Anti-B --> X(s) gamma in two universal extra dimensions
aa r X i v : . [ h e p - ph ] J a n ANL-HEP-PR-08-04; MZ-TH/08-03; ZU-TH-01-08 ¯ B → X s γ in two universal extra dimensions Ayres Freitas and Ulrich Haisch Department of Physics and Astronomy, University of Pittsburgh, PA 15260, USAand Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USAand HEP Division, Argonne National Laboratory, Argonne, IL 60439, USA Institut f¨ur Physik (THEP), Johannes Gutenberg-Universit¨at D-55099 Mainz, Germanyand Institut f¨ur Theoretische Physik, Universit¨at Z¨urich, CH-8057 Z¨urich, Switzerland (Dated: November 11, 2018)We calculate the leading order corrections to the ¯ B → X s γ decay in the standard model withtwo large flat universal extra dimensions. We find that the contributions involving the exchange ofKaluza-Klein modes of the physical scalar field a ± ( kl ) depend logarithmically on the ultraviolet cut-offscale Λ. We emphasize that all flavor-changing neutral current transitions suffer from this problem.Although the ultraviolet sensitivity weakens the lower bound on the inverse compactification radius1 /R that follows from ¯ B → X s γ , the constraint remains stronger than any other available directmeasurement. After performing a careful study of the potential impact of cut-off and higher-ordereffects, we find 1 /R >
650 GeV at 95% confidence level if errors are combined in quadrature. Ourlimit is at variance with the parameter region 1 /R .
600 GeV preferred by dark matter constraints.
PACS numbers: 12.15.Lk, 12.60.-i, 13.25.Hw
I. INTRODUCTION
The branching ratio of the inclusive radiative ¯ B -mesondecay is known to provide stringent constraints on vari-ous non-standard physics models at the electroweak scale[1], because it is accurately measured and its theoreticaldetermination is rather precise.The present experimental world average, which in-cludes the latest measurements by CLEO [2], Belle [3],and BaBar [4], is performed by the Heavy Flavor Av-eraging Group [5] and reads for a photon energy cut of E γ > E with E = 1 . B -meson rest-frame B ( ¯ B → X s γ ) exp = (3 . ± . +0 . − . ± . × − . (1)Here the first error is a combined statistical and sys-tematic one, while the second and third are system-atic uncertainties due to the extrapolation from E =(1 . − .
0) GeV to the reference value and the subtrac-tion of the ¯ B → X d γ event fraction, respectively.After a joint effort [7, 8, 9], the first theoretical es-timate of the total ¯ B → X s γ branching ratio at next-to-next-to-leading order (NNLO) in QCD has been pre-sented recently in Refs. [8, 10]. For E = 1 . B ( ¯ B → X s γ ) SM = (3 . ± . × − , (2) The very recent measurement of BaBar [6] that gives B ( ¯ B → X s γ ) = (3 . ± . stat ± . syst ) × − for E = 1 . The small NNLO corrections related to the four-loop b → sg mix-ing diagrams [9] and from quark mass effects to the electromag-netic dipole [11] and current-current operator [12] contributionsare not included in Eq. (2). where the uncertainties from hadronic power corrections( ± ± ± ± . σ . Potential beyond SM contributions should nowbe preferably constructive, while models that lead to asuppression of the b → sγ amplitude are more severelyconstrained than in the past, where the theoretical de-termination used to be above the experimental one.As emphasized in Refs. [13, 14, 15], among the lat-ter category is the model with a flat, compactified extradimension where all of the SM fields are allowed to prop-agate in the bulk [16], known as minimal universal extradimensions or UED5. Since Kaluza-Klein (KK) modesin the UED5 model interfere destructively with the SM b → sγ amplitude, the B ( ¯ B → X s γ ) constraint leads toa very powerful bound on the inverse compactificationradius of 1 /R >
600 GeV at 95% confidence level (CL)[15]. This exclusion is independent from the Higgs massand therefore stronger than any limit that can be derivedfrom electroweak precision measurements [17].The purpose of this article is to study the phenomenol-ogy of ¯ B → X s γ in the SM with two universal extradimensions [18, 19] or UED6. In contrast to UED5, theUED6 model has additional KK particles in its spectrum.An interesting feature of this model is the fact that darkmatter constraints suggest a rather small KK mass scale.Therefore it is very interesting to derive a bound on thisscale from b → sγ in UED6, taking into account the newKK modes. In this context, several questions will need tobe answered: Does the leading order (LO) result dependon the cut-off scale, in contrast to UED5 where no cut-offdependence was found? If so, is this a generic feature ofall flavor-changing neutral current (FCNC) amplitudesin the UED6 model? What is the theoretical uncertaintystemming from the unknown ultraviolet (UV) dynamics?This article is organized as follows. In Secs. II andIII we describe, first, the model itself and, second, thecalculation of the one-loop matching corrections to theWilson coefficients of the electro- and chromomagneticdipole operators in UED6. Sec. IV contains a numericalanalysis of B ( ¯ B → X s γ ) and the lower bound on the com-pactification scale 1 /R in the UED6 model. Concludingremarks are given in Sec. V. In App. A we show how tocompute the double sums over KK modes appearing inthe calculation of ¯ B → X s γ . II. MODEL
Here we briefly summarize the main features of theUED6 scenario. All SM fields propagate in two flat extradimensions, compactified on a square with side length L = πR and adjacent sides being identified [20]. Thiscompactification, aptly dubbed chiral square, leads tochiral fermion zero modes, while the higher KK modes ofthe fermions are vector-like as usual. Since the geometryis invariant under rotations by 180 ◦ about the center ofthe square, the model respects an additional Z symme-try. It implies that the lightest KK-odd particle is stableand could provide a viable dark matter candidate for asmall KK scale 1 /R .
600 GeV [21].Solving the six-dimensional equations of motion leadsto an orthonormal set of functions, which depend on twoKK indices k, l corresponding to the two extra dimen-sions, with k ≥ , l ≥ k = l = 0 [18]. The model be-comes strongly interacting at high energy scales, so thatit is viewed as a low-energy effective theory which is validup to some cut-off scale Λ. From naive dimensional anal-ysis (NDA) [19], this scale is estimated to be Λ ≈ /R ,corresponding to an upper limit N KK ≤ k + l ≈
10 forthe KK indices.Before electroweak symmetry breaking, all ( kl ) modeshave degenerate tree-level masses m ( kl ) = √ k + l /R .The degeneracy is lifted by loop corrections, which leadto mass operators localized at the corners of the chi-ral square [19, 22]. Additional flavor diagonal and non-diagonal contributions can originate from physics at theUV cut-off scale. Since flavor non-universal operatorswould in general lead to unacceptably large FCNC transi-tions, we will assume that the localized operators are fla- vor conserving, so that the Cabibbo-Kobayashi-Maskawa(CKM) matrix remains the only source of flavor viola-tion. In this work, we concentrate on the leading ordercontributions from the UED6 model to ¯ B → X s γ , us-ing tree-level masses for those KK excitations which re-ceive only logarithmic corrections from loop correctionsand boundary terms localized at the orbifold fixed points[19, 22, 23, 24]. This is justified since these terms are ofone-loop order, thus leading to next-to-leading order ef-fects for ¯ B → X s γ .Upon compactification, the six-component gauge fields W aM , M = 0 , . . . ,
5, decompose into four-component mas-sive KK vector bosons W aµ ( kl ) , µ = 0 , . . . ,
3, and twoscalar KK fields W a , kl ) . Here a denotes the adjointgroup index. Following Refs. [18, 25], a covariant gaugefixing is introduced, such that W aµ ( kl ) do not mix with W a , kl ) . In the six-dimensional formulation, the gaugefixing-term reads L GF = − ξ (cid:2) ∂ µ W aµ − ξ ( ∂ W a + ∂ W a − g v χ a ) (cid:3) − ξ ′ (cid:2) ∂ µ B µ − ξ ′ ( ∂ B + ∂ B + g ′ v χ ) (cid:3) , (3)where W, B are the uncompactified SU (2) and U (1)gauge fields with the six-dimensional gauge couplings g ( ′ )6 ,and ξ ( ′ ) are the gauge parameters. The χ a are the com-ponents of the six-dimensional Higgs doublet H = 1 √ (cid:18) χ + iχ v + h + iχ (cid:19) . (4)The six-dimensional gauge couplings and vacuum expec-tation value are related to the four-dimensional values by g ( ′ )6 = g ( ′ ) πR and v = v/R .The Higgs scalars mix with the fourth and fifth compo-nent of the gauge fields to form the would-be Goldstonebosons G a ( kl ) of the massive vector bosons W aµ ( kl ) , andtwo physical scalars a a ( kl ) and W aH ( kl ) . Only the would-be Goldstone bosons have zero modes G a (00) , which corre-spond to the usual components of the SM Higgs doublet.For k + l ≥
1, the G a ( kl ) are dominated by the scalar ad-joints W a , kl ) and B , kl ) while the a a ( kl ) are composedmostly of the Higgs doublet elements. For the chargedfields one finds G ± ( kl ) = 1 M ( kl ) W (cid:20) R (cid:16) l W ± kl ) − k W ± kl ) (cid:17) + M W χ ± ( kl ) (cid:21) ,a ± ( kl ) = 1 M ( kl ) W h m ( kl ) χ ± ( kl ) − M W m ( kl ) R (cid:16) l W ± kl ) − k W ± kl ) (cid:17) i ,W ± H ( kl ) = 1 √ k + l h k W ± kl ) + l W ± kl ) i , (5)where X ± = X ∓ iX √ , X = W, χ, G, a, W H . (6)Here M W ( kl ) = m kl ) + M W is the tree-level squaredmass of the W ± µ,H ( kl ) and a ± ( kl ) . The would-be Gold-stone bosons G ± ( kl ) receive the unphysical squared mass ξM W ( kl ) from gauge fixing. Similar expressions hold forthe neutral fields, taking into account a small mixing be-tween W kl ) and B ( kl ) . However, since they do not con-tribute to the process ¯ B → X s γ at leading order in theelectroweak interactions we do not give them here.As mentioned above, the masses of the KK modes re-ceive corrections from loop and UV effects, which are de-pendent on the cut-off scale Λ. Since G ± ( kl ) and W ± µ,H ( kl ) are protected by gauge invariance, the dependence onΛ is only logarithmic [22], so that the mass correctionsare small compared to 1 /R and can be neglected in aLO calculation. The a ± ( kl ) scalars, however, can receivecontributions proportional to Λ to both their bulk andboundary mass terms [24].In order to obtain a small mass term for the zero modeHiggs doublet, the bulk and boundary mass terms needto be tuned to cancel to a large extent. However, inde-pendent of this tuning, the higher KK modes can receivesizeable contributions from these terms. As a result, the a ± ( kl ) scalars can be heavier or lighter than the other parti-cles of the same KK level. We include the Λ correctionsto the a ± ( kl ) masses based on the following parametriza-tion of the UV-induced mass terms: L ⊃ (cid:20) L (cid:0) δ ( x ) δ ( x ) + δ ( L − x ) δ ( L − x ) (cid:1) m H, + L δ ( x ) δ ( L − x ) m H, + m H, bulk (cid:21) | H | . (7)Although the UV physics is not specified, these mass pa-rameters are expected to stem from loop contributions ofthe UV dynamics, so that m H,i = h i π Λ = h i π N R , i = 1 , , (8)with h , = O (1). Using the explicit form of the KKwave functions from Refs. [18, 20] and tuning the bulkmass m H, bulk to exactly cancel the Λ correction to thezero mode of the Higgs doublet, the masses of the a ± ( kl ) scalars are found to be M a ( kl ) = M W ( kl ) + 3 h + (cid:0) − k + l (cid:1) h π N R . (9) This problem already arises in UED5, but was not discussed inprevious analyses of ¯ B → X s γ for this model [13, 14, 15]. b sγG − ( kl ) Q i ( kl ) Q i ( kl ) b sγ Q i ( kl ) G − ( kl ) G − ( kl ) b sγW − µ ( kl ) Q i ( kl ) Q i ( kl ) b sγ Q i ( kl ) G − ( kl ) W − µ ( kl ) b sγ Q i ( kl ) W − µ ( kl ) G − ( kl ) b sγ Q i ( kl ) W − µ ( kl ) W − µ ( kl ) b sγa − ( kl ) Q i ( kl ) Q i ( kl ) b sγ Q i ( kl ) a − ( kl ) a − ( kl ) b sγW − H ( kl ) Q i ( kl ) Q i ( kl ) b sγ Q i ( kl ) W − H ( kl ) W − H ( kl ) FIG. 1:
One-loop corrections to the b → sγ amplitude in theUED6 model involving the KK modes of the would-be Gold-stone, G ± ( kl ) , the W -boson, W ± µ ( kl ) , and the scalar fields a ± ( kl ) and W ± H ( kl ) . Diagrams where the SU (2) quark doublets Q i ( kl ) are replaced by the SU (2) quark singlets U i ( kl ) are not shown.Here i = u, c, t . See text for details. We will estimate the theoretical uncertainty from the un-specified UV physics by varying the coupling constants h , of the boundary mass terms independently in therange [0 ,
1] which corresponds to either decoupling orstrong coupling.The boundary mass terms could cause mixing amongKK modes and one would need to re-diagonalize the massmatrix to find the eigenstates if they are large. To havea light Higgs boson, we assume that these mixing massterms are tuned to be much smaller than 1 /R , so thatwe can treat them as small perturbations and ignore thehigher-order mixing effects.The small KK scale suggested by dark matter con-straints would lead to interesting signals at the Fer-milab Tevatron and the CERN Large Hadron Collider[19, 26, 27] as well as the International Linear Collider[28]. However, strong bounds on the compactification ra-dius can arise from heavy flavor physics. In particular,the FCNC decay ¯ B → X s γ , which shall be studied in thefollowing, is known to put stringent constraints on vari-ous beyond the SM physics scenarios at the electroweakscale. III. CALCULATION
We work in an effective theory with five active quarks,photons and gluons obtained by integrating out the elec-troweak bosons, the top quark, and all the heavy KKmodes. Adopting the operator basis of Ref. [29], the ef-fective Lagrangian relevant for the b → sγ ( g ) transitionsat a scale µ reads L eff = L QED × QCD + 4 G F √ V ∗ ts V tb X i =1 C i ( µ ) Q i , (10)where the first term is the conventional QED and QCDLagrangian for the light SM particles. In the second term G F and V ij denotes the Fermi coupling constant and theelements of the CKM matrix, respectively, while C i ( µ )are the Wilson coefficients of the corresponding operators Q i build out of the light fields. Terms proportional to thesmall V ub mixing, which will be included in our numericalresults, have been neglected above for simplicity. Thesame refers to higher-order electroweak corrections [30].The operators Q ,..., are the usual four-quark opera-tors whose explicit form can be found in Ref. [29]. Theremaining two operators, characteristic for the b → sγ ( g )transitions, are the dipole operators Q = em b π (¯ s L σ µν b R ) F µν ,Q = gm b π (¯ s L σ µν T a b R ) G aµν . (11)Here e ( g ) is the electromagnetic (strong) coupling con-stant, q L,R are chiral quark fields, F µν ( G aµν ) is the elec-tromagnetic (gluonic) field strength tensor, and T a arethe color generators normalized such that Tr( T a T b ) = δ ab /
2. The factor m b in the definition of Q , denotesthe bottom quark MS mass renormalized at µ .The relevant quantity entering the calculation of B ( ¯ B → X s γ ) is not C ( µ ) but a linear combination C eff7 ( µ ) of this Wilson coefficient and of the coefficients ofthe four-quark operators. The so-called effective Wilsoncoefficients relevant for b → sγ ( g ) are [31] C eff i ( µ ) = C i ( µ ) for i = 1 , . . . , C ( µ ) + P j =1 y j C j ( µ ) for i = 7, C ( µ ) + P j =1 z j C j ( µ ) for i = 8, (12)where y j and z j are chosen so that the LO b → sγ ( g ) ma-trix elements of the effective Lagrangian are proportionalto the LO terms C eff(0)7 , ( µ ). In the MS scheme with fullyanticommuting γ , one has ~y = (0 , , − , − , − , − )and ~z = (0 , , , − , , − ) [29].We further decompose the effective coefficients into aSM and a new physics part C eff i ( µ ) = C eff i SM ( µ ) + ∆ C eff i ( µ ) , i = 1 , . . . , , (13) and expand the latter contribution in powers of α s asfollows ∆ C eff i ( µ ) = ∞ X n =0 (cid:18) α s ( µ )4 π (cid:19) n ∆ C eff( n ) i ( µ ) . (14)In the case of UED6, new physics affects the initialconditions of the Wilson coefficients of the operatorsin the low-energy effective theory while it does not in-duce new operators besides those already present in theSM. To find the LO corrections from the UED6 modelto B ( ¯ B → X s γ ) one has to consider all the one-loopone-particle-irreducible diagrams contributing to the pro-cesses b → sγ ( g ). The one-loop b → sγ diagrams areshown in Fig. 1. Before performing the loop integration,the Feynman integrands are Taylor-expanded up to sec-ond order in the off-shell external momenta and to firstorder in the bottom quark mass. Thereby only termswhich project on Q after the use of the equations ofmotion are retained. The calculation for the b → sg am-plitude proceeds in the same way. The relevant Feynmanrules have been derived from Ref. [18] and implementedinto a model file for FeynArts 3 [32], which has beenused to generate the necessary amplitudes. At tree-level,the interactions between SM and KK fields preserve bothKK numbers. Consequently, only diagrams where all par-ticles in the loop have the same KK index ( kl ) have tobe taken into account.At the matching scale µ = O ( m t ) the LO results forthe UED6 initial conditions read∆ C eff(0) i ( µ ) = i = 1 , . . . , − P k,l ′ A (0) ( x kl ) for i = 7, − P k,l ′ F (0) ( x kl ) for i = 8, (15)where the ′ superscript in the summation indicates thatthe KK sums run only over the restricted range k ≥ l ≥ i.e. P ′ k,l = P k ≥ P l ≥ .We decompose the Inami-Lim functions as X (0) ( x kl ) = X I = W ,a,H X (0) I ( x kl ) , X = A, F , (16)where the function X (0) W ,a,H ( x kl ) describes the contribu-tion due to the exchange of KK modes of the would-beGoldstone, G ± ( kl ) , and the W -bosons, W ± µ ( kl ) , the scalarfields a ± ( kl ) and W ± H ( kl ) . Here x kl = ( k + l ) / ( R M W ).Our results for the LO Inami-Lim functions enteringEq. (16) are given by A (0) W ( x kl ) = x t (cid:0) x t − x t + 3) x kl − x t − x t + 6) x kl + x t (8 x t + 5) − (cid:1) x t − + 12 ( x kl − x kl ln (cid:18) x kl x kl + 1 (cid:19) − ( x kl + x t ) ( x kl + 3 x t − x t − ln (cid:18) x kl + x t x kl + 1 (cid:19) , (17) F (0) W ( x kl ) = x t (cid:0) − x t − x t + 3) x kl − x t − x t + 6) x kl + ( x t − x t − (cid:1) x t − −
32 ( x kl + 1) x kl ln (cid:18) x kl x kl + 1 (cid:19) + 3( x kl + 1)( x kl + x t ) x t − ln (cid:18) x kl + x t x kl + 1 (cid:19) , (18) A (0) a ( x kl ) = x t (cid:0) x kl − x t (2 x t −
9) + 3) x kl + (29 − x t ) x t − (cid:1) x t − −
16 ( x kl − x kl ln (cid:18) x kl x kl + 1 (cid:19) − ( x kl + 3 x t − x t + x kl (( x kl − x t + 4) x t − x t − ln (cid:18) x kl + x t x kl + 1 (cid:19) , (19) F (0) a ( x kl ) = x t (cid:0) − x kl + (cid:0) x t − x t − (cid:1) x kl + (7 − x t ) x t − (cid:1) x t − + 12 x kl ( x kl + 1) ln (cid:18) x kl x kl + 1 (cid:19) + ( x kl + 1)( x t + x kl (( x kl − x t + 4) x t − x t − ln (cid:18) x kl + x t x kl + 1 (cid:19) , (20) A (0) H ( x kl ) = x t (cid:0) (cid:0) x t − x t + 3 (cid:1) x kl − (cid:0) x t − x t + 2 (cid:1) x kl − x t + 29 x t − (cid:1) x t − + 16 x kl (cid:0) x kl − x kl − (cid:1) ln (cid:18) x kl x kl + 1 (cid:19) − ( x kl + 1) (cid:0) x kl + (4 x t − x kl + x t (3 x t − (cid:1) x t − ln (cid:18) x kl + x t x kl + 1 (cid:19) , (21) F (0) H ( x kl ) = − x t (cid:0) (cid:0) x t − x t + 3 (cid:1) x kl + 3 (cid:0) x t − x t + 10 (cid:1) x kl + 2 x t − x t + 11 (cid:1) x t − − x kl ( x kl + 1) ln (cid:18) x kl x kl + 1 (cid:19) + ( x kl + x t )( x kl + 1) x t − ln (cid:18) x kl + x t x kl + 1 (cid:19) . (22)Here x t = m t ( µ ) /M W . Our results for the sums X (0) W ( x kl ) + X (0) a ( x kl ), X = A, F , agree with the expres-sions for the one-loop dipole functions given in Ref. [14].We note that there is a misprint in the last line ofEq. (3.33) of the latter paper. Obviously, the termln(( x n + x t ) / (1 + x t )) should read ln(( x n + x t ) / (1 + x n ))with x n = n / ( R M W ) and n the single KK index ap-pearing in UED5.For the numerical analysis, the results in Eqs. (17) to(22) need to be summed over the KK indices k, l . Thissummation can be performed analytically employing anexpansion for large 1 /R , as explained in App. A. Forzero boundary mass contributions, h , = 0, we obtainthe following approximate formulas X k,l ′ A (0) W ( x kl ) ≈ . . x t x , (23) X k,l ′ F (0) W ( x kl ) ≈ . . x t x , (24) X k,l ′ A (0) a ( x kl ) ≈ − π x t x ln(Λ R ) − . . x t x , (25) X k,l ′ F (0) a ( x kl ) ≈ − π x t x ln(Λ R ) − . . x t x , (26) X k,l ′ A (0) H ( x kl ) ≈ − . . x t x , (27) X k,l ′ F (0) H ( x kl ) ≈ − . . x t x , (28)where x = 1 / ( R M W ) and ∆ x t = x t − (165 / . . Notethat in the above formulas we have only kept the leadingterms in the 1 /x expansion for simplicity. The coeffi-cients of the logarithms in Eqs. (25) and (26) are exactin the limit of an infinite number of KK modes. We em-phasize that the given approximations are for illustrativepurpose only. In our numerical analysis we will through-out employ the exact double series P k,l X (0) I ( x kl ) , X = A, F, I = W, a, H, summed over the restricted range k ≥ l ≥
0, and l + k ≤ N KK .We see from the latter equations that while the one-loop G ± ( kl ) and W ± µ,H ( kl ) corrections to ∆ C eff(0)7 , ( µ ) areinsensitive to the UV cut-off scale Λ or, equivalently, N KK , the contributions due to a ± ( kl ) exchange depend log-arithmically on Λ. The different convergence behavioris closely connected to the unitarity of the CKM matrixwhich results in a Glashow-Iliopoulos-Maiani (GIM) sup-pression [33] of the higher KK mode contributions to thedouble sums in Eqs. (23) to (28). In the case at hand,the GIM mechanism leads to a hierarchy of the variouscontributions to ∆ C eff(0)7 , ( µ ), with X (0) W ,H ( x kl ) propor-tional to 1 / ( k + l ) and X (0) a ( x kl ) , X = A, F, scalinglike 1 / ( k + l ) for large values of l, k . The extra powerof k + l in the contribution from diagrams with a ± ( kl ) exchange, that leads to the logarithmic divergent results,stems from the left- (right-handed) top quark Yukawacoupling enhanced part of the a +( kl ) ¯ U t ( kl ) b ( a − ( kl ) ¯ s U t ( kl ) )tree-level vertex. No such terms are present in the flavor-changing vertices involving G ± ( kl ) and W ± µ,H ( kl ) .The logarithmic divergences appearing in Eqs. (25) and(26) would be cancelled by counterterms at the scaleΛ at which perturbativity is lost in the higher dimen-sional theory. Our calculation only determines the lead-ing logarithmic corrections associated with the renormal-ization group (RG) running between Λ and 1 /R . Thecorresponding initial conditions contain incalculable fi-nite matching corrections from the unknown UV physics.Assuming that the RG effects dominate over the finitematching corrections and that the UV completion ofthe UED6 model has a CKM-type flavor structure, theUV sensitivity can be absorbed into a logarithmic de-pendence on Λ R or, equivalently, N KK . To gauge thetheoretical uncertainty associated with the unknown UVcompletion we will vary N KK in the range [5 ,
15] around N KK = Λ R ≈
10 as estimated by NDA. The choice of thelower value of N KK is motivated by the observation thatfor N KK < P k,l X (0) a ( x kl ), X = A, F , become numerically of the same size as thelogarithmic ones. Since the choice of the upper value of N KK has no impact on our conclusions we choose it sym-metrically. We mention that the requirement of unitarityof gauge boson scattering at high energies [34] genericallyleads to values of Λ R notably below the NDA estimate N KK ≈ I C eff(0)7 , ( µ ), I = W, a, H , to the UED6 initial conditions of the dipole op-erators as a function of 1 /R are shown in Fig. 2. Thecontribution due to the exchange of G ± ( kl ) and W ± µ ( kl ) and W ± H ( kl ) (green/medium gray) KK modes are de-picted as yellow (light gray) and green (medium gray)curves, while the red (dark gray) bands and the black (cid:144) R @ TeV D D I C e ff H L H Μ L (cid:144) R @ TeV D D I C e ff H L H Μ L FIG. 2: ∆ I C eff(0)7 , ( µ ) as a function of /R . The different curvescorrespond to the individual contributions due to the exchange ofKK modes of the would-be Goldstone, G ± ( kl ) , and the W -bosons, W ± µ ( kl ) (yellow/light gray), the scalar fields a ± ( kl ) (black) and W ± H ( kl ) (green/medium gray), respectively. The lower (upper)borders of the red (dark gray) bands correspond to N KK = 5 (15) while the black lines represent the results for N KK = 10 . Seetext for details. lines illustrate the a ± ( kl ) corrections. The lower (up-per) borders of the red (dark gray) bands correspond to N KK = 5 (15) while the black lines represent the resultsfor N KK = 10. We see that in both cases the contribu-tion involving a ± ( kl ) exchange is by far dominant and itsvariation with N KK is non-negligible. Nevertheless, thelarge positive corrections to ∆ C eff(0)7 , ( µ ) already startto exceed the SM values C eff(0)7 , ( µ ) ≈ − . , − .
10 inmagnitude for 1 /R ≈ ,
335 GeV in the most conser-vative case N KK = 5. The observed strong enhancementof the initial conditions C eff(0)7 , ( µ ) will play the key rolein our phenomenological applications. For compactification scales 1 /R ≈
100 GeV it would even bepossible to reverse the sign of C eff7 ( µ b ) with respect to its SMvalue C eff7 ( µ b ) ≈ − .
37. This possibility is disfavored on generalgrounds by the experimental information on ¯ B → X s l + l − [35]. (cid:144) R @ TeV D B I B ® X s Γ M A - E FIG. 3: B ( ¯ B → X s γ ) for E = 1 . as a function of /R .The red (dark gray) band corresponds to the UED6 result. The CL range and central value of the experimental/SM resultis indicated by the yellow/green (light/medium gray) band un-derlying the straight solid line. See text for details.
Another main observation of our work is, that in theUED6 model the Z - (∆ C ), photon (∆ D ), gluon pen-guin (∆ E ), and the | ∆ F | = 2 boxes (∆ S ) all behaveas 1 / ( k + l ) for large values of k, l . In contrast, | ∆ F | = 1 boxes (∆ B νν,ll ) show an asymptotic 1 / ( k + l ) behavior after GIM. The corresponding UED6 Inami-Lim functions therefore exhibit the following behavior:∆ C, ∆ S ∝ x t /x ln(Λ R ), ∆ D, ∆ E ∝ x t /x ln(Λ R ),and ∆ B νν,ll ∝ x t /x . This implies that the logarith-mic cut-off sensitivity first seen in Eqs. (25) and (26) isa generic feature of all FCNC transitions in the UED6model. A dedicated study of neutral meson mixing, rare K - and B -decays in UED6 is left for further work. IV. NUMERICS
The UED6 prediction of B ( ¯ B → X s γ ) for E =1 . /R is displayed by the red (darkgray) band in Fig. 3. The yellow (light gray) and green(medium gray) band in the same figure shows the experi-mental and SM result as given in Eqs. (1) and (2), respec-tively. In all three cases, the middle line is the centralvalue, while the widths of the bands indicate the uncer-tainties that one obtains by adding errors in quadrature.The central value of the UED6 prediction correspondsto N KK = 10 and h , = 0. The strong suppression of B ( ¯ B → X s γ ) in the UED6 model with respect to theSM expectation and the slow decoupling of KK modes isclearly seen in Fig. 3.In our numerical analysis, matching of the UED6 Wil-son coefficients at the electroweak scale is complete up toleading logarithmic order, while terms beyond that orderinclude SM contributions only. For the reference values ofthe renormalization scales µ , µ b , µ c = 160 , . , .
25 GeV, we utilize the formula B ( ¯ B → X s γ ) = h . ± . − .
03 ∆ C eff(0)7 ( µ ) − .
92 ∆ C eff(0)8 ( µ )+ 4 . (cid:0) ∆ C eff(0)7 ( µ ) (cid:1) + 0 . (cid:0) ∆ C eff(0)8 ( µ ) (cid:1) + 2 .
33 ∆ C eff(0)7 ( µ )∆ C eff(0)8 ( µ ) i × − , (29)which has been derived based on the NNLO SM resultsof Refs. [8, 10, 36]. For the remaining input parameterswe adopt the central values and error ranges that can befound in Ref. [8].The theoretical uncertainty in the UED6 model is esti-mated by scanning N KK , the couplings h , of the bound-ary mass terms, and the matching scale µ in the range[5 , , , +17 − % for 1 /R in the range [0 . , .
0] TeV. Larger rela-tive errors of above +55 − % appear for 1 /R = 300 GeV.Whether the quoted numbers provide a reliable estimateof the cut-off and higher-order corrections to B ( ¯ B → X s γ ) in the UED6 model can only be seen by perform-ing a next-to-leading order (NLO) matching calculation.Such a calculation seems worthwhile but is beyond thescope of this work. The parametric uncertainty due tothe error on the top quark mass is below +1 − % for 1 /R in the range [0 . , .
0] TeV and thus notably smaller thanthe combined theory uncertainty.Since the experimental result is at present above theSM one and KK modes in the UED6 model necessarilyinterfere destructively with the SM b → sγ amplitude,the lower bound on 1 /R following from B ( ¯ B → X s γ )turns out to be much stronger than what one can derivefrom any other currently available direct measurement[26]. If experimental, parametric, and theory uncertain-ties are treated as Gaussian and combined in quadrature,the 95% CL bound amounts to 650 GeV. In contrast tothe upper limit coming from the dark matter abundancethe latter exclusion is almost independent of the Higgsmass because genuine electroweak effects related to Higgsboson exchange enter B ( ¯ B → X s γ ) first at the two-looplevel. In the SM these corrections have been calculated[30] and amount to around − .
5% in the branching ra-tio. They are included in Eq. (29). Neglecting the cor-responding two-loop Higgs effects in the UED6 modelcalculation should therefore have practically no influenceon the derived limits.The upper (lower) contour plot in Fig. 4 shows the 95%CL bound of 1 /R as a function of the experimental (SM)central value and error. The current experimental worldaverage and SM prediction of Eqs. (1) and (2) are indi-cated by the black squares. These plots allow to monitorthe effect of future improvements in both the measure-ments and the SM prediction. Of course, one shouldkeep in mind that the derived bounds depend in a non-negligible way on the treatment of theoretical uncertain-ties. Furthermore, the found limits could be weakened by < > B I B ® X s Γ L exp A - E D B I B ® X s Γ L e xp A - E < > B I B ® X s Γ L SM A - E D B I B ® X s Γ L S M A - E FIG. 4:
The upper/lower panel displays the
CL limits on /R as a function of the experimental/SM central value (hori-zontal axis) and total error (vertical axis). The experimental/SMresult from Eq. (1)/Eq. (2) is indicated by the black square. Thecontour lines represent values that lead to the same bound in TeV . See text for details. the NLO matching corrections in the UED6 model whichremain unknown.
V. CONCLUSIONS
We have calculate the leading order corrections to theinclusive radiative ¯ B → X s γ decay in the standard modelwith two universal extra dimensions. While the one-loop matching corrections associated to the exchange ofKaluza-Klein modes of the would-be Goldstone, G ± ( kl ) ,the W -boson, W ± µ ( kl ) , and the physical scalar W ± H ( kl ) areinsensitive to the ultraviolet physics, we find that contri-butions involving a ± ( kl ) scalars depend logarithmically onthe cut-off scale Λ. We have emphasized that in the con-sidered model all flavor-changing neutral current transi-tions suffer from this problem already at leading order.Moreover, we have included formally next-to-leading, butsizeable mass corrections to the Kaluza-Klein scalars thatdepend quadratically on the scale Λ. Although the ultra-violet sensitivity weakens the lower bound on the inversecompactification radius 1 /R that can be derived from themeasurements of the ¯ B → X s γ branching ratio, a strongconstraint of 1 /R >
650 GeV at 95% confidence level isfound if errors are added in quadrature. Our bound ex-ceeds by far the limits that can be derived from any otherdirect measurement, and is at variance with the parame-ter region preferred by the dark matter abundance. Thisonce again underscores the outstanding role of the inclu-sive radiative ¯ B -meson decay in searches for new physicsclose to the electroweak scale. Acknowledgments
We are grateful to Miko laj Misiak and Matthias Stein-hauser for private communications concerning Eq. (29).Helpful discussions with Bogdan Dobrescu and Giu-lia Zanderighi are acknowledged. ANL is supported bythe U.S. Department of Energy, Division of High EnergyPhysics, under Contract DE-AC02-06CH11357. Thiswork was initiated when U. H. was supported by theSwiss Nationalfonds. He is grateful to the University ofZ¨urich for the pleasant working environment during thattime.
APPENDIX A: EVALUATION OF KK SUMS
Here we show how to approximate the double sum over KK levels ( kl ) appearing in Eq. (15). Following Ref. [25],we first introduce the integrals I n ( a ) = ( − n a n +1 Z dy y n ay + x kl , (A1)where n = 0 , , . . . , and x kl = ( k + l ) x with x = 1 / ( R M W ). Obviously, I n (0) = 0. These integrals allow use toexpress the logarithms appearing in Eqs. (17) to (22) asln (cid:18) x kl + ax kl + 1 (cid:19) = I ( a ) − I (1) ,x kl ln (cid:18) x kl + ax kl + 1 (cid:19) = I ( a ) − I (1) − a ,x kl ln (cid:18) x kl + ax kl + 1 (cid:19) = I ( a ) − I (1) + 12 − x kl + x kl a − a ,x kl ln (cid:18) x kl + ax kl + 1 (cid:19) = I ( a ) − I (1) −
13 + 12 x kl − x kl + x kl a − x kl a + 13 a , (A2)with a = 0 or x t . We note that Eq. (D.3) of Ref. [25] is missing an overall minus sign on its right-hand side.Since the individual building blocks I n ( a ) behave as 1 / ( k + l ) for large k, l , the corresponding double series overthe KK levels diverge logarithmically. We regulate the appearing divergence analytically I δn ( a ) = ( − n a n +1 Z dy y n ( ay + x kl ) δ , (A3)with δ >
0. Then one has X k,l ′ I δn ( a ) = ( − n a n +1 ∞ X k =1 ∞ X l =0 Z dy y n ( ay + x kl ) δ = ( − n a n +1 Γ(1 + δ ) ∞ X k =1 ∞ X l =0 Z dy y n Z ∞ dt t δ e − ( ay + x kl ) t = ( − n a n +1 δ ) Z dy y n Z ∞ dt t δ (cid:16) ϑ (cid:0) , e − xt (cid:1) − (cid:17) e − ayt = ( − n δ ) Z ∞ dt t − − n + δ (cid:16) ϑ (cid:0) , e − xt (cid:1) − (cid:17) (cid:0) Γ(1 + n ) − Γ(1 + n, at ) (cid:1) , (A4)where in the first step we have used the Mellin-Barnes representation1 s δ = 1Γ(1 + δ ) Z ∞ dt t δ e − st . (A5)Here ϑ ( u, q ) = 1 + 2 P ∞ m =1 q m cos(2 mu ), Γ( z ) = R ∞ dt t z − e − t and Γ( u, z ) = R ∞ z dt t u − e − t , denotes the elliptictheta, the Euler gamma, and the plica function, respectively.The integration over t in the last line of Eq. (A4) cannot be performed analytically. Yet using ϑ (cid:0) , e − z (cid:1) ≈ r πz , for z ≤ √ π ,1 + 2 n +1 X m =1 e − m z , for z > √ π , (A6)and expanding the integrand in powers of 1 /t in the latter case, we can perform the integration piecewise andapproximate the double series as X k,l ′ I δn ( a ) ≈ l δn ( a ) + h n ( a ) . (A7)0The integration over t ∈ [0 , √ π/x ] leads to the relatively compact formulas l δn ( a ) = ( − n πa n +1 n + 1) x δ + x " √ πxE (cid:18) a √ πx (cid:19) − x (cid:18) (cid:18) , a √ πx (cid:19) + ln (cid:18) a πx (cid:19)(cid:19) + 2 (cid:0) πa (1 − ln a ) − (cid:0) √ π + γ E (cid:1) x (cid:1) , for n = 0,( − n n ( n + 1) x " e − a √ πx ( n + 1) xa n − a n (cid:18) x ( n + 1) + πan ( n + 1) (cid:18) Γ (cid:18) , a √ πx (cid:19) + ln( a ) (cid:19) − πan (cid:19) + ( n + 1) (cid:0) n (cid:0) − √ π (cid:1) + 1 (cid:1) π − n/ x n +1 × (cid:18) Γ( n + 1) − Γ (cid:18) n + 1 , a √ πx (cid:19)(cid:19) , for n = 1 , , . . . , (A8)where we have expanded the result around δ = 0 and dropped all terms that vanish in the limit δ →
0. Furthermore, E m ( z ) = R ∞ dt t − m e − zt and γ E ≈ . t ∈ ( √ π/x, ∞ ) is finite in the limit δ →
0. For all double sums P ′ k,l I n ( a ) appearing in Eq. (A2)we were able to find analytic expressions. Since the results turn out to be rather lengthy and not very informativewe refrain from giving them here. Short numerical expressions for the h n ( a ) can be obtained in the large x limit.Keeping terms up to third order in 1 /x , we find h n ( a ) = . ax − . a x + 0 . a x , for n = 0, − . a x + 0 . a x − . a x , for n = 1,0 . a x − . a x + 0 . a x , for n = 2, − . a x + 0 . a x − . a x , for n = 3. (A9)Combining Eqs. (A8) and (A9) we finally arrive at the following large x approximations X k,l ′ I δn ( a ) ≈ ( − n πa n +1 n + 1) x (cid:18) δ − ln x (cid:19) + . ax − . a x + 0 . a x , for n = 0, − . a x + 0 . a x − . a x , for n = 1,0 . a x − . a x + 0 . a x , for n = 2, − . a x + 0 . a x − . a x , for n = 3. (A10)The term 1 /δ − ln x in Eq. (A10) implies that one should include counterterm contributions from physics at the UVcut-off scale Λ that cancel the divergences. Our calculation only determines the RG running contribution between Λand 1 /R , given initial conditions at Λ. Assuming that the unknown finite matching corrections are small and have aCKM-type flavor structure, the divergences can be absorbed into a cut-off dependence by switching from analytic tocut-off regularization employing the approximation1 δ − ln x ≈ ln(Λ R ) , (A11)1with Λ not much larger than 1 /R . We remark that the latter assumptions are self-consistent because the finitematching corrections are formally of next-to-leading logarithmic order. [1] For a recent review see U. Haisch, 0706.2056 [hep-ph].[2] S. Chen et al . [CLEO Collaboration], Phys. Rev. Lett. (2001) 251807.[3] P. Koppenburg et al . [Belle Collaboration], Phys. Rev.Lett. , 061803 (2004).[4] B. Aubert et al . [BaBar Collaboration], Phys. Rev. Lett. , 171803 (2006).[5] E. Barberio et al et al . [BaBar Collaboration], 0711.4889 [hep-ex].[7] K. Bieri, C. Greub and M. Steinhauser, Phys. Rev. D , 114019 (2003); M. Misiak and M. Steinhauser, Nucl.Phys. B , 277 (2004); M. Gorbahn and U. Haisch,Nucl. Phys. B , 291 (2005); M. Gorbahn, U. Haischand M. Misiak, Phys. Rev. Lett. , 102004 (2005);K. Melnikov and A. Mitov, Phys. Lett. B , 69 (2005);I. Blokland et al ., Phys. Rev. D , 033014 (2005);H. M. Asatrian et al ., Nucl. Phys. B , 325 (2006)and , 212 (2007).[8] M. Misiak and M. Steinhauser, Nucl. Phys. B , 62(2007).[9] M. Czakon, U. Haisch and M. Misiak, JHEP , 008(2007).[10] M. Misiak et al ., Phys. Rev. Lett. , 022002 (2007).[11] H. M. Asatrian et al ., Phys. Lett. B , 173 (2007).[12] R. Boughezal, M. Czakon and T. Schutzmeier, JHEP , 072 (2007).[13] K. Agashe, N. G. Deshpande and G. H. Wu, Phys. Lett.B , 309 (2001).[14] A. J. Buras et al ., Nucl. Phys. B , 455 (2004).[15] U. Haisch and A. Weiler, Phys. Rev. D , 034014 (2007).[16] T. Appelquist, H. C. Cheng and B. A. Dobrescu, Phys.Rev. D , 035002 (2001).[17] I. Gogoladze and C. Macesanu, Phys. Rev. D , 093012(2006). [18] G. Burdman, B. A. Dobrescu and E. Ponton, JHEP , 033 (2006).[19] G. Burdman, B. A. Dobrescu and E. Ponton, Phys. Rev.D , 075008 (2006).[20] B. A. Dobrescu and E. Ponton, JHEP , 071 (2004);M. Hashimoto and D. K. Hong, Phys. Rev. D , 056004(2005).[21] B. A. Dobrescu et al ., JCAP , 012 (2007).[22] E. Ponton and L. Wang, JHEP , 018 (2006).[23] H. Georgi, A. K. Grant and G. Hailu, Phys. Lett. B ,207 (2001).[24] H. C. Cheng, K. T. Matchev and M. Schmaltz, Phys.Rev. D , 036005 (2002).[25] A. J. Buras, M. Spranger and A. Weiler, Nucl. Phys. B , 225 (2003).[26] B. A. Dobrescu, K. Kong and R. Mahbubani, JHEP , 006 (2007).[27] B. A. Dobrescu, K. Kong and R. Mahbubani, 0709.2378[hep-ph].[28] A. Freitas and K. Kong, 0711.4124 [hep-ph].[29] K. G. Chetyrkin, M. Misiak and M. M¨unz, Phys. Lett. B , 206 (1997) [Erratum-ibid. B , 414 (1998)].[30] P. Gambino and U. Haisch, JHEP , 001 (2000) and , 020 (2001).[31] A. J. Buras et al ., Nucl. Phys. B , 374 (1994).[32] T. Hahn, Comput. Phys. Commun. , 418 (2001).[33] S. L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D , 1285 (1970).[34] R. S. Chivukula et al ., Phys. Lett. B , 109 (2003).[35] P. Gambino, U. Haisch and M. Misiak, Phys. Rev. Lett. , 061803 (2005); C. Bobeth et al ., Nucl. Phys. B ,252 (2005); U. Haisch and A. Weiler, Phys. Rev. D76