Completing the four-body contributions to B ¯ → X s γ at NLO
SSI-HEP-2019-24, P3H-19-055
Completing the four-body contributions to ¯ B → X s γ at NLO Tobias Huber
Naturwissenschaftlich-Technische Fakultät, Universität Siegen,Walter-Flex-Str. 3, 57068 Siegen,E-mail: [email protected]
Lars-Thorben Moos ∗ Naturwissenschaftlich-Technische Fakultät, Universität Siegen,Walter-Flex-Str. 3, 57068 Siegen, GermanyE-mail: [email protected]
We report on the status of the ongoing calculation of multiparticle contributions to theinclusive radiative ¯ B → X s γ decay at next-to leading order. This effort amounts to theevaluation of the four-particle process b → s ¯ qqγ at the one-loop level, supplemented bythe corresponding five-particle tree-level cuts b → s ¯ qqγ + g of the gluon bremsstrahlung.Knowledge of these pieces will formally complete the ¯ B → X s γ decay at the next-to-leading order. The different steps such as the generation of the amplitude, its renor-malization, and the treatment of occuring IR-divergences are discussed. Moreover, weelaborate on the subtleties that arise when treating γ in D dimensions. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - ph ] D ec ompleting the four-body contributions to ¯ B → X s γ at NLO Lars-Thorben Moos
1. Introduction
The inclusive radiative decay of the B meson ¯ B → X s γ constitutes one of the mostprecise tests of the Standard Model (SM) in the quark flavor sector and represents astandard candle in the search for New Physics.At the partonic level, the main contribution comes from the two-particle b → sγ process,which is a flavor-changing neutral current (FCNC) and hence forbidden at tree-level in theSM. Being loop-induced, the process is very sensitive to virtual contributions from newparticles running in the loop.The value for the branching fraction has been measured very precisely. The currentexperimental value of the CP- and isospin-averaged branching ratio of ¯ B → X s γ with aphoton-energy cut of E γ > E = 1 . ∼
5% [1] B expsγ = (3 . ± . · − . (1.1)With uncertainties on the experimental side that are this small, the results needs to besupplemented accordingly by a theoretical value that is determined with a comparableprecision. The work on the theoretical prediction for this process has been carried outfor the last twenty years, see e.g. [2–5]. This program includes corrections up to next-to-next-to-leading order (NNLO), and resulted in the current SM prediction of the aboveobservable [6], B SMsγ = (3 . ± . · − , (1.2)which is in very good agreement with the experimental measurement.With the upcoming run of Belle II and the combination with data from the other B-factories, the uncertainties on the experimental side are expected to decrease further, callingfor increased effort also on the theory side. In this work, we will focus on the last pieces thatare missing in order to formally complete ¯ B → X s γ at the next-to-leading order (NLO).These are multiparticle contributions at the one-loop level, which are suppressed by smallCKM factors or Wilson coefficients. To be precise, we elaborate on the one-loop calculationof those four-particle b → s ¯ qqγ diagrams that must be supplemented by the correspondingfive-particle tree-level cuts b → s ¯ qqγ + g from gluon bremsstrahlung. After describing thetheoretical framework, the different steps of the computation will be discussed. Theseinclude the generation of the diagrams, the Dirac algebra, the reduction of the resultingintegrals and their computation, the renormalization and finally the treatment of infrared(IR) divergent collinear pieces that are visible in the final result as logarithms of quark-massratios.
2. Theoretical Framework
The interactions that are relevant for the process at hand are incorporated in thefollowing effective Lagrangian, L eff = L QED + QCD + 4 G F √ h V ∗ us V ub X i =1 C ui P ui + V ∗ ts V tb X i =3 C i P i i . (2.1)1 ompleting the four-body contributions to ¯ B → X s γ at NLO Lars-Thorben Moos
Here, L QED + QCD is the QED and QCD Standard Model Lagrangian, the V ij are the entriesof the CKM-matrix and the P i are effective four-fermion operators with Wilson coefficients C i . The operators are given by P u = (¯ s L γ µ T a u L )(¯ u L γ µ T a b L ) P u = (¯ s L γ µ u L )(¯ u L γ µ b L ) P = (¯ s L γ µ b L ) X q (¯ qγ µ q ) P = (¯ s L γ µ T a b L ) X q (¯ qγ µ T a q ) P = (¯ s L γ µ γ ν γ ρ b L ) X q (¯ qγ µ γ ν γ ρ q ) P = (¯ s L γ µ γ ν γ ρ T a b L ) X q (¯ qγ µ γ ν γ ρ T a q ) . (2.2)The sum over q runs over five flavours in principle. However, since by definition the X s system does not contain any charm- or anti-charm quarks and bottom is forbiddenkinematically, we can resrict the sum to run over q = u, d, s , which we treat as massless. In a previous calculation [7], part of the NLO four-body contribution has already beencalculated, namely those pieces that do not require the inclusion of gluon bremsstrahlung.The pieces that remain can be seen in Fig. 1. For these diagrams, the four-body correctionsincluding the gluon-loop (top left panel) need to be supplemented by the tree-level five-bodycuts depicted in the top right panel in order to cancel the IR-divergences in the final statethat are induced by the gluon. The lower panel shows an additional operator insertionthat appears in the b → s ¯ ssγ (+ g ) channel. Note that this procedure will not cancel allIR divergences, since there are additional IR divergent pieces that remain because of thephoton in the final state. The treatment of these will be discussed in a later section.In our setup, the diagrams are generated with QGRAF [8] and their computation iscarried out in
FORM [9]. After the Dirac algebra and the calculation of the traces, we usePassarino-Veltman decomposition to simplify the results, which is carried out in
FeynCalc [10]. The next step is the further reduction of the result via IBP relations in
FIRE [11] andthen finally the phase-space integration of the master integrals in
Mathematica and theexpansion of the resulting functions in
HypExp [12]. γ If we use the operators of Eq. (2.2) in their original form, we encounter in the squaredmatrix element products of two traces containing up to two γ each. In this case, anunambiguous treatment in D dimensions is complicated. We avoid these problems byusing Fierz transformations on the operators P u and P u [7, 13], P u = − P u + 19 P u + 127 P u − P u + O ( (cid:15) ) , (2.3)with the notation P u = (¯ s L γ µ b L )(¯ uγ µ u ) etc. These relations trade the occurence of anadditional projector P L in the current-current operators for a linear combination of physicalpenguin operators plus evanescent operators stemming from the fierzing of the fermion lines.The necessary evanescent structure has for example been calculated in Ref. [13] and looksas follows, E = (¯ s L γ µ γ ν γ ρ T a u L )(¯ u L γ µ γ ν γ ρ T a b L ) − (16 + 4 (cid:15) ) P u . (2.4)2 ompleting the four-body contributions to ¯ B → X s γ at NLO Lars-Thorben Moos b bs ¯ qqP P b bs ¯ qqP P b bs ¯ ssP P Figure 1:
Sample diagrams of four-body contributions to the process ¯ B → X s + γ (top left) andtheir five-body counterparts (top right). The lower diagram shows the case of a single trace, thatoccurs when operators with three strange quarks are inserted. The corresponding operator for P u only differes by a color factor.After the Fierz transformation of the operators P u and P u , at most a single tracecontains zero, one or two occurrences of γ . Despite the fact that γ is only well-defined infour dimensions we can nevertheless apply the scheme of Naive Dimensional Regularization(NDR) to treat γ consistently in D dimensions. To this end we use the relationTr( γ µ ...µ m γ γ µ m +1 ...µ n γ ) = ( − n − m Tr( γ µ ...µ m µ m +1 ...µ n ) (2.5)by using { γ , γ µ } = 0 in traces with an even number of them, together with ( γ ) = 1.Traces that do not contain any γ are then evaluated as products of metric tensors in theusual way. For the traces with only a single γ the case is not as simple. However, since weare computing a squared matrix element, the final result will not have any open Lorentzindices. Using this feature and the cyclicity of the trace (but no anticommutation of γ inthis case!) we can infer that in any term of the squared matrix element γ appears at mostin a single place in a term which has the structureTr( /p /p /p /p γ ) , (2.6)where the p i are the light-like momenta of the final state particles. Traces with fewer thanfour γ µ and a γ are consistently set to zero. Subsequently, we integrate over the phasespace whose measure, even in the presence of a photon-energy cut (see below), is sufficientlysymmetric to make the antisymmetric structure (2.6) vanish.3 ompleting the four-body contributions to ¯ B → X s γ at NLO Lars-Thorben Moos k − p − p p p k + p p p k p k − p − p p p p k + p Figure 2:
Four-loop topologies that occur before (left) and after (right) the reduction. If anypropagator with momentum p i gets contracted, the corresponding diagram can be set to zero.
3. Reduction via IBP relations
To reduce the number of integrals that need to be solved in the end, we employ anintegration-by-parts (IBP) reduction. In order to convert the phase space into a loopintegral we employ the relation [14] − πiδ ( p ) = 1 p + iε − p − iε , (3.1)which results in four-loop propagator diagrams like the one on the left in Fig. 2. Therelation (3.1) is not only used for the on-shell-conditions p i = 0 of the outgoing particles,but also for imposing the condition for the photon-energy cut, of which we give more detailsin the next section. The reduction then does not care about the sign of the iε -prescriptionand we can run it with any one of the above propagators. After the reduction procedure,we substitute the occurring propagators back to the δ -functions. It then becomes clearthat in case a propagator with momentum p i gets contracted, the corresponding integralcan be immediately set to zero due the relation δ ( p ) p = 0.
4. Phase-space integration
After the computation of the diagrams, we integrate the kernels K ( s ij ) = |M ( s ij ) | over the four- respectively five-particle phase space in D = 4 − (cid:15) dimensions. For thiswe introduce the momentum invariants s ij that are defined by s ij = 2 p i · p j /m b , with themomenta labelled by b ( p b ) → q ( p )¯ q ( p ) s ( p ) γ ( p ) g ( p ) . After the change of variables, the phase-space integral for the four-particle cuts looks asfollows [15], Z [ ds ij ] δ (1 − X s ij ) K ( s ij )( − ∆ ) D − Θ( − ∆ ) . (4.1)In the formula above, the δ (1 − P s ij ) incorporates the momentum conservation, while ∆ represents the Gram determinant ∆ = λ ( s s , s s , s s ) with the Källen function λ ( x, y, z ) = x + y + z − xy − xz − yz . The Gram determinant can be parametrizedaccording to − ∆ = (¯ z − s ) ( a + − s )( s − a − ) , (4.2)4 ompleting the four-body contributions to ¯ B → X s γ at NLO Lars-Thorben Moos where the roots a ± are given by a ± = z (cid:2) ¯ vwx + ¯ x ¯ w ± vw ¯ wx ¯ x ) / (cid:3) . (4.3)Here z is the parameter related to the photon energy which is, alike the other variables,a function of the s ij , see below. Moreover, we use ¯ z = 1 − z and similar for the othervariables. A parametrization of the five-particle phase space in D dimensions, includingtransformations that lead to a factorization of the Gram determinant, are given in [16]. E γ Since the measurement of the energy E γ of the photon poses a problem in the lower partof the spectrum, we impose a cut on the photon energy to make the prediction compatiblewith experimental results. In the restframe of the b quark, we have for the photon energy2 E γ /m b = 2 p b · p /m b = s + s + s ≡ − z , (4.4)and the inequality that incorporates the energy cut is expressed as E γ > E ≡ m b (1 − δ ),leading to the relation 1 − z = s + s + s > − δ. (4.5)To take all this into account in the phase-space integral, the delta function δ (1 − z − s − s − s ) is added, together with an additional integration over z , running from 0 to δ , Z δ dz Z [ ds ij ] δ (1 − z − s − s − s ) δ ( z − s − s − s ) K ( s ij )( − ∆ ) D − Θ( − ∆ ) . (4.6) To illustrate the procedure, we will now sketch the computation of the resulting phasespace integrals in the four-particle case. Going through the steps mentioned in the previoussections, one arrives at expressions such as˜ I = Z dP S Z d D ‘ (4 π ) D s s ‘ ( ‘ + k + k + k ) s ( − ∆ ) D − Θ( − ∆ ) . (4.7)After the loop integration this evaluates to I = Z dP S Γ( (cid:15) )Γ(1 − (cid:15) ) Γ(2 − (cid:15) ) s s ( s + s + s ) − (cid:15) s ( − ∆ ) D − Θ( − ∆ ) . (4.8)To make the subsequent steps easier, one can use the symmetry of (4.6) in the momentaof the light quarks (here 1 → → →
1) prior to conducting the change of variables s = z − s − s , s = ¯ z − s − s ,s = vwz , s = ¯ z ¯ v ,s = ¯ zvx , s = ( a + − a − ) u + a − . (4.9)5 ompleting the four-body contributions to ¯ B → X s γ at NLO Lars-Thorben Moos b bs ¯ qqP P b bs ¯ ssP P Figure 3:
Tree-level counterterm diagrams. Insertions of evanescent operators must also be in-cluded.
In the above equation, the first two kinematic invariants are fixed by the δ -functions fromEq. (4.6) and the rest is chosen such that the Gram determinant factorizes. This substi-tution leads to the integral I = δ Z dz ( z ¯ z ) D − Z du dv dx dw ( u ¯ u ) D − v D − (¯ vw ¯ wx ¯ x ) D − × h ( a + − a − ) u + a − i x ¯ x − h v ( wz + ¯ z ) i − (cid:15) . (4.10)The evaluation of I leads to a sum of hypergeometric functions, I = δ Z dz c ( (cid:15) ) ¯ z − (cid:15) z − (cid:15) F ( − (cid:15), − (cid:15) ; 3 − (cid:15) ; z )+ c ( (cid:15) ) ¯ z − (cid:15) z − (cid:15) F ( − (cid:15), − (cid:15) ; 3 − (cid:15) ; z ) , (4.11)where the c i are functions of the dimensional regulator (cid:15) . Note that the above expressionsare still differential in the photon energy since the integral over z has not yet been carriedout. In the case when a fully analytic expressions to all orders in (cid:15) can be achieved, e.g.in terms of hypergeometric and Γ-functions, the integration over z and the expansion in (cid:15) can be interchanged, provided the z -integration does not lead to further poles in (cid:15) . Thefinal result can then be obtained as a function of δ to the desired order in (cid:15) . In cases wherean all-order result is not possible, one can derive Mellin-Barnes representations, which canbe analytically continued to (cid:15) = 0. After expanding in (cid:15) and carrying out all remainingintegrations, analytic results as functions of δ can also be achieved in this case.
5. UV renormalization
After the phase-space integration, the result has to be renormalized. For this thecounterterm insertions in the corresponding four-particle cut diagrams have to be included,see Fig 3. In this step also evanescent operators become important since they can lead tofinite pieces in the final result by multiplying 1 /(cid:15) -poles from renormalization constants.6 ompleting the four-body contributions to ¯ B → X s γ at NLO Lars-Thorben Moos
6. Treatment of the collinear IR-divergences
In regions where the photon is collinear to one of the quarks, we run into collineardivergences. In the massive case these divergences are regulated naturally by the quarkmasses. Since we work in the case where all the outgoing quarks are massless, these IR-divergences are regularized dimensionally and show up as poles in (cid:15) . In our case we relatethe collinear 1 /(cid:15) poles to logarithms of quark masses by employing the splitting functions.These can be used because in the quasi-collinear limit the amplitude factorizes, b → q q ¯ q γ ⇒ b → X i q q ¯ q × f i . (6.1)In this framework, the f i is a DGLAP splitting function describing the emission of a photon γ from the quark-line q i .A comparison of the splitting functions in the two different schemes yields a shiftrelation. This relation can then be used to switch from dimensional regularization to thescheme of mass regularization and vice versa, d Γ m dz = d Γ (cid:15) dz + d Γ shift dz . (6.2)The shift part has contributions from three- and four-particle cut diagrams. The shiftinduced by the three-particle cut diagrams can be computed by means of the followingformula, Γ shift dz = 12 m b N c Z dP S K ( s ij ) α e π ¯ z ( Q " z − s ) (1 − s ) × " (cid:15) − − s ) µm q (1 − z ) Θ( z − s ) + (cyclic) ) . (6.3)Sample three- and four-particle cut diagrams and the necessary counterterms can be foundin Fig. 4. Through this shift, we trade the 1 /(cid:15) terms coming from IR divergences forlog( m q /m b ) terms. In these logarithms, the m q is not the physical mass of the quarks,but can be varied in a typical range of O (100 MeV) to get an estimate of the size of thecollinear logarithms. Acknowledgments
We would like to thank the organisers of “RADCOR 2019” for creating a very pleasantand inspiring atmosphere, and Mikołaj Misiak for useful correspondence. This researchwas supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Founda-tion) under grant 396021762 - TRR 257 “Particle Physics Phenomenology after the HiggsDiscovery”. 7 ompleting the four-body contributions to ¯ B → X s γ at NLO Lars-Thorben Moos b bs ¯ qqP P b bs ¯ qqP P b bs ¯ qqP P Figure 4:
Upper two panels: Three and four-particle cut diagrams contributing to the computationof the shift induced by the splitting function. The lower panel shows the relevant counterterminsertions.
References [1] HFLAV Collaboration,
Averages of b -hadron, c -hadron, and τ -lepton properties as ofsummer 2016 , Eur. Phys. J.
C 77 , 895 (2017), arXiv:1612.07233 [hep-ex] .[2] A. J. Buras, A. Czarnecki, M. Misiak and J. Urban,
Completing the NLO QCD calculationof ¯ B → X S γ ) , Nucl. Phys.
B 631 , 219-238 (2002), hep-ph/0203135 .[3] M. Misiak et al.,
Estimate of B ( ¯ B → X s γ ) at O ( α s ), Phys. Rev. Lett. , 022002 (2007), hep-ph/0609232 .[4] M. Misiak and M. Poradziński, Completing the Calculation of BLM corrections to ¯ B to X s γ , Phys. Rev.
D 83 , 014024 (2011), arXiv:1009.5685 [hep-ph] .[5] M. Kamiński, M. Misiak and M. Poradziński,
Tree-level contributions to B → X s γ , Phys. Rev.
D 86 , 094004 (2012), arXiv:1209.0965 [hep-ph] .[6] M. Misiak et al.,
Updated NNLO QCD predictions for the weak radiative B-meson decays , Phys. Rev. Lett. , 221801 (2015), arXiv:1503.01789 [hep-ph] .[7] T. Huber, M. Poradziński and J. Virto,
Four-body contributions to ¯ B → X s γ at NLO , JHEP , 115 (2015), arXiv:1411.7677 [hep-ph] .[8] P. Nogueira, Automatica Feynman graph generation , J. Comp. Phys. , 279 (1993).[9] B. Ruijl, T. Ueda and J. Vermaseren,
FORM version 4.2 , arXiv:2017 [hep-ph] .[10] V. Shtabovenko, New Developments in FeynCalc 9.0 , Comp. Phys. Commun. , 432-444(2016), arXiv:1601.01167 [hep-ph] .[11] A. V. Smirnov and F. S. Chuharev,
FIRE6: Feynman Integral REduction with ModularArithmetic , arXiv:1901.07808 [hep-ph] . ompleting the four-body contributions to ¯ B → X s γ at NLO Lars-Thorben Moos[12] T. Huber and D. Maitre,
HypExp 2, Expanding Hypergeometric Functions about Half-IntegerParameters , Comp. Phys. Commun. , 755-776 (2008), arXiv:0708.2443 [hep-ph] .[13] A. J. Buras, M. Misiak and J. Urban,
Two loop QCD anomalous dimensions of flavorchanging four quark operators within and beyond the standard model , Nucl. Phys.
B 586 ,397-426 (2000), hep-ph/0005183 .[14] C. Anastasiou and K. Melnikov,
Higgs boson production at hadron colliders in NNLO QCD , Nucl. Phys.
B 646 , 220-256 (2002), hep-ph/0207004 .[15] A. Gehrmann-De Ridder, T. Gehrmann and G. Heinrich,
Four particle phase space integralsin massless QCD , Nucl. Phys.
B 682 , 265-288 (2004), hep-ph/0311276 .[16] G. Heinrich,
Towards e + e − → jets at NNLO by sector decomposition , Eur. Phys. J.
C 48 ,25-33 (2006), hep-ph/0601062 ..