Z→b b ¯ in non-minimal Universal Extra Dimensional Model
aa r X i v : . [ h e p - ph ] F e b Z → b ¯ b in non-minimal Universal Extra Dimensional Model Tapoja Jha , Anindya Datta Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road,Kolkata 700009, India
Abstract
We calculate the effective Zb ¯ b coupling at one loop level, in the framework of non-minimalUniversal Extra Dimensional (nmUED) model. Non-minimality in Universal Extra Dimensional(UED) framework is realized by adding kinetic and Yukawa terms with arbitrary coefficientsto the action at boundary points of the extra space like dimension. A recent estimation of theStandard Model (SM) contribution to Zb ¯ b coupling at two loop level, points to a 1 . σ discrep-ancy between the experimental data and the SM estimate. We compare our calculation withthe difference between the SM prediction and the experimental estimation of the above couplingand constrain the parameter space of nmUED. We also review the limit on compactificationradius of UED in view of the new theoretical estimation of SM contribution to Zb ¯ b coupling.For suitable choice of coefficients of boundary-localized terms, 95% C.L. lower limit on R − comes out to be in the ballpark of 800 GeV in the framework of nmUED; while in UED, thelower limit on R − is 350 GeV which is a marginal improvement over an earlier estimate. PACS No:
Extra Dimensional theories can offer unique solutions to many long standing puzzles of StandardModel (SM) such as gauge coupling unifications [1] and fermion mass hierarchy [2]. Most importantlythey can provide a Dark Matter candidate of the universe [3]. In this article, we are interested ina particular incarnation of extra dimensional theory referred as Universal Extra Dimensional Model(UED) where all the SM fields can propagate in 4 + 1 dimensional space-time, the extra dimension(say, y ) being compactified on a circle ( S ) of radius R [4]. The five dimensional action consistsof the same fields of SM and would respect the same SU (3) c × SU (2) L × U (1) Y gauge symmetryalso. R − is the typical energy scale at which the four dimensional effective theory would start toshow up the dynamics of Kaluza-Klein (KK) excitations of SM fields. The masses of KK-modes are m n = m + n R ; where n is an integer, called the KK-number which corresponds to the discretizedmomentum in the compactified dimension, y . m is any mass parameter that has been attributed tothe respective five dimensional field. The n = 0 mode fields in the effective theory could be identifiedwith SM particles.To generate the correct structure of chiral fermions in SM, one needs to impose some extrasymmetry on the action called orbifolding which is nothing but a discrete Z symmetry : y → − y .Fields which have zero modes are chosen to be even under this Z symmetry. There are KK-excitations of other fields which are odd under this transformation. Consequently they cannot have [email protected] [email protected] y is called S /Z orbifold with effective domain of y being from 0 to πR . These two boundary points will be called fixed points of the orbifold.The KK mass spectrum in UED are highly degenerate, and radiative corrections to KK masseslift this degeneracy [5, 6]. Radiative corrections to masses include finite bulk corrections originatedfrom the compactification and boundary corrections due to the orbifolding. Boundary corrections tomasses have logarithmic dependence on the unknown cut-off scale Λ. Furthermore they are localizedat boundary points. In minimal UED (mUED), boundary terms are considered to be vanishing atthe cut-off scale Λ. This is of course a very special assumption and a more general scenario where thisassumption has been relaxed is called non-minimal UED (nmUED) [7]. We will be interested in aparticular non-minimal scenario in which kinetic and Yukawa terms involving fields are added to thefive dimensional action, at boundary points. Coefficients of boundary-localized kinetic terms (BLKT)boundary-localized Yukawa terms (BLYT) and can be chosen as free parameters and experimentaldata can be used to constrain these.Various phenomenological aspects of nmUED have been discussed in [8, 9]. In particular,studies have been made to constrain non-minimality parameters from the perspective of electroweakobservables [8], S, T and U parameters [10], relic density [11] and from the LHC experiment [12].Precision electroweak variables like ρ (T)-parameter, R b ( Z boson decay width to a pair of b quarks normalized to total hadronic decay width), A bF B (forward-backward asymmetry of b quarksat Z pole) always have played the role of a guiding light in search of the new physics. Incidentally,all of these electroweak precision variables are very much sensitive to the radiative corrections andthese quantum corrections themselves are amplified by the large top quark mass. In the same spirit,we would like to investigate how one of the precisely known electroweak variable R b could constrainthe nmUED parameter space.Estimation of radiative corrections to the Zb ¯ b vertex in UED framework has been done pre-viously in Ref. [13, 14]. However, introduction of non-minimality through boundary-localized terms(BLT) would shift masses of KK-excitations in a non-trivial manner from their respective UED val-ues. Moreover, some of couplings involving KK-excitations, in nmUED, are also being modified bysome factors which are nontrivial functions of BLT parameters. So our calculation would not be astraight forward rescaling of earlier calculations of Zb ¯ b vertex done in the context of UED [13, 14].To our knowledge, this is the first effort to estimate the radiative correction to Zb ¯ b interaction innon-minimal UED framework.The plan of the paper is the following. In the next section, we will derive necessary interactionsand vertices in the framework of nmUED with a brief introduction of the model. In section 3, wewill present some calculational details. Section 4 will be devoted to numerical results including theconstraint on the parameter space of nmUED and also a review of Zb ¯ b constraints in UED. Finallyin section 5 we summarize our results and observations. In this section we will very briefly review the non-minimal Universal Extra Dimensional Modelkeeping in mind the necessary masses and couplings which will be used in our calculations of effective Zb ¯ b coupling and we will restrict ourselves to boundary-localized kinetic and Yukawa terms only. Amore detailed account of the model will be found in Ref. [7–9].2e start our discussion with BLKTs for fermions. The resulting action in five dimension isgiven by S quark = Z d x Z πR dy h Qi Γ M D M Q + r f { δ ( y ) + δ ( y − πR ) } Qiγ µ D µ P L Q + U i Γ M D M U + r f { δ ( y ) + δ ( y − πR ) } U iγ µ D µ P R U + Di Γ M D M D + r f { δ ( y ) + δ ( y − πR ) } Diγ µ D µ P R D i , (1)where the four component five dimensional fields can be expressed in terms of two component chiralspinors and their Kaluza-Klein excitations as: Q t,b ( x, y ) = ∞ X n =0 (cid:18) Q nt,bL ( x ) f nL ( y ) Q nt,bR ( x ) g nL ( y ) (cid:19) , U ( x, y ) = ∞ X n =0 (cid:18) U nL ( x ) f nR ( y ) U nR ( x ) g nR ( y ) (cid:19) , D ( x, y ) = ∞ X n =0 (cid:18) D nL ( x ) f nR ( y ) D nR ( x ) g nR ( y ) (cid:19) . (2)In the above expression (and also in the following) M, N = 0 , , , , g MN ≡ diag(+ , − , − , − , − ). The covariant derivative isdefined as D M ≡ ∂ M − i e gW aM T a − i e g ′ B M Y , where e g and e g ′ are the corresponding five dimensional gaugecoupling constants of SU (2) L and U (1) Y respectively. T a and Y are the corresponding generators.Γ M are representations of 4 + 1 dimensional Clifford algebra with Γ µ = γ µ ; Γ = iγ . Since we aredealing with only third generation quark, the compact form of doublet is given as Q = ( Q t , Q b ) T .Upon compactification and orbifolding, this would give rise to the left-handed doublet consisting of t L and b L . U and D are four component fields in five dimension from which t R and b R would emergein the four dimensional effective theory. The y dependent wave functions with appropriate boundaryconditions are given by f L = g R = N Qn cos( M Q n (cid:0) y − πR (cid:1) ) C Q n for n even, − sin( M Q n (cid:0) y − πR (cid:1) ) S Q n for n odd, (3)and g L = f R = N Qn sin( M Q n (cid:0) y − πR (cid:1) ) C Q n for n even,cos( M Q n (cid:0) y − πR (cid:1) ) S Q n for n odd, (4)with C Q n = cos (cid:18) M Q n πR (cid:19) , S Q n = sin (cid:18) M Q n πR (cid:19) . (5)These wave functions satisfy the orthonormality conditions Z dy [1 + r f { δ ( y ) + δ ( y − πR ) } ] k n ( y ) k m ( y ) = δ nm = Z dy l n ( y ) l m ( y ) , (6)where k n ( y ) can be f L or g R and l n ( y ) corresponds to g L or f R . From the above condition, N Qn = r πR q r f M Qn + r f πR . (7)3he mass M Qn of the n th KK-mode is no longer equal to n/R as in UED, rather it satisfiesthe following transcendental equations r f M Qn = − (cid:16) M Qn πR (cid:17) for n even,2 cot (cid:16) M Qn πR (cid:17) for n odd. (8)It is evident that, for zero modes ( n = 0), M Qn vanishes identically.The other required couplings of the theory must be supplemented by the action of gauge fields,Higgs field and the Yukawa interaction between the Higgs and fermions: S A = Z d x Z πR dy h − F MNa F aMN − r g { δ ( y ) + δ ( y − πR ) } F µνa F aµν − B MN B MN − r g { δ ( y ) + δ ( y − πR ) } B µν B µν i , (9) S Φ = Z d x Z πR dy h (cid:0) D M Φ (cid:1) † ( D M Φ) + r φ { δ ( y ) + δ ( y − πR ) } ( D µ Φ) † ( D µ Φ) i , (10) S Y = − Z d x Z πR dy h ˜ y t ¯ Q Φ c U + ˜ y b ¯ Q Φ D + r y { δ ( y ) + δ ( y − πR ) }{ ˜ y t ¯ Q L Φ c U R + ˜ y b ¯ Q L Φ D R } + h.c. i . (11)In the above, F aMN ≡ ( ∂ M W aN − ∂ N W aM + e gf abc W bM W cN ) is the field strength associated with the SU (2) L gauge group ( a is the SU (2) gauge index) and , B MN = ∂ M B N − ∂ N B M is that of the U (1) Y group. Φ and Φ c ( ≡ iτ Φ ∗ ) are the standard Higgs doublet and its charge conjugated field; r φ , r g areare BLKT parameters for the scalar and gauge fields while r y is the coefficient for boundary-localizedYukawa interactions respectively. Five dimensional gauge couplings ˜ g and ˜ g ′ are connected to theirfour dimensional counterparts via the following relation: g ( g ′ ) = e g ( e g ′ ) p r g + πR . (12)In the limit, r g = r φ (which we will be using throughout our analysis), gauge and scalar fieldshave the same y dependent profile as f L (and g R ) given in eq.3 and their KK-excitations have masses M gn (= M Φ n ) which follow from the same transcendental equation given in eq.8 with r f replaced by r g (= r φ ). e y t and e y b denote the Yukawa interactions strengths for the third generation quarks in thefive dimensional theory.Finally, one must note down the gauge fixing action, which is very crucial for the calculationat our dispense, as we would proceed with our calculation in ’t-Hooft Feynman gauge. FollowingRef. [15] one can have, S W GF = − ξ y Z d x Z πR (cid:12)(cid:12)(cid:12) ∂ µ W µ + + ξ y ( ∂ W − iM W φ + { r φ ( δ ( y ) + δ ( y − πR )) } ) (cid:12)(cid:12)(cid:12) , In general when Higgs and gauge BLKTs are unequal, the differential equation governing the dynamics of gaugeprofile in y direction contains a term proportional to r φ due to breakdown of electroweak symmetry [15] and solutionswill be different from those given in eq.3. M W is the W boson mass and ξ y is related to physical gauge fixing parameter ξ (takingvalues 1 in Feynman gauge and 0 in Landau gauge) via ξ = ξ y { r φ ( δ ( y ) + δ ( y − πR )) } . (14)Before delving into the interactions needed for the calculation, let us spend some time discussingthe physical eigenstates which are the outcome of mixing of some of the states originally present inthe four dimensional effective theory. These are quite similar but not exactly the same as in UED.So we have decided to make a dedicated discussion on this issue. There are two such cases relevantfor our calculation. Let us first focus on the mixing in the quark sector. This mixing is driven bythe Yukawa coupling thus it is only important and relevant for top quarks.Substituting the modal expansions for fermions given in eq.2, in actions given in eq.1 and eq.11one can easily find the bilinear terms involving the doublet and singlet states of the quarks. In n thKK-level, mass matrix reads as (cid:16) ¯ Q ( m ) t L ¯ U ( m ) L (cid:17) (cid:18) − M Qn δ mn m t I mn m t I mn M Qn δ mn (cid:19) Q ( n ) t R U ( n ) R ! + h . c ., (15)where M Qn are the solutions of transcendental equations as in eq.8. I mn is an overlap integral of theform Z πR [1 + r y δ ( y ) + r y δ ( y − πR )] f mL ( y ) g nR ( y ) dy. This integral is in general, non-zero for both n = m and n = m . The second case would lead tothe (KK-)mode mixing among the quark of a particular flavour. However, the choice r y = r f wouldmake this integral equal to 1 (when m = n ) or 0 ( m = n ). So by choosing equal fermion and YukawaBLKTs one could easily avoid the mode mixing and end up in a simpler form of the fermion mixingmatrix. In the following we will stick to the choice of equal r y and r f .One can note that strength (off-diagonal terms) of the mixing is proportional to quark mass(denoted by m t here), hence the mixing is only important for top quark (and we will denote topquark mass by m t in the following). The resulting matrix can be diagonalized by separate unitarytransformations for the left- and right-handed fields respectively: U ( n ) L = (cid:18) − cos α n sin α n sin α n cos α n (cid:19) , U ( n ) R = (cid:18) cos α n − sin α n sin α n cos α n (cid:19) , (16)where α n = tan − (cid:16) m t M Qn (cid:17) is the mixing angle. Gauge eigenstates Q ( n ) t and U ( n ) and mass eigenstates Q ′ ( n ) t and U ′ ( n ) are related by, Q ( n ) t L/R = ∓ cos α n Q ′ ( n ) t L/R + sin α n U ′ ( n ) L/R , (17) U ( n ) L/R = ± sin α n Q ′ ( n ) t L/R + cos α n U ′ ( n ) L/R , (18)where the mass eigenstates share the same mass eigenvalue m Q ′ ( n ) t = m U ′ ( n ) = q m t + M Qn = M u (say). 5he four dimensional effective Lagrangian would also contain bilinear terms involving the KK-excitations (starting from KK-level n = 1 and above) of the 5th components of W ± ( Z ) bosonsand the KK-excitations of φ ± ( χ ) of the Higgs doublet field [16]. In the following we note downthe bilinear terms involving the KK-modes of W ± n and φ ± n , which are relevant for our calculation.Using eqs.9,10, and eq.13 one can write in the R ξ gauge, L W n ± φ n ∓ = − (cid:16) W ( n ) − φ ( n ) − (cid:17) (cid:18) M W + ξM n − i (1 − ξ ) M W M Φ n i (1 − ξ ) M W M Φ n M n + ξM W (cid:19) (cid:18) W ( n )+5 φ ( n )+ (cid:19) . (19)The above mass matrix upon diagonalization would lead to a tower of charged Goldstone bosons(with mass square ξ ( M n + M W )), G ± ( n ) = 1 M W n (cid:0) M Φ n W ± n ) ∓ iM W φ ± ( n ) (cid:1) , and a physical charged Higgs pair (with mass square M n + M W ): H ± ( n ) = 1 M W n (cid:0) M Φ n φ ± ( n ) ∓ iM W W ± n ) (cid:1) . So the fields W µ ( n ) ± , G ( n ) ± and H ( n ) ± share the common mass eigenvalue M W n ≡ p M n + M W in’t-Hooft Feynman gauge ( ξ = 1). These combinations of charged Higgs and Goldstone ensure thevanishing coupling of γH n ± W n ∓ ν as it should be with a doublet Higgs at our dispense.Necessary interactions involving the Z -boson, fermions and scalars in the four dimensionaleffective theory can be derived from the above action by simply inserting the appropriate y dependentprofile for the respective five dimensional fields and then integrating over the extra direction, y . Incontrast to mUED, where y dependent profiles are either sin( nyR ) or cos( nyR ), some of the couplingsin nmUED are hallmarked by the presence of few overlap integrals of the form : I mn = Z πR dy f nα ( y ) f mβ ( y ) f pρ ( y ) (20)Here, greek indices refer to the kind of fields involved in the coupling while roman indices refer tothe KK-level of respective fields.At this end, let us pay some attention to a pair of overlap integrals I mn and I mn which arerelevant for our calculation appearing in the interactions listed in appendix A. I mn and I mn are thefollowing overlap integrals: I mn = Z πR dy [1 + r f { δ ( y ) + δ ( y − πR ) } ] f ( m ) Q tL f ( n ) φ f (0) b L , (21) I mn = Z πR dy f ( m ) Q tR f ( n ) W f (0) b L . (22)These integrals are non-zero when n + m is even. Integrals and interactions among KK-stateswith odd n + m identically vanish due to a conserved KK-parity. Even in the former case, theintegrals are of the order 1 when n = m . When m differs from n (in the case of even n + m ) valuesof the integrals diminish generally by an order of magnitude than the m = n case . Keeping this As for example, when r f = 1 and r φ = 2: I = 0 . I = 0 . I = 0 . I = 0 . I = 0 . I = 0 . I = 0 . I = 0 . I = 0 . I ∼ I ∼ I ∼ I ∼ I = 0 .
99 and I = 0 . I = 0 . I = 0 .
6n mind we will be only considering the interactions with n = m neglecting the other sub dominantcontributions coming from interactions in which n = m . The expressions for the integrals (uponintegrating over y ) are given in appendix A along with the necessary Feynman rules. Z b ¯ b vertex: We are now all set to discuss the detail of the calculation leading to the correction of the Zb ¯ b vertexin the framework of nmUED. However, as a preamble we will first briefly discuss the meaning of R b and its correlation to Zb ¯ b coupling in the SM. The tree level Zb ¯ b coupling, in the SM, can be definedas g cos θ W ¯ b γ µ ( g L P L + g R P R ) b Z µ , (23)where Z µ and b ’s are SM fields, P R,L = (1 ± γ ) / g L = −
12 + 13 sin θ W , (24) g R = 13 sin θ W . (25)Any higher order quantum corrections either from SM or from new physics (NP) can be incorporateduniformly as the modification to this tree level couplings given as g L = g L + δg SM L + δg NP L , (26) g R = g R + δg SM R + δg NP R , (27)where δg SM L/R are the radiative corrections from SM and δg NP L/R are that of NP [13]. These correctionscan modify the Z decay width to b quarks normalized to the total hadronic decay width of Z , definedby a dimensionless variable, R b ≡ Γ( Z → b ¯ b )Γ( Z → hadrons) . (28)We will only be considering the effect due to the third generation quarks. Normally, at the one looporder (SM & also in NPs) only the g L receives correction proportional to m t , and the g R receivescorrection proportional to m b (due to the difference in couplings between two chiralities) where m t ( m b ) is the zero mode top (bottom) quark mass. We have neglected the b mass in our calculationand thus a shift δg NP L translates into a shift in R b given by, δR b = 2 R b (1 − R b ) ˆ g L ˆ g L + ˆ g R δg NP L , (29)with ˆ g L and ˆ g R given by ˆ g bL = √ ρ b ( −
12 + κ b
13 sin θ W ) , ˆ g bR = 13 √ ρ b κ b sin θ W , after incorporating the SM electroweak corrections only [17]. Here, ρ b = 0 . κ b = 1 . µ b ¯ b H n ± , G n ± Q ′ nt , U ′ n Q ′ nt , U ′ n (a) Z µ b ¯ b H n ± , G n ± Q ′ nt , U ′ n U ′ n , Q ′ nt (b) Z µ b ¯ b H n ± , G n ± H n ± , G n ± Q ′ nt , U ′ n (c) Z µ b ¯ b H n ± , G n ± G n ± , H n ± Q ′ nt , U ′ n (d) Z µ b ¯ b Q ′ nt , U ′ n H n ± , G n ± (e) Z µ b ¯ b Q ′ nt , U ′ n H n ± , G n ± (f) Figure 1: Loop involving KK-mode of scalar and fermion propagators.In general, the g NP L is calculable in a given framework while R b is an experimentally measurablequantity. Thus eq.29 can be used to constrain the parameters of the model. We would exactly liketo do this exercise in the framework of nmUED in the following.Since we have neglected the interactions involving KK-states with unequal KK-numbers in aninteraction vertex, the number of diagrams contributing to radiative corrections of the Zb ¯ b vertexin nmUED are same as that of minimal UED. Fig.1 shows the Feynman diagrams involving KK-excitations of top quarks, charged Higgs/Goldstone bosons in the loop. The contribution comingfrom the diagrams of Fig.1 is dominant for the presence of Yukawa coupling which is proportionalto m t . In our calculations, we have considered momentum of each external leg to be zero and haveneglected the b quark mass. The amplitude of each diagram, for n th KK-mode, can be expressed interms of a single function, f n ( r n , r ′ n , M ′ ), defined as, i M ( n ) = i g cos θ W u ( p , s ) f n ( r n , r ′ n , M ′ ) γ µ P L v ( p , s ) ǫ µ ( q ) , (30)where r n ≡ m t /M Qn , r ′ n ≡ M W /M Qn , M ′ ≡ M n /M Qn .Amplitudes of different diagrams of Fig.1 (evaluated in ’t-Hooft- Feynman gauge) are given by, f n a ) ( r n , r ′ n , M ′ ) = β (4 π ) g {−
43 sin θ W (cid:18) I + I m t M W (cid:19) + I (cid:0) cos α n + sin α n (cid:1) +2 I m t M W sin α n cos α n } h δ n − { r n + 1) + 3( r ′ n + M ′ ) r n + 1)( r ′ n + M ′ ) − r n ) ln(1 + r n ) − M ′ + r ′ n ) ln( M ′ + r ′ n ) + 4(1 + r n )( M ′ + r ′ n ) ln( M ′ + r ′ n ) } / { ( r n + 1) − ( M ′ + r ′ n ) } i , (31) f n b ) ( r n , r ′ n , M ′ ) = β (4 π ) g { I sin α n cos α n − I m t M W sin α n cos α n } h δ n − {− r n + 1) + 3( r ′ n + M ′ ) − r n ) ln(1 + r n ) − M ′ + r ′ n ) ln( M ′ + r ′ n )+8(1 + r n )( M ′ + r ′ n ) ln(1 + r n ) − r n )( M ′ + r ′ n ) ln( M ′ + r ′ n ) } / { ( r n + 1) − ( M ′ + r ′ n ) } i , (32) f n c + d ) ( r n , r ′ n , M ′ ) = β (4 π ) g { (cid:0) − θ W (cid:1) (cid:18) I + I m t M W (cid:19) − I } h δ n + { r n + 1) + ( r ′ n + M ′ ) − r n + 1)( r ′ n + M ′ ) − r n ) ln(1 + r n ) − M ′ + r ′ n ) ln( M ′ + r ′ n ) + 4(1 + r n )( M ′ + r ′ n ) ln( M ′ + r ′ n ) } / { ( r n + 1) − ( M ′ + r ′ n ) } i , (33) f n e + f ) ( r n , r ′ n , M ′ ) = β (4 π ) g (cid:18) −
23 sin θ W (cid:19) (cid:18) I + I m t M W (cid:19)h δ n + { r n + 1) + ( r ′ n + M ′ ) − r n + 1)( r ′ n + M ′ ) − r n ) ln(1 + r n ) − M ′ + r ′ n ) ln( M ′ + r ′ n ) + 4(1 + r n )( M ′ + r ′ n ) ln( M ′ + r ′ n ) } / { ( r n + 1) − ( M ′ + r ′ n ) } i , (34)where δ n ≡ /ǫ − γ + log(4 π ) + log( µ /M Qn ) , and µ is the ’t-Hooft mass scale; β ≡ πR + r φ πR + r f . I and I stand for the overlap integrals given in equations (20) and (21) respectively for n = m . Amplitudesof diagrams ( e ) and ( f ) are multiplied by a factor of which comes from the usual conventionof contributing one-half of this correction into self-energy and the other half in the wave functionrenormalization. Total amplitude ( i M ( n )1 ) of diagrams in Fig.1 is obtained by adding the individualamplitudes for each diagram and is given by the following expression: i M ( n )1 = i (4 π ) g θ W u ( p , s ) r n β { ( r n + 1) − ( M ′ + r ′ n ) } (cid:18) − I + I m t M W (cid:19)h (1 + r n ) − ( M ′ + r ′ n ) + ( M ′ + r ′ n ) ln (cid:18) M ′ + r ′ n r n (cid:19) i γ µ P L v ( p , s ) ǫ µ ( q ) . (35)From the above expression, it is evident that terms proportional to δ n , as well as sin θ W cancelamong themselves. Here, any correction proportional to sin θ W in the Zb ¯ b coupling must be reflectedin the renormalization of charge (of b quark). This implies that any finite renormalization to the γb ¯ b vertex must be the same to any correction proportional to sin θ W in the Zb ¯ b vertex. We have9xplicitly checked that both of these corrections coming from diagrams of the same topology depictedin Fig.1 identically vanishes. Z µ b ¯ b W n ± Q ′ nt , U ′ n Q ′ nt , U ′ n (a) Z µ b ¯ b W n ± Q ′ nt , U ′ n U ′ n , Q ′ nt (b) Z µ b ¯ b W n ± W n ± Q ′ nt , U ′ n (c) Z µ b ¯ b Q ′ nt , U ′ n W n ± (d) Z µ b ¯ b Q ′ nt , U ′ n W n ± (e) Z µ b ¯ b H n ± , G n ± W n ± Q ′ nt , U ′ n (f) Figure 2: Loop involving KK-mode of W and Goldstone propagators.There is a second set of diagrams contributing to effective Zb ¯ b interaction mainly arisingfrom the KK-excitations of W bosons and quarks. These are sub dominant with respect to thecontributions coming from Fig.1.In the following we present the amplitudes of all the individual diagrams given in Fig.2 : f n a ) ( r n , r ′ n , M ′ ) = I β (4 π ) g {−
43 sin θ W + cos α n + sin α n } h δ n − { r n + 1) + 3( r ′ n + M ′ ) − r n + 1)( r ′ n + M ′ ) − r n ) ln(1 + r n ) − M ′ + r ′ n ) ln( M ′ + r ′ n ) + 4(1 + r n )( M ′ + r ′ n ) ln( M ′ + r ′ n ) } / { ( r n + 1) − ( M ′ + r ′ n ) } i , (36) f n b ) ( r n , r ′ n , M ′ ) = I β (4 π ) g { α n cos α n } h δ n − {− r n + 1) + 3( r ′ n + M ′ ) − r n ) ln(1 + r n ) − M ′ + r ′ n ) ln( M ′ + r ′ n )+8(1 + r n )( M ′ + r ′ n ) ln(1 + r n ) − r n )( M ′ + r ′ n ) ln( M ′ + r ′ n ) } { ( r n + 1) − ( M ′ + r ′ n ) } i , (37) f n c ) ( r n , r ′ n , M ′ ) = − I β (4 π ) g (cid:0) θ W (cid:1) h δ n −
23 + { r n + 1) + ( r ′ n + M ′ ) − r n + 1)( r ′ n + M ′ ) − r n ) ln(1 + r n ) − M ′ + r ′ n ) ln( M ′ + r ′ n ) + 4(1 + r n )( M ′ + r ′ n ) ln( M ′ + r ′ n ) } / { ( r n + 1) − ( M ′ + r ′ n ) } i , (38) f n d + e ) ( r n , r ′ n , M ′ ) = I β (4 π ) g (cid:18) −
23 sin θ W (cid:19) h δ n − { r n + 1) + ( r ′ n + M ′ ) − r n + 1)( r ′ n + M ′ ) − r n ) ln(1 + r n ) − M ′ + r ′ n ) ln( M ′ + r ′ n ) + 4(1 + r n )( M ′ + r ′ n ) ln( M ′ + r ′ n ) } / { ( r n + 1) − ( M ′ + r ′ n ) } i , (39) f n f ) ( r n , r ′ n , M ′ ) = I β (4 π ) g { ( r n + 1) sin θ W − }{− (1 + r n ) + ( M ′ + r ′ n ) + (1 + r n ) ln (cid:18) r n M ′ + r ′ n (cid:19) } / { ( r n + 1) − ( M ′ + r ′ n ) } i . (40)In diagrams of Fig.2, divergences along with the terms proportional to sin θ W do not cancelamong themselves. The divergent terms are r n independent. Following the prescription in Ref. [18],we can write the renormalized amplitude as: i M ( n )2 R ( r n , r ′ n , M ′ ) = i M ( n )2 ( r n , r ′ n , M ′ ) − i M ( n )2 ( r n = 0 , r ′ n , M ′ ) . (41)Finally summing up contributions from all diagrams we have, i M ( n )total = i M ( n )1 + i M ( n )2 R = i (4 π ) g θ W u ( p , s ) r n β { ( r n + 1) − ( M ′ + r ′ n ) } " (cid:26) − I + I (cid:18) − m t M W (cid:19)(cid:27) (cid:26) (1 + r n ) − ( r ′ n + M ′ ) + ( r ′ n + M ′ ) ln (cid:18) r ′ n + M ′ r n (cid:19)(cid:27) +4 I (cid:26) − (1 + r n ) + ( r ′ n + M ′ ) + (1 + r n ) ln (cid:18) r n r ′ n + M ′ (cid:19)(cid:27) γ µ P L v ( p , s ) ǫ µ ( q ) . (42)Therefore for each mode, correction in g L : δg ( n )NP L = Σ i f ni ( r n , r ′ n , M ′ ) = √ G F m t π F ( n )nmUED ( r n , r ′ n , M ′ ) , (43)where F ( n )nmUED ( r n , r ′ n , M ′ ) = r n β [(1 + r n ) − ( r ′ n + M ′ )] " (cid:26) I (cid:18) − M W m t (cid:19) − I M W m t (cid:27) (cid:26) (1 + r n ) − ( r ′ n + M ′ ) + ( r ′ n + M ′ ) ln (cid:18) r ′ n + m ′ r n (cid:19)(cid:27) + 4 M W m t I (cid:26) − (1 + r n ) + ( r ′ n + M ′ ) + (1 + r n ) ln (cid:18) r n r ′ n + M ′ (cid:19)(cid:27) . (44)Total new physics contribution δg NP L (and similarly F nmUED ) can be obtained by summing δg ( n )NP L over KK-modes ( n ). It can be checked that the new physics contribution δg NP L and hence F nmUED goes to zero when R − → ∞ , as expected in a decoupling theory. Let us begin the discussion of our results with the present status of experimental and theoreticalestimation of the Zb ¯ b coupling. Following Gfitter Collaboration [19] and an improved estimation [20]of R b incorporating higher order effects in the framework of SM, the experimental and the theoretical(SM) values are R exp b = 0 . ± . R SM b = 0 . ± . . Above results imply an 1.2 standard deviation discrepancy between the experimental value of R b andits SM estimate. Thus, Eq.43, 44 in conjunction with Eq.29 could be used to translate this 1.2 σ discrepancy on R b to an allowed range for F nmUED : − . ± . F nmUED comes from Feynman graphs listed in Fig.1; as all of theamplitudes listed in Fig.1 contain terms proportional to gy t . While the contributions from diagramsin Fig.2 are proportional only to g , with an exception to the diagram 2( f ); which has some termsproportional to g y t (here, y t is the top quark Yukawa coupling). Total contribution of diagrams inFig.1 is nearly 1 . R − in mUED from R b Before we present our main results in the framework of nmUED, we would like to give a look at thelimit on the R − , in case of UED, keeping in mind the new estimate of SM radiative corrections to the Zb ¯ b vertex at two loop level [20]. One can easily retrieve the UED contributions to δg NP L by simplysetting BLKT parameters to zero. In this limit, overlap integrals ( I and I ) used in the couplingsbecome unity and M Qn , M gn and M Φ n all become equal to nR in the n th KK-level; the ratios β , M ′ will be unity and our expressions completely agree with those given in Ref. [14]. One can define afunction F ( n )UED in the same spirit following Eq.44: F ( n )UED ( r n , r ′ n ) = r n ( r n − r ′ n ) " (cid:18) − M W m t (cid:19) { ( r n − r ′ n ) + (1 + r ′ n ) ln (cid:18) r ′ n r n (cid:19) } + 4 M W m t { r ′ n − r n + (1 + r n ) ln (cid:18) r n r ′ n (cid:19) } . (45)Here, r n ≡ m t /m n , r ′ n ≡ M W /m n and m n = nR .In Fig.3, we plot F UED with R − , the only free parameter in the model after summing con-tributions ( F ( n )UED ) coming from 5 KK-levels starting from n = 1. This has been done in the view12 U E D Figure 3: Variation of F UED with R − in UED model. The horizontal line represents the 95 % C.L.upper limit on the value of F UED calculated from the difference between the experimental value of R b and its theoretical (SM) estimate.of recently discovered Higgs mass and its implication on the cut-off scale of UED [21]. Masses ofthe KK-excitations increase with R − . This in turn set in a decrement of the magnitude of F UED due to the higher values of the masses of the propagators in the loop. One can easily check fromEq.45, that in the limit r ′ n , r n → ∞ , F UED is also vanishing, telling us the decoupling nature ofthe theory. The horizontal line in Fig.3 represents the 95% C.L. upper limit on the value of F UED calculated from difference between the experimental value of R b and its theoretical (SM) estimate( F UED : − . ± . F UED would lead us to the present lower bound on R − from R b . It clearly points that at95% C.L. R − must be greater than 350 GeV, which is a nominal improvement over the earlier limitwhich was 300 GeV [13]. If we ignore the correlation between the Higgs mass and the cut-off scaleof UED, then one could sum upto 20-40 levels. This would slightly push up the magnitude of F UED5 which in turn results into a higher value of the lower limit of R − (370 GeV). However, this limitis still not competitive to the bound derived from experimental data on SM Higgs production andits subsequent decay to W W [22] . At this point it would not be irrelevant to discuss very brieflythe bounds on R − in mUED scenario from other experimental data, to put our result into propercontext. Constraints from ( g − µ [23], FCNC processes [24], ρ -parameter [25] and other electroweakprocesses [26] would result into R − ≥
300 GeV. While projected bounds from the LHC is in theballpark of a TeV [27]. However, presently the strongest bound on R − comes from the considerationof Higgs boson production and decay [22] or from the consideration of relic density [28]. In the lasttwo cases, the derived limits are comparable and yield R − ≥ . R b Now we are ready to discuss the main results of our analysis. Contribution to F nmUED , comingfrom each KK-level are already listed in the previous section. One has to sum over KK-levels toget the total contribution. We have taken into consideration the first 5 levels into the summation. For R − = 1 TeV values of F UED after summing upto 5 levels and 20 levels are 0.0267 and 0.0292 respectively. This is due to the fact that experimental data from LHC on Higgs boson production and subsequent decay to
W W is more consistent to the SM than R b in which there is 1.2 σ new physics window. f = 1.5 R φ = 1.0R φ = 3.0R φ = 5.0R φ = 7.0R φ = 9.0 F n m U E D (a) R f = φ = 1.0R φ = 3.0R φ = 5.0R φ = 7.0R φ = 9.0 F n m U E D (b) R φ = f = 1.0R f = 3.0R f = 5.0R f = 7.0R f = 9.0 F n m U E D (c) R φ = f = f = f = 5. f = f = F n m U E D (d) Figure 4: Variation of F nmUED with R − for different values of BLKT parameters. The horizontalline represents the 95 % C.L. upper limit of the value of F nmUED calculated from difference betweenthe experimental value of R b and its theoretical (SM) estimate.And we have explicitly checked that taking 20 levels into the summation, would not change theresults . But before presenting the result, we would like to comment about the values of the BLTparameters used in our analysis. The fermion and the gauge BLKT can be positive or negative. Acareful look at eq.7 tells us that once the scaled BLT parameters R φ,f ( ≡ r f,φ R − ) < − π , the zeromodes become ghost-like i.e. its norm becomes imaginary (fermion or gauge). Furthermore, onecan also see from eq.12 that for R φ < − π , gauge coupling becomes imaginary. So, negative valuesof R f,φ below − π would be physically unacceptable. Apart from this, all other values of R f,φ aretheoretically acceptable. However, one has to be careful in choosing the values of R f,φ , so that theoverlap integrals and couplings should not be too large to break the perturbative unitarity. A fullanalysis imposing unitarity constraints must be carried out to get an idea about this which is beyond For R − = 1 TeV and r φ = 1 . r f = 1, values of F nmUED after summing upto 5 levels and 20 levels are 0.0439and 0.0472 respectively. R f,φ suchthat the resulting effective couplings never become larger than unity.Figure 5: Contours of constant F nmUED corresponding to 95% C.L. upper limit in R φ − R f plane.Different lines (marked with 400, 500, 600, 700 and 800) represent different values of R − . Regionright to a particular contour is being ruled out at 95% C.L. from the consideration of R b for agiven value of R − (in GeV) on each contour. We also present contours of the W ± t b couplingcorresponding to three different values (0.4, 0.45 and 0.5) on the same plot for a same set of values of R − . Numbers along the top axis and right hand axis correspond to dimensionless quantities M Q R and M Φ1 R respectively.In Fig.4, we have presented the variation of F nmUED as a function of R − for some representativevalues of the scaled BLKT parameters R φ and R f . One common feature that comes out from all ofthe plots, namely the monotonic decrement of F nmUED with increasing R − , showing the decouplingnature of the new physics under our consideration, which has been pointed out earlier in the caseof UED . Panels (a and b) show the dependence of F nmUED on R f keeping the value of R φ fixed to1.5 and 4.5 respectively. While in the lower panels of Fig.4, we have presented how F nmUED changeswith varying R φ with two fixed values of R f namely 1.5 (c) and 4.5 (d) respectively.From the figures it is evident that R φ and R f have more or less same effects on F nmUED and δg NP L . While the effect of R φ is somewhat modest, F nmUED is being more sensitive to any changeof R f . So by increasing the BLT parameters one could enhance the radiative effects on the effective Zb ¯ b coupling. Consequently, in nmUED, one could significantly shift the lower bound on R − , fromits UED value. As for example, for R φ = 1 . R f = 9, the 95 % C.L. lower bound on R − isaround 700 GeV. This limit comes down to 448 GeV for R f = 1.The role of R − in the framework of nmUED is similar as in the case of UED and has beenexplained above. We would also like to understand the role of R f and R φ . However we will do so alittle later.Finally in Fig.5, we present the allowed parameter space in R φ − R f plane for several values of R − . We plot the contours of constant F nmUED which corresponds to the 95 % C.L. upper limit. Theregion right to a particular line is ruled out from the consideration of R b according to our analysis.Near vertical nature of the contours at lower values of R f points out to the modest dependence15f F nmUED on R φ already exhibited in Fig.4( a ) and ( b ). It has been revealed from Fig.4 that withhigher values of BLKT parameters R f and R φ , F nmUED is being increased in magnitude. So as wego towards the right with increasing R f and fixed R φ for a particular value of R − , F nmUED wouldincrease. Furthermore the higher value of R − decreases F nmUED showing the decoupling nature ofnew physics. Thus the increment of F nmUED (with R f ) has been nullified by higher values of R − corresponding to different lines. Thus to compensate one must tune R − to a comparatively highervalues. Furthermore, we have marked axes of Fig.5 with scaled masses m Q ( ≡ M Q R ) and m Φ1 ( ≡ M Φ1 R ). This would facilitate one to read off the bounds on masses of the n = 1 KK-excitationsdirectly from this plot. As for example, the line corresponding to R − = 700 GeV intersects, the M Q axis at around 0.5, which implies, that for this particular value of R − , masses of n = 1 KK-excitations of top quarks below 350 Gev are not allowed by the data. While the correspondinglower bound for W mass for R − = 700 GeV is close to 560 GeV which can be read off from theintersection of the same line with the m Φ1 axis.In Fig.5, contours for constant (for three different) values of W ± f f couplings have alsobeen presented. One can get several important messages from these contours. Primarily the abovecoupling has a minimal dependence on R − . Secondly, BLT parameters R f and R φ have oppositeeffects on the above interaction. While this coupling increases with R φ , increasing values of R f would try to decrease the strength of this interaction. Similar conclusion can be drawn to H ± f f and G ± f f interactions. BLT parameters have another bearing on F nmUED through the masses ofKK-excitations. Heavier KK-masses would tend to decrease the magnitude of F nmUED . It has beenpointed out earlier that KK-masses are decreasing function of respective BLT parameters. ThusBLT parameters have dual role to play in the dynamics of F nmUED . Let us state them one by one.An increasing R φ would increase F by increasing the relevant couplings and at the same time bydecreasing the relevant KK-masses. On the other hand an increasing R f would decrease the massesbut it also decreases the couplings. These two effects play in opposite direction in determining thevalue of F . However, rate at which F increases with decreasing KK-mass, overcome the decrementof F due to decreasing coupling (with increasing R f ).Before we conclude, we would like to comment on the terms which we have neglected byonly considering interactions of SM particles with two KK-excitations of same KK-number. As aconsequence of these our calculation and results presented above do not take into account a numberof Feynman graphs in which propagators in the loop corresponds to KK-excitations of different KK-numbers. To advocate our assumption, we present the values of F nmUED for several vales of R − andfor fixed values of R φ and R f in Table.1. While presenting these numbers we have summed upto 5KK-levels as before. In the second column of Table.1 we present the values of F nmUED when onlyKK-number conserving interactions are taken into account. While in the third column, values of F nmUED correspond to the case when all possible Feynman graphs involving KK-number violatinginteractions have been taken into account. It is evident from the numerical values of F nmUED that ourassumption was realistic and the corrections coming from Feynman graphs involving the KK-numbernon-conserving interactions are minuscule. In summary, we have estimated one loop contribution to the Zb ¯ b vertex in the framework of anUniversal Extra Dimensional Model where kinetic and Yukawa terms are added to the fixed pointsof the extra space like dimension. These boundary-localized terms, with their coefficients as free16 − (GeV) F nmUED ( n = m terms only) F nmUED (all interactions )250 0.5442 0.5481350 0.3127 0.3148450 0.2003 0.2016550 0.1384 0.1393650 0.1009 0.1016750 0.0767 0.0773850 0.0602 0.0606Table 1: Values of F nmUED when only the KK-number conserving interactions are taken (secondcolumn) and when all possible interactions are taken into account (third column) to calculate theeffective Zb ¯ b vertex at one loop. Numbers are presented for several values of R − (first column) andfor R f =1 and R φ =1.5.parameters, parametrize the quantum corrections to the masses of the KK-excitations and theirmutual interactions. We have calculated the interactions necessary for our calculation. Some ofthese interactions are very similar to those in UED. However, some of the interactions are modifiedin comparison to their UED counterparts by some overlap integrals (in extra dimension) involvingthe extra dimensional profiles of the fields present in an interaction vertex.The effect of BLKTs on the masses of KK-modes and their interactions can be summarizedas the following. For a given R − , increasing BLKT parameter would drive the respective massesto lower values. Strength of an interaction does not have such a simple dependence on the BLKTparameters. We have derived all the necessary interactions involving the KK-excitations of topquarks, W bosons, charged Higgs and Goldstone bosons in the framework of nmUED with theassumption of equal gauge and Higgs BLKT along with equal fermion and Yukawa BLKTs. Gaugeand Higgs BLKTs have been chosen to be equal to avoid the unnecessary complication created bythe presence of r φ in equation of motion of gauge fields. While unequal fermion BLKTs and YukawaBLT would lead to the KK-mode mixing in the definition of physical states of KK-excitations of topquarks. So for the sake of a relatively simpler calculation we stick to the choice of equal fermionBLKTs and Yukawa BLTs.In general, coupling of a b quark to the Z boson involves both the left- and right-chiral pro-jectors. However, quantum corrections which go into the coefficient of the left-chiral projector areproportional to m t while the m b proportional terms go into the coefficient of the right-chiral projec-tor. We have done the calculation in the limit where m b →
0. There are two main classes of Feynmandiagrams contributing to δg NP L , (the contribution to Zb ¯ b vertex in nmUED framework) in ’t-HooftFeynman gauge. First set of diagrams listed in Fig.1, captures the dominant contribution (becauseof Yukawa coupling which is proportional to m t ) coming from the participation of KK-excitations oftop quarks and charged Higgs boson/Goldstones in the loops. The remaining set consists of contri-bution mainly coming from the KK-excitations of W bosons and top quarks inside the loops. Thesediagrams are listed in Fig.2.The explicit expressions for the contributions coming from each of the diagrams are listed inthe section 3. Sum of the contributions to δg NP L from the diagrams in Fig.1 is finite and independentof sin θ W . While the second set of diagrams needs to be regularized, after summing up, it is still17ltraviolet divergent and also contains a term which grows with R − . We have used a regularisationscheme following Ref. [14, 18], upon which the total contribution becomes finite and also becomesindependent of sin θ W .A recent theoretical estimation of the Zb ¯ b vertex in the framework of SM at two loop levelhas squeezed the window for new physics that might be operating at TeV scale. The experimentallymeasured value of R b differs from the SM prediction at 1.2 σ level. We have used the experimentaldata and the recent results from the SM on R b , to constrain the parameter space of non-minimalUniversal Extra Dimensional Model. We have relooked into the UED by setting the BLKT parame-ters to zero in our calculation. The resulting expressions can be used to put bound on R − in UEDmodel using the same experimental data and the SM estimations of R b . It has been found that R − in UED model should be greater than 350 GeV at 95 % C.L.Next we focus into our main result. Comparing the numerical estimation of F nmUED with thedifference between experimental data and SM estimation we have constrained the parameters innmUED. First we look into the limits on R − . Both the BLKT parameters have positive effects on F nmUED . This function is very sensitive to any change in R f while the effect of R φ is very mild. Thebottom line is that both the BLKT parameters can push the allowed value of R − to higher values.Depending on magnitude of BLKT parameters R φ and R f (which we have chosen to be positive),lower limit on R − could be close to 800 GeV. Finally, we plot contours of constant F nmUED havingthe 95 % C.L. upper limit value for different value of R − in R φ − R f plane. As for a fixed value of R − i.e. for a fixed curve the value of the function F nmUED increases with increase of R f the left sideof that curve represents the allowed region of this function for respective R − . Acknowledgements:
AD acknowledges partial financial support from BRNS-DAE, Govt. ofIndia. TJ acknowledges financial support from CSIR in terms of a JRF. Both the authors are gratefulto Gautam Bhattacharyya, Amitava Raychaudhuri and Avirup Shaw for many useful discussions.
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In the following Feynman rules, all momenta and fields are assumed to flow into the vertices. W n ± µ F ¯ F : ig √ γ µ C L P L W n + ¯ Q ′ nt b L : C L = − I p β cos α n , W n − ¯ b L Q ′ nt : C L = − I p β cos α n ,W n + ¯ U ′ n b L : C L = I p β sin α n , W n − ¯ b L U ′ n : C L = I p β sin α n . H n ± /G n ± F ¯ F : g √ M W n C L/R P L/R H n + ¯ Q ′ nt b L : C L = − i p β (cid:18) I m t M Φ n M W sin α n − I M W cos α n (cid:19) ,H n − ¯ b L Q ′ nt : C R = − i p β (cid:18) I m t M Φ n M W sin α n − I M W cos α n (cid:19) ,G n + ¯ Q ′ nt b L : C L = p β ( I m t sin α n + I M Φ n cos α n ) ,G n − ¯ b L Q ′ nt : C R = − p β ( I m t sin α n + I M Φ n cos α n ) ,H n + ¯ U ′ n b L : C L = i p β (cid:18) I m t M Φ n M W cos α n + I M W sin α n (cid:19) , n − ¯ b L U ′ n : C R = i p β (cid:18) I m t M Φ n M W cos α n + I M W sin α n (cid:19) ,G n + ¯ U ′ n b L : C L = − p β ( I m t cos α n − I M Φ n sin α n ) ,G n − ¯ b L U ′ n : C R = p β ( I m t cos α n − I M Φ n sin α n ) . Z µ W n + β W n − α : ig cos θ W { ( p − p ) µ g αβ + ( p − q ) α g βµ + ( q − p ) β g αµ } p p qZ µ F n ¯ F n : ig θ W γ µ ( C L P L + C R P R ) Z ¯ Q ′ nt Q ′ nt : (cid:26) C L = − θ W + 3cos α n C R = − θ W + 3cos α n , Z ¯ U ′ n U ′ n : (cid:26) C L = − θ W + 3sin α n C R = − θ W + 3sin α n ,Z ¯ Q ′ nt U ′ n : (cid:26) C L = − α n cos α n C R = − α n cos α n , Z ¯ U ′ n Q ′ nt : (cid:26) C L = − α n cos α n C R = − α n cos α n . Z µ H n − , G n − H n + , G n + : g θ W M ′ Wn ( p − p ) µ C p p Z H n + H n − : C = i { ( − θ W ) M n − θ W M W } ,Z G n + G n − : C = i { ( − θ W ) M W − θ W M n } ,Z H n − G n + : C = − M Φ n M W ,Z G n − H n + : C = M Φ n M W . µ W n ± ν H n ∓ , G n ∓ : gg µν cos θ W M Wn C Z W n + G n − : C = (cid:0) − M W sin θ W + M n cos θ W (cid:1) ,Z W n − G n + : C = (cid:0) M W sin θ W − M n cos θ W (cid:1) ,Z W n + H n − : C = − iM Φ n M W ,Z W n − H n + : C = − iM Φ n M W .I and I have the following form: I = 2 πR q r f M Qn + r f πR q r φ M n + r φ πR M n ( − r f + r φ ) (cid:0) M Qn − M n (cid:1) ,I = 2 πR q r f M Qn + r f πR q r φ M n + r φ πR M Φ n M Qn ( − r f + r φ ) (cid:0) M Qn − M n (cid:1) ..