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ANL-HEP-PR-08-04;MZ-TH/08-03; ZU-TH-01-08 B¯ → X γ in two universal extra dimensions s Ayres Freitas1 and Ulrich Haisch2 1 Department of Physics and Astronomy, University of Pittsburgh, PA 15260, USA and Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA and HEP Division, Argonne National Laboratory, Argonne, IL 60439, USA 2Institut fu¨r Physik (THEP), Johannes Gutenberg-Universität D-55099 Mainz, Germany and Institut fu¨r Theoretische Physik, Universität Zu¨rich, CH-8057 Zu¨rich, Switzerland (Dated: February 5, 2008) We calculate the leading order corrections to the B¯ → Xsγ decay in the standard model with two large flat universal extra dimensions. We findthat thecontributions involving theexchange of Kaluza-Kleinmodesofthephysicalscalarfielda± dependlogarithmicallyontheultravioletcut-off (kl) scaleΛ. Weemphasizethatallflavor-changingneutralcurrenttransitionssufferfromthisproblem. 8 Althoughtheultravioletsensitivityweakensthelowerboundontheinversecompactificationradius 00 1/R that follows from B¯ → Xsγ, the constraint remains stronger than any other available direct measurement. After performing a careful study of the potential impact of cut-off and higher-order 2 effects, we find 1/R >650GeV at 95% confidence level if errors are combined in quadrature. Our n limitisatvariancewiththeparameterregion1/R.600GeVpreferredbydarkmatterconstraints. a J PACSnumbers: 12.15.Lk,12.60.-i,13.25.Hw 8 2 I. INTRODUCTION where the uncertainties from hadronic power corrections ] ( 5%), higher-order perturbative effects ( 3%), the in- h ± ± p ThebranchingratiooftheinclusiveradiativeB¯-meson terpolation in the charm quark mass (±3%), and para- - decay is known to provide stringent constraints on vari- metric dependences ( 3%) have been added in quadra- p ture to obtain the tot±al error. ousnon-standardphysicsmodelsattheelectroweakscale e h [1], because it is accurately measured and its theoretical Compared with the experimental world average of [ determination is rather precise. Eq.(1), the new SM prediction of Eq.(2) is lower by The present experimental world average, which in- 1.2σ. Potential beyond SM contributions should now 1 v cludes the latest measurements by CLEO [2], Belle [3], be preferably constructive, while models that lead to a 6 and BaBar [4], is performed by the Heavy Flavor Av- suppression of the b sγ amplitude are more severely → 4 eraging Group [5] and reads for a photon energy cut of constrained than in the past, where the theoretical de- 3 E >E with E =1.6GeV in the B¯-meson rest-frame1 termination used to be above the experimental one. γ 0 0 4 As emphasized in Refs. [13, 14, 15], among the lat- . 1 (B¯ X γ) =(3.55 0.24+0.09 0.03) 10−4. (1) ter category is the model with a flat, compactified extra 0 B → s exp ± −0.10± × dimensionwhereallofthe SMfieldsareallowedtoprop- 8 Here the first error is a combined statistical and sys- agate in the bulk [16], known as minimal universalextra 0 tematic one, while the second and third are system- dimensions or UED5. Since Kaluza-Klein (KK) modes : v atic uncertainties due to the extrapolation from E = in the UED5 model interfere destructively with the SM 0 Xi (1.8 2.0)GeV to the reference value and the subtrac- b sγ amplitude, the (B¯ Xsγ) constraint leads to tion−of the B¯ X γ event fraction, respectively. a→very powerful boundBon t→he inverse compactification ar After a join→t effdort [7, 8, 9], the first theoretical es- radius of 1/R > 600GeV at 95% confidence level (CL) timate of the total B¯ X γ branching ratio at next- [15]. This exclusion is independent from the Higgs mass s to-next-to-leading order→(NNLO) in QCD has been pre- andthereforestrongerthananylimitthatcanbederived sented recently in Refs. [8, 10]. For E = 1.6GeV the from electroweak precision measurements [17]. 0 result of the improved standard model (SM) evaluation Thepurposeofthisarticleistostudythephenomenol- is given by2 ogy of B¯ X γ in the SM with two universal extra s → dimensions [18, 19] or UED6. In contrast to UED5, the (B¯ X γ) =(3.15 0.23) 10−4, (2) UED6modelhasadditionalKKparticlesinitsspectrum. s SM B → ± × An interesting feature of this model is the fact that dark matterconstraintssuggestarathersmallKKmassscale. Therefore it is very interesting to derive a bound on this 1 The very recent measurement of BaBar [6] that gives B(B¯ → scalefromb sγ in UED6,taking into accountthe new Xsγ)=(3.66±0.85stat±0.60syst)×10−4 for E0 =1.9GeV is KKmodes. I→nthiscontext,severalquestionswillneedto nottakenintoaccount intheaverageofEq.(1). be answered: Does the leading order (LO) result depend 2 ThesmallNNLOcorrectionsrelatedtothefour-loopb→sgmix- onthecut-offscale,incontrasttoUED5wherenocut-off ingdiagrams[9]andfromquarkmasseffects totheelectromag- dependence was found? If so, is this a generic feature of neticdipole[11]andcurrent-currentoperator [12]contributions arenotincludedinEq.(2). all flavor-changing neutral current (FCNC) amplitudes 2 intheUED6model? Whatisthetheoreticaluncertainty vorconserving,sothattheCabibbo-Kobayashi-Maskawa stemmingfromtheunknownultraviolet(UV)dynamics? (CKM) matrix remains the only source of flavor viola- This article is organized as follows. In Secs.II and tion. In this work, we concentrate on the leading order III we describe, first, the model itself and, second, the contributions from the UED6 model to B¯ X γ, us- s → calculation of the one-loop matching corrections to the ing tree-level masses for those KK excitations which re- Wilson coefficients of the electro- and chromomagnetic ceive only logarithmic corrections from loop corrections dipole operators in UED6. Sec.IV contains a numerical andboundarytermslocalizedattheorbifoldfixedpoints analysisof (B¯ X γ)andthelowerboundonthecom- [19, 22, 23, 24]. This is justified since these terms are of s B → pactification scale 1/R in the UED6 model. Concluding one-loop order, thus leading to next-to-leading order ef- remarks are given in Sec.V. In App.A we show how to fects for B¯ X γ. s → compute the double sums over KK modes appearing in Uponcompactification,thesix-componentgaugefields the calculation of B¯ →Xsγ. WMa ,M =0,...,5,decomposeintofour-componentmas- sive KK vector bosons Wa , µ = 0,...,3, and two µ(kl) scalar KK fields Wa . Here a denotes the adjoint II. MODEL 4,5(kl) group index. Following Refs. [18, 25], a covariant gauge fixing is introduced, such that Wa do not mix with µ(kl) Here we briefly summarize the main features of the Wa . In the six-dimensional formulation, the gauge UED6scenario. AllSMfieldspropagateintwoflatextra 4,5(kl) fixing-term reads dimensions, compactified on a square with side length L = πR and adjacent sides being identified [20]. This compactification, aptly dubbed chiral square, leads to LGF =− 21ξ ∂µWµa−ξ(∂4W4a+∂5W5a−g6v6χa) 2 chiralfermionzeromodes,whilethehigherKKmodesof (3) (cid:2) (cid:3) the fermionsarevector-likeasusual. Since the geometry 1 ∂µB ξ′(∂ B +∂ B +g′v χ3) 2 , is invariant under rotations by 180◦ about the center of − 2ξ′ µ− 4 4 5 5 6 6 the square, the model respects an additional Z symme- (cid:2) (cid:3) 2 try. Itimplies thatthe lightestKK-oddparticleis stable where W,B are the uncompactified SU(2) and U(1) and could provide a viable dark matter candidate for a gaugefieldswiththesix-dimensionalgaugecouplingsg(′), 6 small KK scale 1/R.600GeV [21]. and ξ(′) are the gauge parameters. The χa are the com- Solving the six-dimensional equations of motion leads ponents of the six-dimensional Higgs doublet to anorthonormalsetoffunctions, whichdepend ontwo KK indices k,l corresponding to the two extra dimen- 1 χ2+iχ1 H = . (4) csioomnse,swstirthonkgl≥y i1n,tle≥rac0tionrgkat=hlig=h0en[1e8rg].yTschaelems,osdoelthbaet- √2(cid:18)v6+h+iχ3(cid:19) itisviewedasalow-energyeffectivetheorywhichisvalid The six-dimensionalgaugecouplings and vacuum expec- uptosomecut-offscaleΛ. Fromnaivedimensionalanal- tationvaluearerelatedtothefour-dimensionalvaluesby ycosirsre(sNpDonAd)in[1g9t],otahnisuspcapleerislimesittimNated tokb+e Λl ≈1100/fRor, g6(′) =g(′)πR and v6 =v/R. KK ≤ ≈ TheHiggsscalarsmixwiththefourthandfifthcompo- the KK indices. nent of the gauge fields to form the would-be Goldstone Before electroweaksymmetry breaking,all (kl) modes bosons Ga of the massive vector bosons Wa , and have degenerate tree-level masses m = √k2+l2/R. (kl) µ(kl) (kl) two physical scalars aa and Wa . Only the would- The degeneracy is lifted by loop corrections, which lead (kl) H(kl) beGoldstonebosonshavezeromodesGa ,whichcorre- to mass operators localized at the corners of the chi- (00) ral square [19, 22]. Additional flavor diagonal and non- spondtothe usualcomponentsofthe SMHiggsdoublet. diagonal contributions can originate from physics at the For k+l ≥1, the Ga(kl) are dominated by the scalar ad- UV cut-off scale. Since flavor non-universal operators joints W4a,5(kl) and B4,5(kl) while the aa(kl) are composed wouldingeneralleadtounacceptablylargeFCNCtransi- mostly of the Higgs doublet elements. For the charged tions,wewillassumethatthelocalizedoperatorsarefla- fields one finds 1 1 G± = lW± kW± +M χ± , (kl) M(kl) R 4(kl)− 5(kl) W (kl) W (cid:20) (cid:16) (cid:17) (cid:21) 1 M a± = m χ± W lW± kW± , (5) (kl) M(kl) (kl) (kl)− m(kl)R 4(kl)− 5(kl) W h (cid:16) (cid:17)i 1 W± = kW± +lW± , H(kl) √k2+l2 4(kl) 5(kl) h i 3 where G− s i s W− s (kl) Q(kl) µ(kl) X± = X1∓iX2 , X =W,χ,G,a,WH. (6) b Qi(kl) b G−(kl) b Qi(kl) √2 i G− i Q(kl) γ (kl) γ Q(kl) γ Here M2 = m2 + M2 is the tree-level squared W(kl) (kl) W s s s mass of the W± and a± . The would-be Gold- Qi(kl) Qi(kl) Qi(kl) µ,H(kl) (kl) stone bosons G± receive the unphysical squared mass Wµ−(kl) G−(kl) Wµ−(kl) (kl) b b b ξMW2 (kl) from gauge fixing. Similar expressions hold for G−(kl) γ Wµ−(kl) γ Wµ−(kl) γ the neutralfields,takingintoaccountasmallmixingbe- tween W(3kl) and B(kl). However, since they do not con- a−(kl) s Qi(kl) s tribute to the process B¯ Xsγ at leading order in the i a− electroweak interactions w→e do not give them here. b Q(kl) b (kl) i a− As mentioned above, the masses of the KK modes re- Q(kl) γ (kl) γ ceivecorrectionsfromloopandUVeffects,whicharede- pendent on the cut-off scale Λ. Since G± and W± W− s i s (kl) µ,H(kl) H(kl) Q(kl) are protected by gauge invariance, the dependence on i W− Λ is only logarithmic [22], so that the mass corrections b Q(kl) b H(kl) i W− are small compared to 1/R and can be neglected in a Q(kl) γ H(kl) γ LO calculation. The a± scalars, however, can receive (kl) contributions proportional to Λ2 to both their bulk and FIG. 1: One-loop corrections to the b → sγ amplitude in the boundary mass terms [24]. UED6 model involving the KK modes of the would-be Gold- Inordertoobtainasmallmasstermforthezeromode stone, G± , the W-boson, W± , and the scalar fields a± (kl) µ(kl) (kl) Higgs doublet, the bulk and boundary mass terms need and W± . Diagrams where the SU(2) quark doublets Qi H(kl) (kl) to be tuned to cancel to a large extent. However, inde- are replaced by the SU(2) quark singlets Ui are not shown. (kl) pendentofthis tuning, the higherKKmodescanreceive Here i=u,c,t. Seetext for details. sizeable contributions from these terms. As a result, the a± scalarscanbeheavierorlighterthantheotherparti- (kl) clesofthesameKKlevel.3 WeincludetheΛ2corrections Wewillestimatethetheoreticaluncertaintyfromtheun- to the a± masses based on the following parametriza- specified UV physics by varying the coupling constants (kl) h of the boundary mass terms independently in the tion of the UV-induced mass terms: 1,2 range [0,1] which corresponds to either decoupling or L2 strong coupling. δ(x )δ(x )+δ(L x )δ(L x ) m2 L ⊃ 2 4 5 − 4 − 5 H,1 The boundary mass terms could cause mixing among (cid:20) (cid:0) (cid:1) (7) KKmodesandonewouldneedtore-diagonalizethemass L2 + δ(x )δ(L x )m2 +m2 H 2. matrix to find the eigenstates if they are large. To have 2 4 − 5 H,2 H,bulk | | a light Higgs boson, we assume that these mixing mass (cid:21) terms are tuned to be much smaller than 1/R, so that AlthoughtheUV physicsis notspecified, thesemasspa- we cantreat them as smallperturbations and ignore the rametersareexpectedtostemfromloopcontributionsof higher-order mixing effects. the UV dynamics, so that The small KK scale suggested by dark matter constraints would lead to interesting signals at the Fer- h2 h2 N2 m2 = i Λ2 = i KK , i=1,2, (8) milab Tevatron and the CERN Large Hadron Collider H,i 16π2 16π2 R2 [19, 26, 27] as well as the International Linear Collider [28]. However,strongboundsonthecompactificationra- with h = (1). Using the explicit form of the KK 1,2 O dius can arise from heavy flavor physics. In particular, wave functions from Refs. [18, 20] and tuning the bulk theFCNCdecayB¯ X γ,whichshallbestudiedinthe mass m2H,bulk to exactly cancel the Λ2 correction to the following, is known→to pust stringent constraints on vari- zero mode of the Higgs doublet, the masses of the a± (kl) ous beyond the SM physics scenarios at the electroweak scalars are found to be scale. 3h2+ 1+( 2)k+l h2 N2 M2 =M2 + 1 − 2 KK . (9) a(kl) W(kl) (cid:0) 16π2 (cid:1) R2 III. CALCULATION We workinaneffective theorywithfiveactive quarks, 3 This problem already arises inUED5, but was not discussed in photonsandgluonsobtainedbyintegratingoutthe elec- previousanalyses ofB¯→Xsγ forthismodel[13,14,15]. troweak bosons, the top quark, and all the heavy KK 4 modes. Adopting the operator basis of Ref. [29], the ef- and expand the latter contribution in powers of α as s fective Lagrangianrelevant for the b sγ(g) transitions follows → at a scale µ reads ∞ n α (µ) 8 ∆Ceff(µ)= s ∆Ceff(n)(µ). (14) Leff =LQED×QCD+ 4√G2FVt∗sVtb Ci(µ)Qi, (10) i nX=0(cid:18) 4π (cid:19) i i=1 X In the case of UED6, new physics affects the initial where the first term is the conventional QED and QCD conditions of the Wilson coefficients of the operators LagrangianforthelightSMparticles. Inthesecondterm in the low-energy effective theory while it does not in- G andV denotesthe Fermicouplingconstantandthe F ij duce new operators besides those already present in the elements of the CKM matrix, respectively, while C (µ) i SM. To find the LO corrections from the UED6 model aretheWilsoncoefficientsofthecorrespondingoperators to (B¯ X γ) one has to consider all the one-loop s Q buildoutofthelightfields. Termsproportionaltothe B → i one-particle-irreduciblediagramscontributingtothepro- smallV mixing,whichwillbeincludedinournumerical ub cesses b sγ(g). The one-loop b sγ diagrams are results, have been neglected above for simplicity. The → → shown in Fig.1. Before performing the loop integration, same refers to higher-order electroweak corrections [30]. the Feynman integrands are Taylor-expanded up to sec- The operators Q are the usual four-quark opera- 1,...,6 ond order in the off-shell external momenta and to first tors whose explicit form can be found in Ref. [29]. The order in the bottom quark mass. Thereby only terms remainingtwooperators,characteristicfortheb sγ(g) → which project on Q7 after the use of the equations of transitions, are the dipole operators motion are retained. The calculation for the b sg am- → em plitudeproceedsinthesameway. TherelevantFeynman Q = b (s¯ σµνb )F , 7 16π2 L R µν rules have been derived from Ref. [18] and implemented (11) into a model file for FeynArts 3 [32], which has been gm Q8 = 16πb2(s¯LσµνTabR)Gaµν. usedto generatethe necessaryamplitudes. Attree-level, theinteractionsbetweenSMandKKfieldspreserveboth Here e (g) is the electromagnetic (strong) coupling con- KKnumbers. Consequently,onlydiagramswhereallpar- stant, q are chiral quark fields, F (Ga ) is the elec- ticles in the loop have the same KK index (kl) have to L,R µν µν tromagnetic (gluonic) field strength tensor, and Ta are be taken into account. the color generators normalized such that Tr(TaTb) = At the matching scale µ = (m ) the LO results for 0 t O δab/2. The factor mb in the definition of Q7,8 denotes the UED6 initial conditions read the bottom quark MS mass renormalized at µ. The relevant quantity entering the calculation of 0 for i=1,...,6, (B¯ X γ) is not C (µ) but a linear combination CtBh7eeff(foµ→u)ro-qfuthasirskWopilesroantocrose7.ffiTchieentsoa-ncdalolefdtheeffeccoteiffiveciWenitlssoonf ∆Cieff(0)(µ0)=−211Pk,l′′AF((00))((xxkl)) ffoorr ii==87,, (15) coefficients relevant for b sγ(g) are [31] −2 kl → k,l Ci(µ) for i=1,...,6,  P where the superscript in the summation indicates that ′  6 the KK sums run only over the restricted range k 1 Cieff(µ)=C7(µ)+jP=61yjCj(µ) for i=7, (12) anWd le≥de0c,omi.ep.ose′kt,hle=Inamk≥i-1Liml≥f0u.nctions as ≥ C (µ)+ z C (µ) for i=8, P P P 8 j j wherey andz arechojPs=e1nsothattheLOb sγ(g)ma- X(0)(xkl)= XI(0)(xkl), X =A,F , (16) trixelemjentsojftheeffectiveLagrangianare→proportional I=XW,a,H to the LO terms Ceff(0)(µ). In the MS scheme with fully 7,8 where the function X(0) (x ) describes the contribu- anticommuting γ , one has ~y = (0,0, 1, 4, 20, 80) W,a,H kl 5 −3 −9 − 3 − 9 tion due to the exchange of KK modes of the would-be anWd ~ze =fur(t0h,e0r,1d,e−co61m,2p0o,s−e 1t30h)e[e2ff9e].ctive coefficients into a Goldstone, G±(kl), and the W-bosons, Wµ±(kl), the scalar fields a± and W± . Here x =(k2+l2)/(R2M2 ). SM and a new physics part (kl) H(kl) kl W Our results for the LO Inami-Lim functions entering Ceff(µ)=Ceff (µ)+∆Ceff(µ), i=1,...,8, (13) Eq.(16) are given by i iSM i 5 x 6((x 3)x +3)x2 3(5(x 3)x +6)x +x (8x +5) 7 A(0)(x )= t t− t kl− t− t kl t t − W kl 12(x 1)3 (cid:0) t− (cid:1) 1 x (x +x )2(x +3x 2) x +x + (x 2)x2 ln kl kl t kl t− ln kl t , (17) 2 kl− kl x +1 − 2(x 1)4 x +1 (cid:18) kl (cid:19) t− (cid:18) kl (cid:19) x 6((x 3)x +3)x2 3((x 3)x +6)x +(x 5)x 2 F(0)(x )= t − t− t kl− t− t kl t− t− W kl 4(x 1)3 (cid:0) t− (cid:1) 3 x 3(x +1)(x +x )2 x +x (x +1)x2 ln kl + kl kl t ln kl t , (18) − 2 kl kl x +1 2(x 1)4 x +1 (cid:18) kl (cid:19) t− (cid:18) kl (cid:19) x 6x2 3(x (2x 9)+3)x +(29 7x )x 16 A(0)(x )= t kl− t t− kl − t t− a kl 36(x 1)3 (cid:0) t− (cid:1) 1 x (x +3x 2)(x +x ((x x +4)x 1)) x +x kl kl t t kl kl t t kl t (x 2)x ln − − − ln , (19) − 6 kl− kl x +1 − 6(x 1)4 x +1 (cid:18) kl (cid:19) t− (cid:18) kl (cid:19) x 6x2 + 6x2 9x 9 x +(7 2x )x 11 F(0)(x )= t − kl t − t− kl − t t− a kl 12(x 1)3 (cid:0) (cid:0) t−(cid:1) (cid:1) 1 x (x +1)(x +x ((x x +4)x 1)) x +x kl kl t kl kl t t kl t + x (x +1)ln + − − ln , (20) 2 kl kl x +1 2(x 1)4 x +1 (cid:18) kl (cid:19) t− (cid:18) kl (cid:19) x 6 x2 3x +3 x2 3 3x2 9x +2 x 7x2+29x 16 A(0)(x )= t t − t kl− t − t kl − t t− H kl 36(x 1)3 (cid:0) (cid:0) (cid:1) (cid:0) t− (cid:1) (cid:1) 1 x (x +1) x2 +(4x 2)x +x (3x 2) x +x + x x2 x 2 ln kl kl kl t− kl t t− ln kl t , (21) 6 kl kl − kl− x +1 − 6(x 1)4 x +1 (cid:18) kl (cid:19) (cid:0) t− (cid:1) (cid:18) kl (cid:19) (cid:0) (cid:1) x 6 x2 3x +3 x2 +3 3x2 9x +10 x +2x2 7x +11 F(0)(x )= t t − t kl t − t kl t − t H kl − 12(x 1)3 (cid:0) (cid:0) (cid:1) (cid:0) t− (cid:1) (cid:1) 1 x (x +x )(x +1)2 x +x x (x +1)2ln kl + kl t kl ln kl t . (22) − 2 kl kl x +1 2(x 1)4 x +1 (cid:18) kl (cid:19) t− (cid:18) kl (cid:19) He(r0e) xt = m(02t)(µ0)/MW2 . Our results for the sums ′A(a0)(xkl)≈ − 2238π8xxt ln(Λ2R2) X (x )+X (x ), X =A,F, agree with the expres- W kl a kl Xk,l sions for the one-loop dipole functions given in Ref. [14]. 0.86695+0.205844∆x t We note that there is a misprint in the last line of , (25) − x Eq. (3.33) of the latter paper. Obviously, the term ln((xn+xt)/(1+xt)) shouldreadln((xn+xt)/(1+xn)) ′F(0)(x ) 7πxt ln(Λ2R2) with x = n2/(R2M2 ) and n the single KK index ap- a kl ≈ − 96 x n W k,l pearing in UED5. X 0.791563+0.187945∆x t For the numerical analysis, the results in Eqs.(17) to , (26) − x (22) need to be summed over the KK indices k,l. This summation can be performed analytically employing an ′A(0)(x ) 0.211118+0.050127∆xt , (27) expansion for large 1/R, as explained in App.A. For H kl ≈ − x2 k,l zero boundary mass contributions, h = 0, we obtain X 1,2 the following approximate formulas ′ 0.158339+0.037595∆x F(0)(x ) t , (28) H kl ≈ − x2 k,l ′ 0.686134+0.162912∆x X A(W0)(xkl)≈ x2 t, (23) wherex=1/(R2MW2 )and∆xt =xt−(165/80.4)2. Note k,l thatinthe aboveformulaswehaveonlykeptthe leading X terms in the 1/x expansion for simplicity. The coeffi- ′ 0.316677+0.075190∆x F(0)(x ) t, (24) cients of the logarithms in Eqs.(25) and (26) are exact W kl ≈ x2 in the limit of an infinite number of KK modes. We em- k,l X 6 phasizethatthegivenapproximationsareforillustrative purpose only. In ournumericalanalysiswe willthrough- 0.4 (0) out employ the exact double series X (x ),X = k,l I kl A,F,I = W,a,H, summed over the restricted range 0.3 k 1, l 0, and l+k NKK. P ΜL0 ≥We se≥e from the latt≤er equations that while the one- 0LH ilnoosepnsGit±(ikvl)e atnodthWeµ±,UHV(kl)cucto-rorffectsicoanles tΛo o∆rC, 7ee,ffq8(u0)iv(µal0e)ntalrye, effHDCI7 0.2 0.1 N ,thecontributionsduetoa± exchangedependlog- KK (kl) arithmically on Λ. The different convergence behavior 0.0 is closely connected to the unitarity of the CKM matrix 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 whichresultsinaGlashow-Iliopoulos-Maiani(GIM)suppression[33]ofthe higherKKmodecontributionstothe 1(cid:144)R @TeVD double sums in Eqs.(23) to (28). In the case at hand, 0.4 the GIM mechanism leads to a hierarchy of the various contributions to ∆Ceff(0)(µ ), with X(0) (x ) propor- 7,8 0 W,H kl 0.3 tional to 1/(k2 +l2)2 and X(0)(x ),X = A,F, scaling a kl L like 1/(k2+l2) for large values of l,k. The extra power Μ0 H eoxfckh2an+gel2,tihnatthleeacdosnttoritbhuetiloongafrriotmhmdiciadgirvaemrgsenwtitrhesau±(lktsl), eff0HLC8 0.2 stems from the left- (right-handed) top quark Yukawa DI 0.1 coupling enhanced part of the a+ ¯t b (a− s¯ t ) (kl)U(kl) (kl) U(kl) tree-levelvertex. Nosuchtermsarepresentintheflavor- changing vertices involving G± and W± . 0.0 (kl) µ,H(kl) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ThelogarithmicdivergencesappearinginEqs.(25)and 1(cid:144)R @TeVD (26) would be cancelled by counterterms at the scale Λ at which perturbativity is lost in the higher dimensional theory. Our calculation only determines the lead- FIG.2: ∆IC7e,ff8(0)(µ0)asafunctionof1/R. Thedifferentcurves correspondtotheindividualcontributionsduetotheexchangeof inglogarithmiccorrectionsassociatedwiththerenormal- KKmodesofthewould-beGoldstone,G± ,andtheW-bosons, ization group (RG) running between Λ and 1/R. The W± (yellow/light gray), the scalar fi(kell)ds a± (black) and corresponding initial conditions contain incalculable fi- µ(kl) (kl) W± (green/medium gray), respectively. The lower (upper) nitematchingcorrectionsfromtheunknownUVphysics. H(kl) bordersofthered(darkgray)bandscorrespondtoN =5(15) Assuming that the RG effects dominate over the finite KK while the black lines represent the results for N = 10. See matching corrections and that the UV completion of KK text for details. the UED6 model has a CKM-type flavor structure, the UV sensitivity can be absorbed into a logarithmic dependence on ΛR or, equivalently, N . To gauge the KK theoreticaluncertaintyassociatedwiththe unknownUV lines illustrate the a± corrections. The lower (up- (kl) completion we will vary N in the range [5,15] around KK per) borders of the red (dark gray) bands correspond to N =ΛR 10asestimatedbyNDA.Thechoiceofthe KK N = 5(15) while the black lines represent the results ≈ KK lower value of N is motivated by the observationthat KK for N = 10. We see that in both cases the contribu- KK forNKK <5thenon-logarithmictermsin k,lXa(0)(xkl), tion involving a±(kl) exchange is by far dominant and its X = A,F, become numerically of the same size as the variation with N is non-negligible. Nevertheless, the P KK logarithmic ones. Since the choice of the upper value of large positive corrections to ∆Ceff(0)(µ ) already start N hasnoimpactonourconclusionswechooseitsym- 7,8 0 mKetKrically. Wementionthattherequirementofunitarity to exceed the SM values C7e,ff8(S0M) (µ0) ≈ −0.19,−0.10 in ofgaugebosonscatteringathighenergies[34]generically magnitude for 1/R 240,335GeV in the most conser- ≈ leads to values of ΛR notably below the NDA estimate vative case NKK =5. The observed strong enhancement NKK ≈10. of the initial conditions C7e,ff8(0)(µ0) will play the key role The individual contributions ∆ Ceff(0)(µ ), I = in our phenomenological applications.4 I 7,8 0 W,a,H, to the UED6 initial conditions of the dipole operators as a function of 1/R are shown in Fig.2. The contribution due to the exchange of G± and W± (kl) µ(kl) and W± (green/medium gray) KK modes are de- 4 For compactification scales 1/R ≈ 100GeV it would even be pcuicrtveeds,Haws(khyli)elellotwhe(lriegdht(dgararky)garanyd)gbraenends(maneddiuthme gbrlaacyk) pgvaroolsuuseinbdCles7effbtoy(µrtbeh)ve≈eersx−ep0et.rh3iem7.esiTngthnailsoifpnofCos7rsemiffb(aiµltibtioy)niwsoidnthisB¯frae→vsopreXecdstlo+tnol−gites[n3eS5rM]a.l 7 we utilize the formula 4.0 (B¯ X γ)= 3.15 0.23 s E 3.5 B → ± -4 h 10A 3.0 −8.03∆C7eff(0)(µ0)−1.92∆C8eff(0)(µ0) (29) ΓM +4.96 ∆Ceff(0)(µ ) 2+0.36 ∆Ceff(0)(µ ) 2 Xs 7 0 8 0 2.5 ® +2.33(cid:0)∆Ceff(0)(µ )∆(cid:1)Ceff(0)(µ(cid:0)) 10−4, (cid:1) B 7 0 8 0 × B I 2.0 i which has been derived based on the NNLO SM results of Refs. [8, 10, 36]. For the remaining input parameters 1.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 we adoptthe centralvalues and errorrangesthat canbe 1(cid:144)R @TeVD found in Ref. [8]. ThetheoreticaluncertaintyintheUED6modelisesti- FIG. 3: B(B¯ →Xsγ) for E0 =1.6GeV as a function of 1/R. matedbyscanningNKK,thecouplingsh1,2ofthebound- The red (dark gray) band corresponds to theUED6 result. The ary mass terms, and the matching scale µ0 in the range 68% CL range and central value of the experimental/SM result [5,15], [0,1], and [80,320]GeV for the largest possible is indicated by the yellow/green (light/medium gray) band un- variations. The combined theory error does not exceed derlying thestraight solid line. Seetext for details. +17% for 1/R in the range [0.4,2.0]TeV. Larger rela- −8 tive errors of above +55% appear for 1/R = 300GeV. −25 Whether the quotednumbers providea reliable estimate Another main observation of our work is, that in the of the cut-off and higher-order corrections to (B¯ B → UED6 model the Z- (∆C), photon (∆D), gluon pen- X γ) in the UED6 model can only be seen by perform- s guin (∆E), and the ∆F = 2 boxes (∆S) all behave ing a next-to-leading order (NLO) matching calculation. | | as 1/(k2 + l2) for large values of k,l. In contrast, Such a calculation seems worthwhile but is beyond the ∆F =1boxes(∆B )showanasymptotic1/(k2+l2)2 scope of this work. The parametric uncertainty due to νν,ll |beha|vior after GIM. The corresponding UED6 Inami- the error on the top quark mass is below +1% for 1/R −3 Lim functions therefore exhibit the following behavior: in the range [0.3,2.0]TeV andthus notably smaller than ∆C,∆S x2/x ln(Λ2R2), ∆D,∆E x /x ln(Λ2R2), the combined theory uncertainty. and ∆B ∝ t x /x2. This implies t∝hattthe logarith- Since the experimental result is at present above the νν,ll t ∝ mic cut-off sensitivity first seen in Eqs.(25) and (26) is SM one and KK modes in the UED6 model necessarily a generic feature of all FCNC transitions in the UED6 interfere destructively with the SM b sγ amplitude, model. A dedicated study of neutral meson mixing, rare the lower bound on 1/R following fro→m (B¯ X γ) s B → K- and B-decays in UED6 is left for further work. turns out to be much stronger than what one can derive from any other currently available direct measurement [26]. If experimental, parametric, and theory uncertain- tiesaretreatedasGaussianandcombinedinquadrature, the 95% CL bound amounts to 650GeV. In contrast to IV. NUMERICS the upper limit coming from the dark matter abundance the latter exclusion is almost independent of the Higgs The UED6 prediction of (B¯ Xsγ) for E0 = massbecausegenuineelectroweakeffectsrelatedtoHiggs B → 1.6GeVasafunctionof1/Risdisplayedbythered(dark boson exchange enter (B¯ X γ) first at the two-loop s gray) band in Fig.3. The yellow (light gray) and green level. In the SM theseBcorr→ections have been calculated (mediumgray)bandinthesamefigureshowstheexperi- [30] and amount to around 1.5% in the branching ra- mentalandSMresultasgiveninEqs.(1)and(2),respec- tio. They are included in E−q.(29). Neglecting the cor- tively. In all three cases, the middle line is the central responding two-loop Higgs effects in the UED6 model value, while the widths of the bands indicate the uncer- calculationshouldthereforehavepracticallynoinfluence tainties that one obtains by adding errorsin quadrature. on the derived limits. The central value of the UED6 prediction corresponds Theupper(lower)contourplotinFig.4showsthe95% to NKK = 10 and h1,2 = 0. The strong suppression of CLboundof1/Rasafunctionoftheexperimental(SM) (B¯ Xsγ) in the UED6 model with respect to the central value and error. The current experimental world B → SM expectationandthe slow decoupling ofKKmodes is average and SM prediction of Eqs.(1) and (2) are indi- clearly seen in Fig.3. catedbythe blacksquares. Theseplotsallowtomonitor Inour numericalanalysis,matchingof the UED6 Wil- the effect of future improvements in both the measure- soncoefficientsattheelectroweakscaleiscompleteupto ments and the SM prediction. Of course, one should leadinglogarithmicorder,whiletermsbeyondthatorder keep in mind that the derived bounds depend in a non- includeSMcontributionsonly. Forthereferencevaluesof negligible wayon the treatmentof theoreticaluncertain- therenormalizationscalesµ ,µ ,µ =160,2.5,1.25GeV, ties. Furthermore,thefoundlimitscouldbeweakenedby 0 b c 8 0.40 theNLOmatchingcorrectionsintheUED6modelwhich remain unknown. <0.3TeV 0.4 0.5 0.6 0.7 0.8 1.0 1.4 0.35 V. CONCLUSIONS E 4 - 0 0.30 1 We have calculate the leading order corrections to the A inclusiveradiativeB¯ X γdecayinthestandardmodel p s x → ΓLe 0.25 with two universal extra dimensions. While the one- Xs loop matching corrections associated to the exchange of ® Kaluza-Klein modes of the would-be Goldstone, G± , (kl) B BI 0.20 the W-boson, Wµ±(kl), and the physicalscalar WH±(kl) are D insensitiveto the ultravioletphysics,wefindthatcontri- butions involving a± scalarsdepend logarithmicallyon 0.15 (kl) the cut-offscaleΛ. We haveemphasizedthatinthe con- >2.0TeV sidered model all flavor-changing neutral current transi- 0.10 tions suffer from this problem already at leading order. 3.0 3.2 3.4 3.6 3.8 4.0 Moreover,wehaveincludedformallynext-to-leading,but B IB®Xs ΓLexp A10-4E sizeablemasscorrectionstotheKaluza-Kleinscalarsthat dependquadraticallyonthescaleΛ. Althoughtheultra- 0.40 violetsensitivityweakensthe lowerboundonthe inverse 1.4 1.0 0.8 0.7 0.6 0.5 0.4 <0.3TeV compactificationradius1/Rthatcanbederivedfromthe 0.35 measurementsof the B¯ Xsγ branching ratio,a strong → constraint of 1/R > 650GeV at 95% confidence level is E found if errors are added in quadrature. Our bound ex- 4 - 0 0.30 ceedsbyfarthelimitsthatcanbederivedfromanyother 1 A directmeasurement,andis atvariancewiththe parame- M terregionpreferredby the darkmatter abundance. This S ΓL 0.25 onceagainunderscoresthe outstanding roleof the inclu- Xs siveradiativeB¯-mesondecayinsearchesfor new physics ® close to the electroweak scale. B B I 0.20 D Acknowledgments 0.15 >2.0TeV WearegratefultoMiko lajMisiakandMatthiasStein- 0.10 hauser for private communications concerning Eq.(29). 2.6 2.8 3.0 3.2 3.4 3.6 Helpful discussions with Bogdan Dobrescu and Giu- B IB®Xs ΓLSM A10-4E lia Zanderighi are acknowledged. ANL is supported by the U.S. Department ofEnergy,DivisionofHigh Energy FIG. 4: The upper/lower panel displays the 95% CL limits on 1/R as a function of the experimental/SM central value (hori- Physics, under Contract DE-AC02-06CH11357. This zontalaxis)andtotalerror(verticalaxis). Theexperimental/SM work was initiated when U. H. was supported by the resultfromEq.(1)/Eq.(2)isindicatedbytheblacksquare. The Swiss Nationalfonds. He is grateful to the University of contour lines represent values that lead to the same bound in Zu¨richforthe pleasantworkingenvironmentduringthat TeV. See text for details. time. 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 1 yn I (a)=( 1)nan+1 dy , (A1) n − ay+x Z0 kl 9 where n = 0,1,..., and x = (k2 +l2)x with x = 1/(R2M2 ). Obviously, I (0) = 0. These integrals allow use to kl W n express the logarithms appearing in Eqs.(17) to (22) as x +a kl ln =I (a) I (1), 0 0 x +1 − (cid:18) kl (cid:19) x +a kl x ln =I (a) I (1) 1+a, kl 1 1 x +1 − − (cid:18) kl (cid:19) (A2) x +a 1 1 x2 ln kl =I (a) I (1)+ x +x a a2, kl x +1 2 − 2 2 − kl kl − 2 (cid:18) kl (cid:19) x +a 1 1 1 1 x3 ln kl =I (a) I (1) + x x2 +x2 a x a2+ a3, kl x +1 3 − 3 − 3 2 kl− kl kl − 2 kl 3 (cid:18) kl (cid:19) with a=0 or x . We note that Eq. (D.3) of Ref. [25] is missing an overall minus sign on its right-hand side. t Since the individual building blocks I (a) behave as 1/(k2+l2) for large k,l, the corresponding double series over n the KK levels diverge logarithmically. We regulate the appearing divergence analytically 1 yn Iδ(a)=( 1)nan+1 dy , (A3) n − (ay+x )1+δ Z0 kl with δ >0. Then one has ′ ∞ ∞ 1 yn Iδ(a)=( 1)nan+1 dy n − (ay+x )1+δ k,l k=1l=0Z0 kl X XX ( 1)nan+1 ∞ ∞ 1 ∞ = − dyyn dttδe−(ay+xkl)t Γ(1+δ) k=1l=0Z0 Z0 (A4) XX = (−1)nan+1 1dyyn ∞dttδ ϑ 0,e−xt 2 1 e−ayt 3 4Γ(1+δ) − Z0 Z0 (cid:16) (cid:0) (cid:1) (cid:17) = (−1)n ∞dtt−1−n+δ ϑ 0,e−xt 2 1 Γ(1+n) Γ(1+n,at) , 3 4Γ(1+δ) − − Z0 (cid:16) (cid:0) (cid:1) (cid:17)(cid:0) (cid:1) where in the first step we have used the Mellin-Barnes representation 1 1 ∞ = dttδe−st. (A5) s1+δ Γ(1+δ) Z0 Here ϑ (u,q) = 1+2 ∞ qm2cos(2mu), Γ(z) = ∞dttz−1e−t and Γ(u,z) = ∞dttu−1e−t, denotes the elliptic 3 m=1 0 z theta, the Euler gamma, and the plica function, respectively. P R R The integration over t in the last line of Eq.(A4) cannot be performed analytically. Yet using π , for z √π, z ≤ r ϑ 0,e−z (A6) 3 ≈1+2n+1e−m2z, for z >√π, (cid:0) (cid:1) m=1  X and expanding the integrand in powers of 1/t in the latter case, we can perform the integration piecewise and approximate the double series as ′ Iδ(a) lδ(a)+h (a). (A7) n ≈ n n k,l X 10 The integration over t [0,√π/x] leads to the relatively compact formulas ∈ 1 a√π a√π a2π 2√πxE x 2Γ 0, +ln 8x" 2(cid:18) x (cid:19)− (cid:18) (cid:18) x (cid:19) (cid:18) x2 (cid:19)(cid:19) for n=0, lnδ(a)= (−4(1n)n+πa1n)x+11δ +4n((+n−+21)(cid:0)1nπ)a2x(1"−e−lna√xaπ)(−n+(cid:0)√1π)2+xaγnE−(cid:1)xa(cid:1)n#(cid:18),x(n+1)2 (A8) a√π +πan(n+1) Γ 0, +ln(a) πan where we have expanded the result aroundδ =+×0(cid:18)(nΓa+n(nd1+d)r(cid:0)1on)p(cid:0)−p1(cid:18)e−Γd(cid:18)(cid:18)a√lnlπt+(cid:1)er+x1m,1sa(cid:1)(cid:19)√txπhπ−ant(cid:19)/v(cid:19)2ax#nni(cid:19)+,sh1−in the(cid:19)limfiotrδn=01.,2F,u.r.t.h,ermore, E (z)= ∞dtt−me−zt and γ 0.577216is the exponential integral function and the Euler cons→tant. m 1 E ≈ ′ Theintegrationovert (√π/x, )isfiniteinthelimitδ 0. Foralldoublesums I (a)appearinginEq.(A2) R ∈ ∞ → k,l n we were able to find analytic expressions. Since the results turn out to be rather lengthy and not very informative P we refrain from giving them here. Short numerical expressions for the h (a) can be obtained in the large x limit. n Keeping terms up to third order in 1/x, we find 0.184616a 0.25221a2 0.259202a3 + , for n=0, x − x2 x3  hn(a)=−00..00691253x308882aa32 + 00..1162x86211405aa34− 0.01.9145x45305222aa45, for n=1, (A9) + , for n=2, x − x2 x3 Combining Eqs.(A8) and (A9)we−fi0n.0a4ll6y1xa5r4r1ivae4a+t t0h.1e0f0xo8l2l8o4wai5ng−la0r.g1e29xx630a1ppar6o,ximatfioonrsn=3. 0.644381a 0.751902a2 0.387481a3 + , for n=0, x − x2 x3  Xk,l′Inδ(a)≈ (−4(1n)n+πa1n)x+1 (cid:18)1δ −lnx(cid:19)+−00..322124x179914aa23 + 00..530715x2926581aa34 −+ 00..229302x6431819aa45,, ffoorr nn==12,, (A10) x − x2 x3 The term 1/δ lnx in Eq.(A10) implies that−on0e.1s6h1ox0u9l5dain4c+lud0e.3c0o0xu72n61tear5te−rm0.c1o9n3xt7r34ib1uat6io,ns fromforpnhy=sic3s. at the UV − cut-off scale Λ that cancel the divergences. Our calculation only determines the RG running contribution between Λ and 1/R, given initial conditions at Λ. Assuming that the unknown finite matching corrections are small and have a CKM-type flavor structure, the divergences can be absorbed into a cut-off dependence by switching from analytic to cut-off regularizationemploying the approximation 1 lnx ln(Λ2R2), (A11) δ − ≈