This diagram, shown in Fig. 5, is the first of the two non-trivial two-loop vertex diagrams we have to consider. There are now five momentum regions to be considered: 1. Hard–hard, k ∼ p ∼ mb , similar to those of M1,2 . 2. Hard–soft, k ∼ mb , p ∼ mc , which now enters only once, when hard momentum flows through the same c-quark line from which the photon is emitted. 3. Collinear–collinear, with k and p having their only large components (O(mb )) aligned with the s-quark momentum q (and with k 2 ∼ p2 ∼ m2c ). 4. Collinear–collinear, but with the alignment with the photon momentum γ . 5. Hard–collinear, where k ∼ mb and p is aligned parallel with γ . The first two regions are treated in an analogous manner as was described for M1,2 . In the hard–hard contribution, after expansion in mc , the integrand is symmetrical (apart from different powers of propagators) under replacement p ↔ k, γ ↔ q. Hence, we can first integrate over k, and then over p. Both integrations are very similar to those in M1,2 . In the second region, we use the “large momentum expansion” mentioned in Section 2.2.2. The collinear–collinear regions require more attention [48]. After expansion of the integrands in the available small quantities, it turns out that the integrals over Feynman parameters have singularities (like 0 dx/x) which are not regularized by our dimensional regulator. It is necessary to introduce additional, analytical regularization on the heavy quark line. Similar phenomena have been observed before (see, e.g., [48] and references
Fig. 5. Diagrams contributing to M3 , with photon emission from two c-quark lines.
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therein). Nevertheless, all integrals over the Feynman parameters can be evaluated analytically without particular difficulties. Dependence on the analytical regulator cancels in the sum of the two doubly-collinear contributions. 2.5. M4 The most complicated diagram is M4 , depicted in Fig. 6. There are six momentum regions: 1. Hard–hard. 2. Hard–soft. 3. Collinear–collinear with alignment along γ . There is no contribution with alignment along q here, because of the different structure of the internal quark propagator, which now contains the large b-quark mass. 4. Hard–collinear, as in M3 . 5. Ultrasoft–collinear [48], with k ∼ m2c /mb and p aligned with γ . 6. Soft–soft. We give a more detailed description of the hard–hard contribution, because the integrals resulting here are rather complicated. First, we consider the scalar integral with all propagators in the first power, J3 =
dD k k 2 (k 2 + 2Q · k)
dD p . p2 (p − k)2 (p + γ )2
(2.21)
We first evaluate the massless (c-quark) loop (integrals over all Feynman parameters run from 0 to 1): dD p dD p = p2 (p − k)2 (p + γ )2 p2 (p2 − 2p · k + k 2 )(p2 + 2p · γ ) dD p = 2(1 − y) dy dx 2 [p − 2yp · k + 2x(1 − y)p · γ + yk 2 ]3 −1−' (1 − y)−' dy dx y = Γ (1 + ')π 2−' (k 2 + 2xk · γ )1+' dx 2−' = Γ (1 + ')B(−', 1 − ')π . (2.22) 2 (k + 2xk · γ )1+'
Fig. 6. Diagram M4 .
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In the process of the integration, we made a shift p → p + yk − x(1 − y)γ . Next, we integrate over k: dD k dx k 2 (k 2 + 2Q · k)(k 2 + 2xk · γ )1+' dD k = (1 + ') dx du (1 − u)' k 2 [k 2 + 2uk · Q + 2(1 − u)xk · γ ]2+' dD k = (2 + ')(1 + ') dz z1+' dx du (1 − u)' 2 2 2 [k + z u + z2 u(1 − u)x]3+' Γ (1 + 2') 2−' z−1−3' u−1−2' (1 − u)' = dz dx du π Γ (1 + ') [u + (1 − u)x]1+2' Γ (1 + 2') 2−' = 2 π du u−1−2' (1 − u)−1+' 1 − u−2' 6' Γ (1 + ') Γ (1 + 2') 2−'
= 2 (2.23) π B(−2', ') − B(−4', ') . 6' Γ (1 + ') We made a shift k → k − zuQ − z(1 − u)xγ and used Q2 = 2Q · γ = −1. Multiplying the result (2.23) with the coefficient Γ (1 + ')B(−', 1 − ') from (2.22) we find
Γ (1 + 2') B(−', 1 − ') B(−2', ') − B(−4', ') 2 6' π2 7 29π 4 1 2 = Γ (1 + ') + + ζ3 + + O('). 24' 4 18' 2 6' 360
J3 =
(2.24)
Now we consider the general case, in which momenta p and k can be present in the numerator, dD k dD p (scalar products involving k and p) (2.25) a a a a . (k 2 ) 1 (k 2 + 2k · Q) 2 (p − k)2a3 (p2 ) 4 (p2 + 2p · γ ) 5 We first combine the propagators 5 (·x) and 4 (·(1 − x)), and then the result (·(1 − y)) with 3 (·y). We shift the integration momentum p = K − x(1 − y)γ + yk and average over K. This again results in extra powers y n and x m , while canceling powers of K 2 changes the power of the denominator. After the integration over K, the denominator becomes [y(1 − y)(k 2 + 2xk · γ )]d−2+' . We see that the dependence on y factorizes and we can integrate over this variable. As a result, we find an expression of the form Γ (a3 + a4 + a5 )Γ (d − 2 + ')B(n + a3 − d + 2 − ', a4 + a5 − d + 2 − ') Γ (a3 )Γ (a4 )Γ (a5 )Γ (d) a5 +m−1 × dx x (1 − x)a4 −1 dD k (scalar products involving k) × . a a d−2+' (k 2 ) 1 (k 2 + 2k · Q) 2 (k 2 + 2xk · γ )
(2.26) (2.27) (2.28)
We now repeat a similar procedure with the variable p. We combine the propagator 2 (·u) with (k 2 + 2xk · γ ) (·(1 − u)), and then the result (·z) with the propagator 1 (·(1 − z)). We change the momentum variable k = K − zuQ − zx(1 − u)γ , average over K and simplify
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powers of K 2 (as a result, the power of the denominator changes from a1 + a2 + d − 2 to some d1 , and we also get extra powers ze and uw ). After integrating over K we get (including the factors (2.26), (2.27)) Γ (a3 + a4 + a5 )Γ (d − 2 + ')B(n + a3 − d + 2 − ', a4 + a5 − d + 2 − ') Γ (a3 )Γ (a4 )Γ (a5 )Γ (d) Γ (d1 − 2 + 2')Γ (a1 + a2 + d − 2 + ')B(a1 , a2 + d + 2 + e − 2d1 − 3') × Γ (d1 + ')Γ (a1 )Γ (a2 )Γ (d − 2 + ') × dx du x a5 +m−1 (1 − x)a4 −1 uw+a2 +1−d1 −2' (1 − u)d−3+'
2−d1 −2' × u + x(1 − u) .
(2.29)
First we integrate over x, using 1 0
j −1−n−µ n
µ uµ+n+1 −1 j n u . dx x n u + x(1 − u) = (−1) n+1 j µ+n+1−j (1 − u)
(2.30)
j =0
This integration is possible because the powers of x and (1 − x) in (2.27) are non-negative integer. The u integration gives a simple Beta function. The techniques described above are sufficient to compute all the remaining contributions. Again, the dimensional regularization alone is insufficient to evaluate the doublycollinear contribution. We introduce an analytical regulator on the heavy quark mass. The resulting singularities vanish when we add the soft–soft contribution. 2.6. Results i are listed below, with z = m2c /m2 and L = ln z. The final results for M b 1 = 37 + − 5 − L z + − 5 + π 2 + L − L2 z2 M 216 2 2 2 17 2 2 10 + − − π − L + L2 z3 27 3 9 3 2 7 11 1 67 + L z6 + − + L z4 + − + L z5 + 12 30 3 270 9
4 1 10 2 7 − z + (1 − 2L)z2 + − + L z3 + z4 + z5 + z6 , (2.31) + iπ 18 9 3 3 9 13 1 1 1 2 2 = M + − + π 2 z − π 2 z3/2 + 3 − 3L + L2 z2 216 2 6 3 2 4 2 3 157 5 2 1 4679 2 2 107 + π + L+ L z + − + π + L + 2L2 z4 + − 108 9 18 3 900 3 30 26185 5 2 277 + π + L + 5L2 z5 + − 2352 3 28 2831737 14 2 16177 + π + L + 14L2 z6 , + − (2.32) 97200 3 540
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3 = − 15 + 3 − 1 π 2 L − 1 π 2 − 2ζ3 + 2L + 1 L2 + 1 L3 z M 16 2 2 4 4 6 1 1 1 1 5 1 2 − π L + π 2 − 2ζ3 + L − L2 + L3 z2 + 4 2 2 2 2 6 17 7 5 3 + − +L z + − + L z4 12 216 9 231 1183 49 8783 5 + L z + + L z6 + 4320 72 12000 200
1 1 2 1 2 1 1 2 2 1 2 1 L + L + 2 − π z + −L + L + − π z + iπ − + 4 2 2 6 2 2 6 5 49 231 z6 , + z3 + z4 + z5 + (2.33) 9 72 200 4 = − 7 + 1 − 1 π 2 L − 1 π 2 − ζ3 + 1 L − 1 L3 z M 8 4 12 2 12 1 2 1 1 3 2 1 1 2 + π + 2ζ3 − L + L − L z + 2 6 2 4 12 5 5 1 1 2 3 58 41 − π + L − L2 z3 + − π 2 − L − L2 z4 + 4 4 4 27 18 72 6 35 6547 35 2 283 − π − L − L2 z5 + 1728 72 144 24 63 2 6 185837 21 2 4227 − π − L− L z . + (2.34) 24000 20 800 20 For the imaginary part (and the leading power of L in the real part) it is possible to guess 1 , beginning with z4 we have the form of the higher order terms. For example, for M 1 = 8z3 Higher orders in Im M
∞ n=1
(2n + 1)! zn . n(n + 1)! (n + 3)!
(2.35)
If mc = mb /2, we expect the imaginary part to vanish. Indeed, this corresponds to z = 1/4, 43 5 − 24 ln 2, which exactly cancels the contribution of the at which point (2.35) gives 288 first 4 terms in the imaginary part of M1 given in (2.31). The results for the leading terms and the powers z, z3/2 , z2 and z3 in (2.31)–(2.34) agree with the corresponding terms in (2.25)–(2.28) of Greub, Hurth and Wyler [28]. The contributions z4 , z5 and z6 are new. Inserting (2.31)–(2.34) into (2.11) we find r2 = Re r2 + i Im r2 with 1666 32 2 3/2 + π z 243 27 8 128 8 2 40 32 32 8 − π L − π 2 − ζ3 + L + L2 + L3 z + 9 3 27 3 3 9 27
Re r2 = −
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8 16 16 2 16 32 − π L + π 2 + L + L3 z2 3 9 27 9 27 56 4 56 728 L − L2 z3 + − − π2 + 9 81 81 9 176 2 4 111748 176 2 3308 − π − L− L z + 6075 81 405 27 380 2 5 816731 380 2 13234 − π − L− L z + 23814 81 567 27 1624 2 6 44551813 1624 2 421121 − π − L− L z , + 546750 135 6075 45
16 16 2 80 16 2 80 16 2 16 2 2 L+ L + − π z+ L − π z Im r2 = π − + 81 9 9 9 27 9 27 448 3 16 4 100 5 5516 6 64 z + z + z + z + − L+ 27 81 81 81 2025 +
(2.36)
(2.37)
confirming the results (2.37) and (2.38) of [28], and generalizing them to include terms O(z4 ), O(z5 ) and to O(z6 ). The contributions of different terms zn for mc /mb = 0.29 and mc /mb = 0.22 are presented in Table 1. The term z3/2 has been added to the term z2 in this table. We observe that the terms O(zn ) with n 4 are negligible. The final result for r2 is given by: −4.093, mc /mb = 0.29, Re r2 = −4.985, mc /mb = 0.22, −0.450, mc /mb = 0.29, Im r2 = (2.38) −0.899, mc /mb = 0.22. The strong dependence of r2 on mc /mb has been pointed out in [40]. With decreasing z the enhancement of B → Xs γ by QCD corrections becomes stronger. Table 1 The numerical value of Coefficient ·zn for the real and imaginary part of r2 are given up to z6 using two different values of mc /mb . The term z3/2 in Re r2 was included in the term proportional to z2 Re r2
Im r2
n
mc mb = 0.29
mc mb = 0.22
mc mb = 0.29
mc mb = 0.22
0 1 2 3 4 5 6
−6.8559671 2.2271232 0.5775520 −0.0402440 −0.0011397 −0.0001703 −0.0000307
−6.8559671 1.6504869 0.2307186 −0.0103793 −0.0002090 −0.0000187 −0.0000020
−3.1028076 2.5193530 0.1121646 0.0213018 0.0000310 0.0000163 0.0000030
−3.1028076 2.1225820 0.0769360 0.0045268 0.0000034 0.0000010 0.0000001
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3. Conclusion In the present paper we have calculated the important two-loop matrix element sγ |Q2 |b contributing to the decay B → Xs γ at the NLO level. Our result for sγ |Q2 |b agrees with the one presented in [28] and used in the literature by many authors. The additional terms in the expansion in z = m2c /m2b , that is O(z4 ) and higher, turn out to amount to at most 0.05% and are negligible. As we have used a completely different method from the one used in [28], the confirmation of the result of these authors is very gratifying.
Acknowledgements This work was supported by Deutscher Akademischer Austauschdienst (DAAD), by the Natural Sciences and Engineering Research Council (NSERC), and by the German Bundesministerium für Bildung und Forschung under contract 05HT9WOA0. M.M. was supported by the Polish Committee for Scientific Research under grant 2 P03B 121 20.
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Two-loop matrix element of the current–current operator in the decay B → Xs γ Andrzej J. Buras a , Andrzej Czarnecki b , Mikołaj Misiak c , Jörg Urban a a Physik Department, Technische Universität München, D-85748 Garching, Germany b Physics Department, University of Alberta, Edmonton, Alberta, T6G 2J1 Canada c Institute of Theoretical Physics, Warsaw University, Ho˙za 69, 00-681 Warsaw, Poland
Received 23 May 2001; accepted 3 July 2001
Abstract We evaluate important two-loop matrix element sγ |Q2 |b of the operator (cγ ¯ µ PL b)(¯s γµ PL b) contributing to the inclusive radiative decay B → Xs γ . The calculation is performed in the NDR scheme, by means of asymptotic expansions method. The result is given as a series in z ≡ m2c /m2b up to O(z6 ). We confirm the result of Greub, Hurth and Wyler obtained by a different method up to O(z3 ). Higher-order terms are found to be numerically insignificant. 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The radiative decay B → Xs γ plays an important role in the present tests of the Standard Model (SM) and of its extensions [1]. In particular, in the supersymmetric extensions of the SM, the best bounds on several new parameters come from the data on B → Xs γ [2]. The short distance QCD effects are very important for this decay. They are known to enhance the branching ratio Br(B → Xs γ ) by roughly a factor of three, as first pointed out in [3,4]. Since these first analyses, a lot of progress has been made in calculating these important QCD effects in the renormalization group improved perturbation theory, beginning with the work in [5,6]. Let us briefly summarize this progress. A peculiar feature of the renormalization group analysis in B → Xs γ is that the mixing under infinite renormalization between the four-fermion operators (Q1 , . . . , Q6 ) and the magnetic penguin operators (Q7γ , Q8G ), which govern this decay, vanishes at the oneloop level. Consequently, in order to calculate the coefficients C7γ (µb ) and C8G (µb ) at µb = O(mb ) in the leading logarithmic approximation, two-loop calculations of O(egs2 ) E-mail address: [email protected] (J. Urban). 0550-3213/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 ( 0 1 ) 0 0 3 3 6 - 4
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and O(gs3 ) are necessary. Such calculations were completed in [7,8] and confirmed in [9–11]. Earlier analyses contained either additional approximations or mistakes. It turns out that the leading order expression for the branching ratio Br(B → Xs γ ) suffers from sizable renormalization scale uncertainties [12,13] implying that a complete NLO analysis including also dominant higher order electroweak effects to this decay is mandatory. By 1998, the main ingredients of such an analysis had been calculated. It was a joint effort of many groups: • The O(αs ) corrections to C7γ (µW ) and C8G (µW ) were first calculated in [14] and subsequently confirmed by several groups [15–18]. • The two-loop mixing involving the four fermion operators Q1 , . . . , Q6 and the two-loop mixing in the sector (Q7γ , Q8G ) was calculated in [19–24] and [25], respectively. The very difficult three-loop mixing between the set (Q1 , . . . , Q6 ) and the operators (Q7γ , Q8G ) was calculated in [11]. • Leading order matrix elements sγ gluon|Qi |b were calculated in [26,27], and the challenging two-loop calculation of sγ |Q2 |b was presented in [28]. • Higher order electroweak corrections were incorporated with increasing level of sophistication in [29–33]. In addition, non-perturbative corrections were calculated in [34–39]. The most recent analysis of B → Xs γ incorporating all these calculations can be found in [40]. Now, among the perturbative ingredients listed above, three have been calculated only by one group. These are: • Two-loop mixing in the sector (Q7γ , Q8G ) [25]; • Three-loop mixing between the set (Q1 , . . . , Q6 ) and the operators (Q7γ , Q8G ) [11]; • Two-loop matrix element sγ |Q2 |b [28]. Moreover, • Two-loop matrix elements of the QCD penguin operators sγ |Qi |b with i = 3, . . . , 6 have not yet been calculated. It should be emphasized that all these four ingredients enter not only the analysis of B → Xs γ in the SM but are also necessary ingredients of any analysis of this decay in the extensions of this model. It is therefore desirable to check the first three calculations and to perform the last one. In the present paper we will make the first step in this direction by calculating the twoloop matrix element sγ |Q2 |b using the method of asymptotic expansions. This matrix element turned out [28,40] to be the most important ingredient of the NLO-analysis for Br(B → Xs γ ) enhancing this branching ratio by roughly 20%. In [28] the matrix element sγ |Q2 |b was found by applying the Mellin–Barnes representation to certain Feynmanparameter integrals, and the result was presented as an expansion in z = m2c /m2b up to and including terms O(z3 ). In order to be sure about the convergence of this expansion, we will include also the terms O(z4 ), O(z5 ) and O(z6 ). Asymptotic expansions [41–43] provide a systematic approach to computing Feynman integrals which depend on two (or more) mass or momentum scales. The general idea is to
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expand the integrand in the small parameters. The coefficients of the resulting expansion depend on a single mass scale and are simpler to evaluate than the original integral, since they are given by numbers rather than potentially complicated functions of dimensionless quantities — ratios of the various scales. Of course, it is well known that if an integral depends on a parameter, an expansion does not, in general, commute with the integration. The “error” one makes can be systematically corrected. Below, these corrections will be reflected in consideration of various momentum regions in each integral. A nice formulation of the expansion procedure can be given in the language of counterterms (and theory of distributions); see [44] for a review and further references.
2. Two-loop contribution to sγ |Q2 |b 2.1. Preface The effective Hamiltonian for the process b → sγ is given by 8 4GF Heff (b → sγ ) = − √ Vt b Vt∗s Ci (µ)Qi . 2 i=1
(2.1)
Here GF is the Fermi constant, Vij are the CKM matrix elements and Ci are the Wilson coefficients of the operators Qi evaluated at µ = O(mb ). We have dropped the negligible ∗ . The full list of operators in (2.1) can be found in [28]. contributions proportional to Vub Vus In the present paper we need only the expressions for two of them. These are µ Q2 = cγ ¯ PL b s¯ γµ PL b , e Q7 = s¯α σ µν mb (µ)PR + ms (µ)PL bα Fµν , 2 16π
(2.2) (2.3)
where PL = (1 − γ5 )/2, PR = (1 + γ5 )/2 and σ µν = 2i [γµ , γν ]. As in [28], we will set ms = 0. In this section we present the details of the calculation of the matrix element sγ |Q2 |b in the NDR scheme. This matrix element vanishes at the one-loop level. Therefore, in order to obtain a non-vanishing contribution, one has to calculate two-loop diagrams. They are shown in Figs. 1, 4, 5, and 6, where the wavy and dashed lines represent the photon and gluon, respectively. Following [28], we have divided the contributing diagrams into four sets: • In sets 1 and 2, the photon is emitted from an internal s-quark (set 1) or an internal b-quark (set 2), which also exchanges a gluon with the charm quark. • Sets 3 and 4 are obtained from sets 1 and 2, respectively, by emitting this time the photon from the charm quark propagator. • Thanks to QED gauge invariance, it is not necessary to consider diagrams with a real photon emission from external quark lines.
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Fig. 1. Diagram M1 . Momentum assignments: Q, q, and γ denote respectively four-momenta of the b and s quarks, and of the photon; p and k are loop momenta. Double lines denote a massive particle (the b quark) and single lines are light particle propagators. The gluon is denoted by a dashed line.
It is convenient to write the regularized contributions from each set of the diagrams as follows: 1 mb −4' αs CF Qd sγ |Q7 |btree , M1 = (2.4) + M1 36' κµ π 5 mb −4' αs M2 = − (2.5) + M2 CF Qd sγ |Q7 |btree , 36' κµ π 1 mb −4' αs M3 = − (2.6) CF Qu sγ |Q7 |btree , + M3 8' κµ π 1 mb −4' αs CF Qu sγ |Q7 |btree . M4 = − (2.7) + M4 4' κµ π We work in D = 4 − 2' dimensions, with κ 2 = 4πe−γE , Qu = 23 , Qd = − 13 , CF = 43 and γE is Euler’s constant. The tree level matrix element of the operator Q7 is given by e sγ |Q7 |btree = mb (µb ) 2 u(p )/'q/ PR u(p). (2.8) 8π In order to make the two-loop matrix element finite, counterterms have to be added. These counterterms can easily be calculated by using the operator renormalization constants that were needed in the context of the calculation of the leading order anomalous dimension matrix. The complete counterterm is found to be [28] Qd mb −2' 3Qu Qd κ 4' αs count − CF sγ |Q7 |btree . = + M2 (2.9) 6 µ 8 18 ' π The above expression includes two-loop counterterms as well as contributions from oneloop diagrams with one-loop counterterm insertions. Adding the contributions from (2.4)–(2.7) and (2.9), we find the two-loop matrix element sγ |Q2 |b in the NDR-scheme: αs 416 mb ln + r2 , sγ |Q2 |b = sγ |Q7 |btree (2.10) 4π 81 µ with r2 = −
16 3 + M 2 + 32 M 4 . M1 + M 9 9
(2.11)
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Fig. 2. One-loop diagram (the k propagator can have fractional power) corresponding to Eq. (2.12).
The rest of this section is devoted to a detailed presentation of the calculation of the i using the method of asymptotic expansions. All the momentum integrals contributions M are performed in the Euclidean space. 2.2. Diagram M1 Of the four sets of diagrams which we have to consider, M1 and M2 shown in Figs. 1 and 4 are relatively simple. In M1 , the photon is emitted from a massless s-quark propagator. There are four momentum regions we have to consider. Denoting the momenta in the cquark and vertex loops by p and k, we have 1. k ∼ p ∼ mb : the “hard–hard” region; 2. k ∼ mb and p ∼ mc : the “hard-soft” region (which comes in two variants, depending on the routing of k through the c-quark loop); 3. a “collinear-soft” region, in which p ∼ mc and k · γ ∼ m2b but k · q ∼ m2c . Below, we describe the treatment of those regions in some detail. 2.2.1. Hard contribution: Taylor expansion in mc Since all scalar products are now large compared with m2c , we can expand the c-quark propagators in mc . This leads to a massless loop integration, which simply modifies the power of the momentum (k) in the gluon propagator (see Fig. 2). After this first integration, the integrand has the following form: dD k (scalar products involving k) . (2.12) (k 2 )a1 +' (k 2 + 2k · Q + Q2 )a2 (k 2 + 2k · q)a3 We combine propagators 1 and 3 using the Feynman parameter x, and then combine the result (·(1 − y)) with propagator 2 (·y). We shift the variable k → K − x(1 − y)q − yQ, and the denominator becomes a power of k 2 + y(1 − y)(x − 1 − i0). Since x − 1 < 0, this diagram has an imaginary part; its sign is determined by assigning a small negative imaginary mass to the massless lines. After the momentum shift, we have additional factors x m and y n in the numerator, and denominator has the power d + '. The result is
−2'
Γ (d − 2 + 2') B(m + a3 , a1 + 2 − d − ') Γ (d + ') B(a3 , a1 + ')B(a2 , a1 + a3 + ') × B(n + a2 + 2 − d − 2', a1 + a3 + 2 − d − '),
exp(−iπ)
(−1)d
where B and Γ are the standard Euler functions Beta and Gamma.
(2.13)
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Fig. 3. Subgraph corresponding to one of the two hard–soft contributions.
2.2.2. Hard–soft contributions from k ∼ mb , p ∼ mc This region can be handled as in the “large momentum expansion” [41–43]. In Fig. 3 we see one of the two configurations to be computed. We first integrate over p, and then evaluate the one-loop vertex. An analogous procedure is executed for the other (left) cquark line. 2.2.3. Collinear region Naively, one might think that we should consider the region k ∼ p ∼ mc , in which (Q + k)2 Q2 , (q + k)2 2k · q. However, this does not lead to a consistent power counting, since we have to shift the momentum k by an amount proportional to q, which is ∼ mb . Instead, we have to consider the collinear configurations [45]. We define k · γ = Q0 k+ ,
k · q = Q0 k− , Q2 = Q20 ,
2 k 2 = k+ k− − k⊥ ,
k · Q = Q0 (k+ + k− ).
(2.14)
The relevant contribution is k+ ∼ Q0 , k− ∼ m2c /Q0 . Now (Q + k)2 Q2 + 2k · γ , while (k + q)2 = k 2 + 2k · q is already homogeneously of order m2c and cannot be expanded. First, we integrate over p; for this we combine the two c propagators (we assume here, for generality, that their powers are a1 and a2 ) using x, shift p → P − xk, and obtain (m and d are exponents arising from the presence of p in the numerator) polynomial(p) Γ (d − 2 + ') x m+a1 +1−d−' (1 − x)a2 +1−d−' → , 2 2 a 2 2 a [(p + k) + mc ] 1 [p + mc ] 2 B(a1 , a2 )Γ (d) (k 2 + m2x )d−2+' m2x ≡
m2c . x(1 − x)
For the k integration, the integrand has the form dD k polynomial(k) . 2 a 2 2 d−2+' (k ) 3 (k + mx ) (k 2 + 2k · q)a4 (Q2 + 2γ · k)a5
(2.15)
(2.16)
Now, we combine the first two propagators multiplying them by 1 − z and z, respectively. We multiply the result by y and combine it with the “a4 ” propagator multiplied by 1 − y. Finally, we use the parameter u to include the last propagator (i = a3 + a4 + d − 2 + '): [k 2
+ yzm2x
1 + 2(1 − y)k · q]i (Q2 + 2k · γ )a5
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1 = B(i, a5 )
∞ du 0
ua5 −1 . [k 2 + yzm2x + 2(1 − y)k · q + 2uk · γ + uQ2 ]i+a5
(2.17)
Now, we shift k → K − (1 − y)q − uγ and, using q · γ = Q2 /2, find that the denominator simplifies to become [K 2 +y(zm2x +uQ2 )]. Because of the presence of K in the numerator, we generate additional powers of Feynman parameters. After integrating over K, we integrate over u, ∞ du 0
un−1 B(n, m − n) = . (zm2 + uQ2 )m (Q2 )n (zm2 )m−n
(2.18)
We see that the dependence on Q separates (so the integral really depends only on a single scale). Also, only mc appears in fractional power, so that there is no imaginary part in this integral, even though Q2 < 0 (after the Wick rotation). 2.3. M2 In this diagram (see Fig. 4) we have again four momentum regions, of which the first three are similar to those discussed for M1 . The fourth one is different: it is a “soft–soft” contribution, with k ∼ p ∼ mc . For the hard–hard contribution, we first integrate over p, just like in M1 . After that, the integrand takes on the following form: dD k (scalar products involving k) (2.19) . 2 a (k ) 1 +' (k 2 + 2k · Q)a2 (k 2 + 2k · q − Q2 )a3 We combine propagators 2 and 3 using a Feynman parameter x, and then combine the result (·y) with the propagator 1 (·(1 − y)). We shift the variable k → K − y((1 − x)Q + xq), and the denominator becomes a power of k 2 + y[y + x(1 − y)]. After shifting the momentum and simplifying the numerator, additional factors x m and y n appear in the numerator, and the denominator has the power d + 2'. After the k integration, we find that the x and y integrations do not separate, and we first integrate over x using Eq. (2.30), to be discussed below. The remaining y integration is a Beta function. The hard–soft contributions are similar to those in M1 and lead to rather trivial products of one-loop integrals.
Fig. 4. Diagram M2 .
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On the other hand, the soft–soft contribution is less standard. We have k ∼ mc and the two b-quark propagators can be expanded. In the lowest order we have 1 1 → , k 2 + 2k · Q 2k · Q
1 1 →− 2. k 2 + 2k · q − Q2 Q
(2.20)
Since the second propagator becomes independent of k, the whole two-loop diagram becomes equivalent to a simpler, two-point function. The resulting integrals are similar, but not the same, as those considered in the eikonal expansion study [46]. Details of their evaluation have been recently described in a completely different context of bound-state calculations [47]. In that work, exactly such type of integrals appeared when energies of bound-states consisting of two particles with very different masses were expressed as expansions in the ratio of those masses. 2.4. M3 This diagram, shown in Fig. 5, is the first of the two non-trivial two-loop vertex diagrams we have to consider. There are now five momentum regions to be considered: 1. Hard–hard, k ∼ p ∼ mb , similar to those of M1,2 . 2. Hard–soft, k ∼ mb , p ∼ mc , which now enters only once, when hard momentum flows through the same c-quark line from which the photon is emitted. 3. Collinear–collinear, with k and p having their only large components (O(mb )) aligned with the s-quark momentum q (and with k 2 ∼ p2 ∼ m2c ). 4. Collinear–collinear, but with the alignment with the photon momentum γ . 5. Hard–collinear, where k ∼ mb and p is aligned parallel with γ . The first two regions are treated in an analogous manner as was described for M1,2 . In the hard–hard contribution, after expansion in mc , the integrand is symmetrical (apart from different powers of propagators) under replacement p ↔ k, γ ↔ q. Hence, we can first integrate over k, and then over p. Both integrations are very similar to those in M1,2 . In the second region, we use the “large momentum expansion” mentioned in Section 2.2.2. The collinear–collinear regions require more attention [48]. After expansion of the integrands in the available small quantities, it turns out that the integrals over Feynman parameters have singularities (like 0 dx/x) which are not regularized by our dimensional regulator. It is necessary to introduce additional, analytical regularization on the heavy quark line. Similar phenomena have been observed before (see, e.g., [48] and references
Fig. 5. Diagrams contributing to M3 , with photon emission from two c-quark lines.
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therein). Nevertheless, all integrals over the Feynman parameters can be evaluated analytically without particular difficulties. Dependence on the analytical regulator cancels in the sum of the two doubly-collinear contributions. 2.5. M4 The most complicated diagram is M4 , depicted in Fig. 6. There are six momentum regions: 1. Hard–hard. 2. Hard–soft. 3. Collinear–collinear with alignment along γ . There is no contribution with alignment along q here, because of the different structure of the internal quark propagator, which now contains the large b-quark mass. 4. Hard–collinear, as in M3 . 5. Ultrasoft–collinear [48], with k ∼ m2c /mb and p aligned with γ . 6. Soft–soft. We give a more detailed description of the hard–hard contribution, because the integrals resulting here are rather complicated. First, we consider the scalar integral with all propagators in the first power, J3 =
dD k k 2 (k 2 + 2Q · k)
dD p . p2 (p − k)2 (p + γ )2
(2.21)
We first evaluate the massless (c-quark) loop (integrals over all Feynman parameters run from 0 to 1): dD p dD p = p2 (p − k)2 (p + γ )2 p2 (p2 − 2p · k + k 2 )(p2 + 2p · γ ) dD p = 2(1 − y) dy dx 2 [p − 2yp · k + 2x(1 − y)p · γ + yk 2 ]3 −1−' (1 − y)−' dy dx y = Γ (1 + ')π 2−' (k 2 + 2xk · γ )1+' dx 2−' = Γ (1 + ')B(−', 1 − ')π . (2.22) 2 (k + 2xk · γ )1+'
Fig. 6. Diagram M4 .
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In the process of the integration, we made a shift p → p + yk − x(1 − y)γ . Next, we integrate over k: dD k dx k 2 (k 2 + 2Q · k)(k 2 + 2xk · γ )1+' dD k = (1 + ') dx du (1 − u)' k 2 [k 2 + 2uk · Q + 2(1 − u)xk · γ ]2+' dD k = (2 + ')(1 + ') dz z1+' dx du (1 − u)' 2 2 2 [k + z u + z2 u(1 − u)x]3+' Γ (1 + 2') 2−' z−1−3' u−1−2' (1 − u)' = dz dx du π Γ (1 + ') [u + (1 − u)x]1+2' Γ (1 + 2') 2−' = 2 π du u−1−2' (1 − u)−1+' 1 − u−2' 6' Γ (1 + ') Γ (1 + 2') 2−'
= 2 (2.23) π B(−2', ') − B(−4', ') . 6' Γ (1 + ') We made a shift k → k − zuQ − z(1 − u)xγ and used Q2 = 2Q · γ = −1. Multiplying the result (2.23) with the coefficient Γ (1 + ')B(−', 1 − ') from (2.22) we find
Γ (1 + 2') B(−', 1 − ') B(−2', ') − B(−4', ') 2 6' π2 7 29π 4 1 2 = Γ (1 + ') + + ζ3 + + O('). 24' 4 18' 2 6' 360
J3 =
(2.24)
Now we consider the general case, in which momenta p and k can be present in the numerator, dD k dD p (scalar products involving k and p) (2.25) a a a a . (k 2 ) 1 (k 2 + 2k · Q) 2 (p − k)2a3 (p2 ) 4 (p2 + 2p · γ ) 5 We first combine the propagators 5 (·x) and 4 (·(1 − x)), and then the result (·(1 − y)) with 3 (·y). We shift the integration momentum p = K − x(1 − y)γ + yk and average over K. This again results in extra powers y n and x m , while canceling powers of K 2 changes the power of the denominator. After the integration over K, the denominator becomes [y(1 − y)(k 2 + 2xk · γ )]d−2+' . We see that the dependence on y factorizes and we can integrate over this variable. As a result, we find an expression of the form Γ (a3 + a4 + a5 )Γ (d − 2 + ')B(n + a3 − d + 2 − ', a4 + a5 − d + 2 − ') Γ (a3 )Γ (a4 )Γ (a5 )Γ (d) a5 +m−1 × dx x (1 − x)a4 −1 dD k (scalar products involving k) × . a a d−2+' (k 2 ) 1 (k 2 + 2k · Q) 2 (k 2 + 2xk · γ )
(2.26) (2.27) (2.28)
We now repeat a similar procedure with the variable p. We combine the propagator 2 (·u) with (k 2 + 2xk · γ ) (·(1 − u)), and then the result (·z) with the propagator 1 (·(1 − z)). We change the momentum variable k = K − zuQ − zx(1 − u)γ , average over K and simplify
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powers of K 2 (as a result, the power of the denominator changes from a1 + a2 + d − 2 to some d1 , and we also get extra powers ze and uw ). After integrating over K we get (including the factors (2.26), (2.27)) Γ (a3 + a4 + a5 )Γ (d − 2 + ')B(n + a3 − d + 2 − ', a4 + a5 − d + 2 − ') Γ (a3 )Γ (a4 )Γ (a5 )Γ (d) Γ (d1 − 2 + 2')Γ (a1 + a2 + d − 2 + ')B(a1 , a2 + d + 2 + e − 2d1 − 3') × Γ (d1 + ')Γ (a1 )Γ (a2 )Γ (d − 2 + ') × dx du x a5 +m−1 (1 − x)a4 −1 uw+a2 +1−d1 −2' (1 − u)d−3+'
2−d1 −2' × u + x(1 − u) .
(2.29)
First we integrate over x, using 1 0
j −1−n−µ n
µ uµ+n+1 −1 j n u . dx x n u + x(1 − u) = (−1) n+1 j µ+n+1−j (1 − u)
(2.30)
j =0
This integration is possible because the powers of x and (1 − x) in (2.27) are non-negative integer. The u integration gives a simple Beta function. The techniques described above are sufficient to compute all the remaining contributions. Again, the dimensional regularization alone is insufficient to evaluate the doublycollinear contribution. We introduce an analytical regulator on the heavy quark mass. The resulting singularities vanish when we add the soft–soft contribution. 2.6. Results i are listed below, with z = m2c /m2 and L = ln z. The final results for M b 1 = 37 + − 5 − L z + − 5 + π 2 + L − L2 z2 M 216 2 2 2 17 2 2 10 + − − π − L + L2 z3 27 3 9 3 2 7 11 1 67 + L z6 + − + L z4 + − + L z5 + 12 30 3 270 9
4 1 10 2 7 − z + (1 − 2L)z2 + − + L z3 + z4 + z5 + z6 , (2.31) + iπ 18 9 3 3 9 13 1 1 1 2 2 = M + − + π 2 z − π 2 z3/2 + 3 − 3L + L2 z2 216 2 6 3 2 4 2 3 157 5 2 1 4679 2 2 107 + π + L+ L z + − + π + L + 2L2 z4 + − 108 9 18 3 900 3 30 26185 5 2 277 + π + L + 5L2 z5 + − 2352 3 28 2831737 14 2 16177 + π + L + 14L2 z6 , + − (2.32) 97200 3 540
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3 = − 15 + 3 − 1 π 2 L − 1 π 2 − 2ζ3 + 2L + 1 L2 + 1 L3 z M 16 2 2 4 4 6 1 1 1 1 5 1 2 − π L + π 2 − 2ζ3 + L − L2 + L3 z2 + 4 2 2 2 2 6 17 7 5 3 + − +L z + − + L z4 12 216 9 231 1183 49 8783 5 + L z + + L z6 + 4320 72 12000 200
1 1 2 1 2 1 1 2 2 1 2 1 L + L + 2 − π z + −L + L + − π z + iπ − + 4 2 2 6 2 2 6 5 49 231 z6 , + z3 + z4 + z5 + (2.33) 9 72 200 4 = − 7 + 1 − 1 π 2 L − 1 π 2 − ζ3 + 1 L − 1 L3 z M 8 4 12 2 12 1 2 1 1 3 2 1 1 2 + π + 2ζ3 − L + L − L z + 2 6 2 4 12 5 5 1 1 2 3 58 41 − π + L − L2 z3 + − π 2 − L − L2 z4 + 4 4 4 27 18 72 6 35 6547 35 2 283 − π − L − L2 z5 + 1728 72 144 24 63 2 6 185837 21 2 4227 − π − L− L z . + (2.34) 24000 20 800 20 For the imaginary part (and the leading power of L in the real part) it is possible to guess 1 , beginning with z4 we have the form of the higher order terms. For example, for M 1 = 8z3 Higher orders in Im M
∞ n=1
(2n + 1)! zn . n(n + 1)! (n + 3)!
(2.35)
If mc = mb /2, we expect the imaginary part to vanish. Indeed, this corresponds to z = 1/4, 43 5 − 24 ln 2, which exactly cancels the contribution of the at which point (2.35) gives 288 first 4 terms in the imaginary part of M1 given in (2.31). The results for the leading terms and the powers z, z3/2 , z2 and z3 in (2.31)–(2.34) agree with the corresponding terms in (2.25)–(2.28) of Greub, Hurth and Wyler [28]. The contributions z4 , z5 and z6 are new. Inserting (2.31)–(2.34) into (2.11) we find r2 = Re r2 + i Im r2 with 1666 32 2 3/2 + π z 243 27 8 128 8 2 40 32 32 8 − π L − π 2 − ζ3 + L + L2 + L3 z + 9 3 27 3 3 9 27
Re r2 = −
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8 16 16 2 16 32 − π L + π 2 + L + L3 z2 3 9 27 9 27 56 4 56 728 L − L2 z3 + − − π2 + 9 81 81 9 176 2 4 111748 176 2 3308 − π − L− L z + 6075 81 405 27 380 2 5 816731 380 2 13234 − π − L− L z + 23814 81 567 27 1624 2 6 44551813 1624 2 421121 − π − L− L z , + 546750 135 6075 45
16 16 2 80 16 2 80 16 2 16 2 2 L+ L + − π z+ L − π z Im r2 = π − + 81 9 9 9 27 9 27 448 3 16 4 100 5 5516 6 64 z + z + z + z + − L+ 27 81 81 81 2025 +
(2.36)
(2.37)
confirming the results (2.37) and (2.38) of [28], and generalizing them to include terms O(z4 ), O(z5 ) and to O(z6 ). The contributions of different terms zn for mc /mb = 0.29 and mc /mb = 0.22 are presented in Table 1. The term z3/2 has been added to the term z2 in this table. We observe that the terms O(zn ) with n 4 are negligible. The final result for r2 is given by: −4.093, mc /mb = 0.29, Re r2 = −4.985, mc /mb = 0.22, −0.450, mc /mb = 0.29, Im r2 = (2.38) −0.899, mc /mb = 0.22. The strong dependence of r2 on mc /mb has been pointed out in [40]. With decreasing z the enhancement of B → Xs γ by QCD corrections becomes stronger. Table 1 The numerical value of Coefficient ·zn for the real and imaginary part of r2 are given up to z6 using two different values of mc /mb . The term z3/2 in Re r2 was included in the term proportional to z2 Re r2
Im r2
n
mc mb = 0.29
mc mb = 0.22
mc mb = 0.29
mc mb = 0.22
0 1 2 3 4 5 6
−6.8559671 2.2271232 0.5775520 −0.0402440 −0.0011397 −0.0001703 −0.0000307
−6.8559671 1.6504869 0.2307186 −0.0103793 −0.0002090 −0.0000187 −0.0000020
−3.1028076 2.5193530 0.1121646 0.0213018 0.0000310 0.0000163 0.0000030
−3.1028076 2.1225820 0.0769360 0.0045268 0.0000034 0.0000010 0.0000001
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3. Conclusion In the present paper we have calculated the important two-loop matrix element sγ |Q2 |b contributing to the decay B → Xs γ at the NLO level. Our result for sγ |Q2 |b agrees with the one presented in [28] and used in the literature by many authors. The additional terms in the expansion in z = m2c /m2b , that is O(z4 ) and higher, turn out to amount to at most 0.05% and are negligible. As we have used a completely different method from the one used in [28], the confirmation of the result of these authors is very gratifying.
Acknowledgements This work was supported by Deutscher Akademischer Austauschdienst (DAAD), by the Natural Sciences and Engineering Research Council (NSERC), and by the German Bundesministerium für Bildung und Forschung under contract 05HT9WOA0. M.M. was supported by the Polish Committee for Scientific Research under grant 2 P03B 121 20.
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