The dynamical fermion mass generation in the 4-dimensional brane is discussed
in a model with 5-dimensional Kaluza-Klein fermions in interaction with
4-dimensional fermions. It is found that the dynamical fermion masses are
generated beyond the critical radius of the compactified extra dimensional
space and may be made small compared with masses of the Kaluza-Klein modes.Comment: 11 pages, 3 figures, REVTeX; typos corrected, comment about the
  symmetry breaking added in section 4, new references for section 1 adde

Abe, Hiroyuki

Miguchi, Hironori

Muta, Taizo

English

arXiv

The dynamical fermion mass generation in the 4-dimensional brane is discussed in a model with 5-dimensional Kaluza-Klein fermions in interaction with 4-dimensional fermions. It is found that the dynamical fermion masses are generated beyond the critical radius of the compactified extra dimensional space and may be made small compared with masses of the Kaluza-Klein modes. 04.50.th, 04.60.-m, 11.15.Pg, 11.30.Q

Name not available

arXiv:hep-ph/0002212 v2   14 Mar 2000HUPD-9922Dynamical fermion masses under the influence ofKaluza-Klein fermions in extra dimensionsHiroyuki Abe, Hironori Miguchi and Taizo MutaDepartment of Physics, Hiroshima UniversityHigashi-Hiroshima, Hiroshima 739-8526, Japan(March 15, 2006)AbstractThe dynamical fermion mass generation in the 4-dimensional brane isdiscussed in a model with 5-dimensional Kaluza-Klein fermions in interactionwith 4-dimensional fermions. It is found that the dynamical fermion massesare generated beyond the critical radius of the compactified extra dimensionalspace and may be made small compared with masses of the Kaluza-Kleinmodes.04.50.th,04.60.-m,11.15.Pg,11.30.QcTypeset using REVTEX1I. INTRODUCTIONIt is an interesting idea to assume an existence of the extra-dimensional space whicheventually compactifies leaving our 4-dimensional space-time as a real world [1]. The recentproposal [2,3] for the mass scale of the compactified space to be much smaller than thePlanck scale gave a strong impact on the onset of studying phenomenological evidences ofextra-dimensional effects. In recent approaches with extra dimensions it is usually assumedthat the standard model particles reside in the 4-dimensional brane while the graviton maymove around the bulk, the space-time with extra dimensions. In our present analysis weintroduce bulk fermions in addition to the graviton and see what effects could be observedon the standard model particles. The bulk fermions interact with themselves as well as withfermions in the 4-dimensional brane through the exchange of the graviton and its Kaluza-Klein excited modes [4], or through the exchange of gauge bosons which may be assumed toexist in the bulk [5]. The interactions among fermions generated as a result of the exchangeof all the Kaluza-Klein excited modes of the graviton or gauge bosons may be expressedas effective four-fermion interactions [4,5]. According to the four-fermion interactions weexpect that the dynamical generation of fermion masses will take place.In the present communication we look for a possibility of the dynamical fermion massgeneration under the influence of the bulk fermions through the effective four-fermion in-teractions. Although our argument is applicable to any higher dimensional models, weconfine ourselves to the 5-dimensional space-time for the convenience of explanations. In 5dimensions fermion mass terms are forbidden if we require the symmetry under the chiralprojection. The possible source of fermion masses in 4 dimensions is two-fold, i. e. thedynamically generated fermion masses and masses of the Kaluza-Klein excited modes of thebulk fermions. The mass of the Kaluza-Klein excited modes is known to be of order 1/Rwhere R is the radius of the compactified fifth dimension.We show in Sec. III that the mixing between the brane fermion and bulk fermion does notlead to the large mass of order 1/R for the fermion in 4 dimensions. We then consider in Secs.IV and V whether the dynamically generated fermion mass can be made small comparedwith the mass of the Kaluza-Klein excited modes. We calculate an effective potential for acomposite operator composed of a fermion and an anti-fermion in the leading order of the1/Nf expansion with Nf the number of fermion species. We find that the mass of fermions inthe 4-dimensional brane is generated dynamically if the compactification radius R passes itscritical value RC and the phase transition associated with the mass generation is of secondorder. This means that the fermion masses in the 4-dimensional brane is small as far as theradius of the compactified fifth dimension is close to its critical value.II. 5-DIMENSIONAL FERMION THEORY AND TORUS COMPACTIFICATIONWe assume an existence of 5-dimensional bulk fermions ψ in interaction with fermionsL on the 4-dimentional brane. Effective interactions among these fermions can be given inthe form of the four-fermion interaction. We imagine that such effective interactions orig-inate from the exchange of the bulk gravitons between fermions. In fact it is known thatthe exchange of the Kaluza-Klein excited modes of the bulk graviton results in effective2four-fermion interactions [4]. After the Fierz transformation on the four-fermion interac-tions we generate the transition-type interactions. Accordingly we start with the followingLagrangian for our modelL(5) = ψ¯iγM∂Mψ + [ L¯iγµ∂µL + g2ψ¯γML L¯γMψ ] δ(x4), (1)where g is the coupling constant with mass dimension -3/2 and index M runs from 0 to 4while index µ runs from 0 to 3. Fermions ψ and L are assumed to be of Nf components.It is easy to see that the Lagrangian is symmetric under the chiral projection x4 → −x4,ψ(−x4) → −iγ4ψ(x4) and L(−x4) → −iγ4L(x4). Thus, if we impose this symmetry ona Lagrangian describing our fermion system, any fermion mass terms are forbidden thusresulting in the above Lagrangian. It should be noted here that, as is easily understood byreferring to the Clifford algebra, an irreducible representation of the 5-dimensional fermionfield is given by a 4-component field just as in the case of 4 dimensions. Hence we can usethe same field both for the 4 and 5 dimensions. In odd dimensions it is well-known thatthere exists no object like γ5 in 4 dimensions which commutes with all the γ matrices. Hencewe do not have such object in 5 dimensions while the fifth component of the γ matrices in5 dimensions, γ4, turns out to be iγ5 in 4 dimensions.For the later convenience we rewrite the above Lagrangian by using auxiliary field σMin the following equivalent form,L(5) = ψ¯iγM∂Mψ + [ L¯i/∂L − σM σ∗M + (g σM ψ¯γML + h.c.) ] δ(x4). (2)Since we are mainly interested in the dynamical fermion mass generation in the leadingorder of the 1/Nf expansion, we neglect the irrelevant terms in the Lagrangian by assuming〈σµ〉 = 0 where µ runs from 0 to 3. After chiral rotation ψ → eipi4 γ5ψ and L → eipi4 γ5L wehaveL(5) = ψ¯i/∂ψ − iψ¯∂4ψ + [ L¯i/∂L − |σ|2 + (g σ ψ¯L + h.c.) ] δ(x4), (3)where σ = −σ4. The Lagrangian (3) is considered to be a composite-Higgs version of themixing interaction adopted in Ref. [6–8].We now consider that the space of the fifth dimension is compactified on a circle withradius R. By adopting the periodic boundary condition at x4 = 0 and x4 = 2piR the bulkfermion field is expressed as a Fourier series,ψ(xµ, x4) = N∞∑n=−∞ψn(xµ) einRx4 , (4)where N is the normalization constant.We define the 4-dimensional Lagrangian in the following way,L(4) ≡∫ 2piR0dx4L(5)=∞∑n=−∞ψ¯ni/∂ψn +∞∑n=−∞nRψ¯nψn + L¯i/∂L− |σ|2 +(m∞∑n=−∞ψ¯nL + h.c.), (5)where m ≡ Ngσ. Note that we have normalized the kinetic terms in Eq. (5) by choosingN ≡ 1/√2piR.3III. MASS SPECTRUM IN 4-DIMENSIONAL SPACE-TIMEIn the following arguments we employ the matrix expressionsΨt ≡ (L , ψ0 , ψ1 , ψ−1 , ψ2 , ψ−2 · · ·) , (6)andM ≡0 m∗ m∗ m∗ m∗ m∗ · · ·m 0 0 0 0 0 · · ·m 0 1R0 0 0 · · ·m 0 0 − 1R0 0 · · ·m 0 0 0 2R0 · · ·m 0 0 0 0 − 2R· · ·................... . .. (7)By using Eqs. (6) and (7) we rewrite the mass term and mixing term in the 4-dimensionalLagrangian such thatL(4)mixing =∞∑n=−∞nRψ¯nψn +(m∞∑n=−∞ψ¯nL + h.c.)= Ψ¯MΨ. (8)If auxiliary field σ acquires a non-vanishing vcuum expectation value, we replace σ in m byits vacuum expectation value 〈σ〉, i. e. m = Ng 〈σ〉. The eigenvalues of matrixM determinethe masses of 4-dimensional fermions. The eigenvalue equation is given by [7]det(M − λI)= ∞∏j=1(λ2 −(jR)2) [λ2 − |m|2 − 2|m|2λ2 ∞∑l=11λ2 − (l/R)2]= 0. (9)It should be noted that the solutions λ = j/R obtained by setting the first factor in themiddle of Eq. (9) to vanish is not the eigenvalues of Eq. (9) since they are canceled by thesame factor in the denominator in the second factor. The real eigenvalues are obtained fromthe second factor in the middle of Eq. (9). The summation in the third term of the secondfactor in Eq. (9) can be performed as it is the Fourier series expansion of the cotangentfunction and thus the eigenvalue equation reduces to the simpler form [7],λR = pi|mR|2 cot(piλR). (10)We would like to see whether we have a possibility of getting the light solution in theeigenvalue equation (10). We confine ourselves to the case |m| ≪ 1/R. By expanding thesolution of Eq. (10) in powers of |mR| we obtain [7]λ±0R = ± |mR|(1− pi26|mR|2 +O(|mR|4)), (11)λ±nR = ± n(1 +|mR|2n2+O(|mR|4))(n 6= 0). (12)4Obviously we find from Eqs. (11) and (12) that the lightest eigenvalue is given byλ±0 = ± |m| for |m| ≪ 1/R. (13)Thus we conclude that within our scheme there is a possibility of having the light fermionmasses which is much smaller than the mass of the Kaluza-Klein modes of the bulk fermion.It should be noted here that in Eq. (13) we have two kinds of fermions with the positive andnegative mass. These fermions are, however, indistinguishable since they have exactly thesame properties. The next step that we have to proceed is to show that this light fermionmass is obtained as a result of the dynamical mass generation mechanism and can really besmall.IV. EFFECTIVE POTENTIAL FOR COMPOSITE FIELDSWe are now interested in the actual value of the lightest fermion mass in the 4-dimensionalbrane as is given in Eq. (13). Thus we have to study the dynamical mechanism of generatingthe non-vanishing vacuum expectation value of composite field σ which is involved in theexpression m = Ng 〈σ〉. For this purpose we would like to calculate the effective potentialfor composite field σ.Our 4-dimensional Lagrangian (5) is rewritten with the matrix representation introducedin the last section as follows,L(4) = Ψ¯(M + Ii/∂)Ψ − |σ|2 . (14)The generating functional Z for our system is given byZ =∫[DΨ¯][DΨ][Dσ][Dσ∗] ei∫d4xL(4) . (15)By performing the path-integration for fermion field Ψ in Eq. (15) we find in the leadingorder of the 1/Nf expansion,Z =∫[Dσ][Dσ∗] e−i∫d4xV (σ), (16)V (σ) ≡ |σ|2 −∫d4ki(2pi)4ln det(M + I/k), (17)where V is the effective potential for the auxiliary field σ. By performing the Wick rotationin the momentum integration in Eq. (17) we rewrite the effective potential such thatV (σ) = |σ|2 − 18pi2∫ Λ20d(k2E) k2E ln det(M2 + Ik2E), (18)where kE stands for the Euclidean momentum and Λ is the momentum cut-off. Note that inaddition to the divergence in the momentum integration the divergence in the Kaluza-Kleinmode sum shows up in general. In our 5-dimensional model this divergence is not presentwhile, if we start from the space-time dimensions higher than six, the divergence does existand we have to rely on the regularization method developed recently in Ref. [9]. The sameargument applies to the case where we derived Eq. (10).5After some calculations we find thatV (σ) = |σ|2 − 12pi2∫ Λ0dx x3 ln[x2 + |m|2(pixR) coth(pixR)]− 12pi2∞∑j=1∫ Λ0dx x3 ln[x2 +(jR)2]. (19)The gap equation to determine the vacuum expectation value < |σ| > of |σ| reads∂V (σ)∂|σ| = 2|σ|{1− g22pi2∫ Λ0dxx32x tanh(pixR) + g2|σ|2}= 0. (20)By numerical observation of Eq. (20) we find that there exists a non-trivial solution for|σ| for a suitable range of parameters g and R and the solution corresponds to the trueminimum of the effective potential. Accordingly the fermion mass is generated dynamically.Here the auxiliary field σ (or the composite field L¯ψ) acquires a vacuum expectation value.The continuous symmetry that is broken by this dynamical process is the U(1) symmetryexisted in the original Lagrangian (1):ψ → eiβψ, L→ e−iβL. (21)Moreover the phase transition associated with this symmetry breaking is of second order asis seen in Fig. 1.In the case of the second order phase transition the critical radius may be obtained bysolving the equation derived from Eq. (20) by setting |σ| = 0,1− g24pi2∫ Λ0dxx2tanh(pixR)= 0. (22)Eq. (22) determines the relation between R and g with the suitable choice of the cut-offparameter Λ.It is interesting to note here that, if we started from 3 dimensions instead of 5 dimensionsand regarded the 2-dimensional world as a physical world, we would have obtained anequation for the critical radius corresponding to Eq. (22) as follows,1− g22pi∫ Λλdx1tanh(pixR)= 0, (23)where parameter λ is the cut-off of the low momentum integration which is needed toremove an infrared divergence created by letting σ → 0. The integration in Eq. (23) iseasily performed and we find thatg22pi2= R/ lnsinh(piΛR)sinh(piλR). (24)Eq. (24), if it is inverted, gives us a formula for the critical radius as a function of thefour-fermion coupling constant. Similarly Eq. (22) gives us a relation between R and g ifwe solve the equation numerically.6V. GENERATION OF DYNAMICAL FERMION MASSESAs was shown in Eq. (13) the lowest fermion mass on the 4-dimensional brane is m =Ng < |σ| > where < |σ| > is determined by solving Eq. (20). By using the numericalestimation we calculate the fermion mass m as a function of radius R for fixed couplingconstant g. The result is shown in Fig. 2 where m, R and g are normalized by the cut-offparameter Λ. It is immediately recognized that the fermion mass generation takes placebelow the critical radius and so is kept small near the critical radius. The critical curvewhich represents the critical radius as a function of the coupling constant is shown in Fig. 3as the curve for m = 0. It is easy to see that g2Λ3 = 12pi2 as R→∞ on the critical curve.In our model the bulk fermions reside in the 5-dimensional space-time and their Kaluza-Klein modes show up as fermions with masses n/R in the 4-dimensional space-time. Sincethose fermions are not observed in the present experimental situation except for the zeromode which mixes with the fermion on the brane, their masses have to be very high and hence1/R should be much higher than several TeV. Thus the compactification scale R for the fifthdimension is considered to be very small. On the other hand the bulk graviton lives in thehigher dimensional space-time and the compactification scale 1/RG for the extra-dimensionalspace associated with the bulk graviton could be smaller than 1/R as is suggested in Ref.2. Our cut-off scale Λ is introduced to suppress the divergence appearing in the integrationin the effective potential. Hence it is to be determined by the region of the validity of oureffective four-fermion theory. We suppose that Λ > 1/R since the fundamental theory whichderives our effective theory is realized at much higher scale than the compactification scalefor the bulk fermions. In Fig. 3 we recognize that, if g2Λ3 ∼ 12pi2, the fermion mass is keptsmall for a wide range of radius R except for the small R region.VI. CONCLUSIONSWe have found within our model that in spite of the presence of the large mass scale1/R in the theory the fermion masses on the 4-dimensional brane can be made small asa consequence of the interaction among the bulk and brane fermions: the mixing of thebrane fermions with the bulk fermions does not lead to the lightest fermion masses of order1/R and also the dynamically generated fermion masses are not of order 1/R. This resultis obtained because the dynamical fermion masses generated under the second-order phasetransition are small irrespectively of 1/R near the critical radius. In our model the possibilityof having low mass fermions resulted from the dynamical origin. This mechanism is quitedifferent from the ones in other approaches in which low mass fermions are expected to showup as a result of the kinematical origins [6–8,10,11].It is tempting to make a final comment on the conjecture that the nature chooses thecompactification scale near the critical radius so as to have low mass fermions. In thisconnection it may be interesting to suppose that the existence of the critical radius is aresult of the physical process that the compactification of the fifth dimensional space isdriven by the force unknown to us and this force is balanced by the pressure coming fromthe fermionic Casimir energy [12,13].7ACKNOWLEDGMENTSThe authors would like to thank Masahiro Yamaguchi (Tohoku U.) and Koichi Yoshioka(Kyoto U.) for fruitful discussions and correspondences. They are also indebted to TakMorozumi for useful comments and to Keith Dienes, Emilian Dudas and Tony Gherghettafor a useful correspondence. The present work is supported financially by the MonbushoGrant, Grant-in-Aid for Scientific Research (C) with contract number 11640280.8REFERENCES[1] Th. Kaluza, Sitzungsber. d. Preuss. Akad. d. Wiss. p.966 (1921); O. Klein, Zeitsch. f.Phys. 37, 895 (1926).[2] I. Antoniadis, Phys. Lett. B246, 377 (1990).[3] N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B429, 263 (1998); Phys.Rev.D59, 086004 (1999); I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. Dvali,Phys. Lett. B436, 257 (1998).[4] T. Han, J. D. Lykken and R-J. Zhang, Phys. Rev. D59, 105006 (1999).[5] H-C. Cheng, B. A. Dobrescu and C. T. Hill, hep-ph/9912343.[6] N. Arkani-Hamed, S. Dimopoulos, G. Dvali and J. March-Russel, hep-ph/9811448.[7] K. R. Dienes, E. Dudas and T. Gherghetta, Nucl. Phys. B557, 25 (1999).[8] A. Das and C. W. Kong, hep-ph/9907272.[9] M. Bando, T. Kugo, T. Noguchi and K. Yoshioka, Phys. Rev. Lett. 83, 3601 (1999).[10] R. N. Mohapatra, S. Nandi and A. Perez-Lorenzana, Phys. Lett. B466, 115 (1999).[11] K. Yoshioka, hep-ph/9904433.[12] S. A. Gundersen and F. Ravndal, Ann. Phys. 182, 90 (1988).[13] H. Abe, J. Hashida, T. Muta and A. Purwanto, Mod. Phys. Lett. A14, 1033 (1999).9FIGURES0 0.5 1 1.5 2m Λ−1−0.0100.01V Λ−4RΛ = 0.1RΛ = 0.2RΛ = 0.3RΛ = 0.4RΛ = 0.5RΛ = 0.6RΛ = 0.7FIG. 1. Typical behavior of the effective potential100 0.5 1 1.5R Λ00.51m Λ−1g2Λ3 /12pi2 = 0.25g2Λ3 /12pi2 = 0.60g2Λ3 /12pi2 = 0.95g2Λ3 /12pi2 = 1.25g2Λ3 /12pi2 = 1.60FIG. 2. Dynamical fermion mass as a function of R with g fixed.0 0.5 1 1.5 2R Λ00.511.522.5g2Λ3/12pi2m = 0.00 Λm = 0.25 Λm = 0.50 Λm = 0.75 Λm = 1.00 ΛFIG. 3. Critical radius as a function of g.11

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Dynamical fermion masses under the influence of Kaluza-Klein fermions in extradimensions

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Dynamical fermion masses under the influence of Kaluza-Klein fermions in extra dimensions

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