A Problem in Virtual Graviton exchanges in Flat Large Extra Dimensions
aa r X i v : . [ h e p - ph ] J a n A Problem in Virtual Graviton exchanges in Flat Large ExtraDimensions
Haruka Namatame ∗ Department of Physics, Tokyo Metropolitan University,Minami-Osawa, Hachioji, Tokyo 192-0397, Japan
Abstract
It is pointed out in a class of models with large extra dimensions that the cross section of processeswith virtual Kaluza-Klein graviton exchanges becomes either much smaller or much larger by manyorders of magnitude than what is expected from that of the on-shell production of the Kaluza-Klein gravitons. We demonstrate how the problem arises using a toy model. The cause of this newproblem lies in the fact that we do not have momentum conservation in the extra dimensions. Tosearch for the signal of the large extra dimensions with high energy collider experiments, we needmore care in interpreting the earlier results on the cross sections of these processes.
PACS numbers: ∗ Electronic address: [email protected]
Typeset by REVTEX 1 . INTRODUCTION
In order to solve the hierarchy problem between the electroweak scale and the Planckscale, extra space dimensions can play an important role. The scenario of large extra dimen-sions, which was proposed by Antoniadis [1–4] is very simple. In this scenario, we assumethat the Standard Model (SM) particles live on a three-dimensional hypersurface and theonly graviton lives in the whole higher dimensional space.The simplest and explicit model with this scenario is the one by Arkani-hamed, Dimopou-los, and Dvali (ADD) [5, 6]. ADD have introduced a δ -dimensional flat extra space whichis compactified on a δ -dimensional torus with common radius R . In this model, the Planckscale M p in our four-dimensional space time is given by the fundamental Plank scale M f inthe whole 4 + δ -dimensional spacetime and radius R of the extra dimensions as M p = 8 πR δ M δf . (1)If R is large enough, we can take the fundamental Planck scale M f to be a few TeV, whichnaturally explains the hierarchy between the energy scale of electroweak interaction and thatof gravitational interaction.Note that in the ADD model, localization of the SM particles on the three-dimensionalhypersurface is described by a delta function. In this way, the three-dimensional hypersurfaceis treated as a complete rigid-body and the momentum in the extra dimensions is notconserved in the interaction between the higher dimensional graviton and the SM particles.The hypothesis of the complete rigid-body hypersurface leads to a explicit breaking of thetranslational symmetry in the extra dimensions [7].Many massive gravitons, the Kaluza-Klein excitation modes of the graviton (KK gravi-tons), are contained in the four-dimensional effective theory of the ADD model. They candecay into the light SM particles. We can test the ADD model by searching for these massivegravitons directly or indirectly. The stringent constraint comes from the supernova SN1987a[8, 9]. Using this constraint, it was shown that δ has to be larger than two in order for thefundamental Plank scale M f to be a few TeV. In the rest of this paper, we will assume that δ satisfies this constraint.To test the ADD model at collider experiments, two types of the process have beenstudied. One is the real KK graviton emission process, and the other one is the virtualKK graviton exchange process. In case of a hadron collider, dominant channels are the realKK graviton emission process pp → jet + (missing) [10–12], and the virtual KK gravitonexchange process pp → l ¯ l, γγ [13–15]. These studies show that the virtual KK gravitonexchange processes are advantageous to search for signals of the ADD model, compared tothe real KK graviton emission processes due to difficulty of identifying the missing energycarried by the KK gravitons.However, as we will show below, the cross section of the virtual KK graviton exchangeprocesses in the ADD model have the following problems: (I) A problem of ultraviolet divergence of momentum integration in extradimensions The origin of this problem lies in the violation of the momentum conservation in extra dimen-sions. In Ref.[7], it was shown that we can naturally regularize this ultraviolet divergenceby introducing a fluctuation of the three dimensional hypersurface.2
II) Breakdown of narrow width approximation This is a new problem that we point out in this paper. As we will see below, the narrowwidth approximation σ (full process) = σ ( X production) × Br ( X decay) + O (Γ X /m X ) (2)is not satisfied for the cross sections of the virtual KK graviton exchange processes. Ineq.(2), σ (full process) and σ ( X production) are the cross sections of the full process andproduction process of decaying particles, respectively, Γ X stands for the decay width of theparticle, and Br ( X decay) is the branching ratio of the decay channel of X which is includedin the full process. Eq.(2) is expected to hold in most processes under certain conditions tobe described later. However, as we will see below, the cross section of such processes in pastworks (for example in Ref.[16, 17]) does not satisfy the approximate expression in eq.(2).The origin of this problem lies in lack of the momentum conservation in extra dimensions.As we will see later, lack of the momentum conservation in the extra dimensions causes overcounting of the extra dimensional phase space of the higher dimensional graviton or thesummation over the KK index, and we can not obtain the on-shell contributions from theKK gravitons in the intermediate state (or the first term in eq.(2)) correctly even if we takethe decay width of the KK gravitons into account. Although the cause of the problem (II)is the same as that of the problem (I), the problem (II) is independent from the problem (I).This is because the problem (II) appears even if we put the momentum cutoff to regularizethe ultraviolet divergence, or even if we use the prescription in Ref.[7]. We will see moredetails of this problem later.The purpose of this paper is to clarify the problem (II). We will show that how the problemappears in the virtual KK graviton exchange processes. For simplicity, we analyze theseprocesses using a toy model in which tensor fields are replaced by scalar fields, because spinof the intermediate particles is not essential for the problem (II). To show that the problem(II) arises in the virtual KK graviton exchanges even if we regularize the pole divergencewith its decay width to take care of the on-shell contribution of the KK gravitons, we analyzethe virtual KK graviton exchange process putting its decay width into the denominator ofits propagator to regularize the on-shell pole divergence.The rest of this paper is organized as follows. In section 2, we introduce the toy model,and show how we can calculate the cross section of virtual KK graviton exchanges in thismodel. In section 3, we explain how the problem occurs in the virtual KK graviton exchangeprocesses. Moreover, we demonstrate that the same problem appears in the virtual KKgraviton exchange processes with three final states. In section 4, we draw our conclusions. II. VIRTUAL KALUZA-KLEIN GRAVITON EXCHANGES AND A TOY MODEL
First of all, we briefly review the KK graviton and its decay width in the ADD model.For more detail, see Ref.[16, 17]. In Ref.[18], the same subject was discussed. However the main point in Ref.[18] is that the narrow widthapproximation would be broken down if the mass of one of the final state particles is close to that of theintermediate particle, because of a technical problem in the approximation method. In the present case,as we will see below, the problem appears irrespective of whether such condition is satisfied or not, andthe situation is completely different from that in Ref.[18].
3n the four-dimensional effective theory for the ADD model, the higher dimensional gravi-ton whose momentum in the extra dimensions is ~n/R is treated as the massive KK gravitonwhose mass is | ~n | /R where ~n is a δ -dimensional number vector. The effective theory isvalid only if the center of mass energy √ s and the mass of the KK graviton | ~n | /R are muchsmaller than the fundamental Planck scale, i.e., only if √ s, | ~n | /R ≪ M f . These conditionsgive the cut off momentum to regularize the ultraviolet divergence of momentum in extradimensions which derives from lack of dynamics for the localization of the SM particles inthree-dimensional space dimension. These conditions also keep perturbation theory withrespect to the gravitational vertex valid.The interactions between the spin-2 KK gravitons and the SM particles are described bythe interaction Lagrangian L int = − M p X ~n G ( ~n ) µν T µν , (3)where ¯ M p ≡ M p / √ π ∼ GeV is the reduced planck mass, G ( ~n ) µν is a field of the KKgraviton whose mass is | ~n | /R , and T µν is the energy-momentum tensor of the SM particles.The summation of eq.(3) extends over all the KK modes whose mass is less than M f .From the higher dimensional point of view, this summation corresponds to the momentumintegration in the extra dimensions.Main features of the KK gravitons in the ADD model are the following:(i) Lack of the momentum conservation in the extra dimensionsIn the interaction Lagrangian 3, the SM particles, which originally have zero momentum inthe extra dimensions, couple to the KK gravitons, which originally have non-zero momentumin the extra dimensions, and the momenta in the extra dimensions are not conserved.(ii) High degeneracy of the KK gravitonsThe mass spectrum of the KK gravitons is highly degenerate. The number of the KKgravitons which have the same mass is equal to a number of KK index ~n which have samelength. For example, the number of KK gravitons whose mass is 100 GeV is 10 .(iii) Smallness of the decay widthThe KK gravitons can decay into standard model particles. Main decay channels are twophotons, two gluons (two light mesons) and two light fermions. The total decay width ofthe KK gravitons Γ( m ) can be written as a following formΓ( m ) = N ( m ) m ¯ M p , (4)where N ( m ) is a step function of the mass of KK graviton m (= | ~n | /R ). The function N ( m ) is determined by the number and phase space of the decay channels, and is at mostone. The total decay width of the KK gravitons are very small compared to their masses(Γ( m ) /m ∼ m / ¯ M p ≪ L = ¯ ψiγ µ ∂ µ ψ + 12 ∂ µ χ∂ µ χ + 12 ∂ µ χ ′ ∂ µ χ ′ + X ~n (cid:20) ∂ µ φ ~n ∂ µ φ ~n − m ~n φ ~n (cid:21) + X ~n " − λ ¯ M p ¯ ψψφ ~n χ ′ + X ~n " − g M p φ ~n ( ∂ µ χ∂ µ χ + ∂ µ χ ′ ∂ µ χ ′ ) , (5)where the fields φ ~n are KK scalars which correspond to the KK gravitons whose mass is | ~n | /R , χ and χ ′ correspond to the standard model gauge bosons, ψ is a massless fermion,and the two coefficients of the interaction terms λ and g are real coupling constants.The main decay modes of the KK scalars φ ~n are φ ~n → χχ, χ ′ χ ′ . A partial decay widthΓ( m ) of these modes is given by Γ( m ) = m π g ¯ M p ! . (6)Another decay mode is φ ~n → ψ ¯ ψχ ′ , but this decay mode is kinematically suppressed:Γ( φ ~n → ψ ¯ ψχ ) / Γ( φ ~n → χχ ) < − .In the toy model, the KK graviton production process pp → jet + G is replaced by a φ ~n production process ψ ¯ ψ → φ ~n χ ′ , and the virtual KK graviton exchange process pp → l ¯ l isby a virtual φ ~n exchange process χχ → χ ′ χ ′ . III. BREAKDOWN OF NARROW WIDTH APPROXIMATION
In this section, to see the problem of the virtual KK graviton exchange processes, wecalculate the cross section of the virtual KK scalar exchange processes in the toy model. Toget a finite value of the cross section, we should analyze these process putting the decaywidth of the KK scalars into denominator of its propagator.In the ADD model or the toy model, we can approximately treat the spectrum of the KKparticle as a continuous one. Therefore for any value of √ s >
0, there are KK states whichcan be on-shell. This result indicates that resonant effects are important for the virtual KKparticle exchange processes.
A. Two body scattering via KK graviton
First, let us analyze the two-body scattering process via KK scalars χχ → χ ′ χ ′ . Theamplitude of this process is given by M Γ ( χχ → χ ′ χ ′ ) = g ¯ M p ! s ! X ~n is − m + im Γ( m ) , (7)and then the cross section of this process is given by σ Γ = 12 π (cid:18) g ¯ M p (cid:19) s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ~n is − m + im Γ( m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (8)5o get the cross section, we have to perform the summation of propagators, D ( √ s ) ≡ X ~n is − m + im Γ( m ) . (9)Mass splittings between different excitation modes are at most 1 /R . If we set M f = 1 TeVand δ = 4, then we have 1 /R = 30 keV, which is very small compared to the center massenergy √ s . Therefore, we can approximate the summation as an intergration over mass andwe can carry it out explicitly: D ( √ s ) ≈ Z αM f dm ρ ( m ) is − m + im Γ( m )= − i S ¯ M p M f (cid:20) α + y ln (cid:20) | y − α | y (cid:21)(cid:21) + O ( x ) , (10)where S is the surface of three-dimensional unit sphere, x = M f / ¯ M p , y = √ s/M f , and ρ ( m ) dm = S ¯ M p M f m dm (11)is the number of KK modes with its mass between m and m + dm . Here we put the cutoffscale to be αM f , where α is a cutoff parameter ( √ s, | ~n | /R ≤ αM f ). After substitutingeq.(10) in eq.(8), we get the cross section σ Γ ≃ S π g M f s (cid:0) α + ln (cid:2)(cid:12)(cid:12) y − α (cid:12)(cid:12) y (cid:3)(cid:1) . (12)More generally, the leading term of the cross section in the case of the δ -dimensional extraspace is σ Γ ≃ π (cid:18) S δ − δ − (cid:19) g M f s α δ − , ( y ≤ α ) (13)if δ >
2. The cross section in eq.(13) reproduces the result that we get using the iǫ method orthe principal value integration of the propagators to regularize the on-shell pole divergenceof KK scalars ( for more details, see Ref. [17]) . This cross section has two problems.Firstly, it strongly depends on the cutoff parameter α . Secondly, this cross section does notcontain the on-shell contribution from KK scalars in intermediate state. The cross section ofthe on-shell contribution to the two-body scattering via the KK scalars is given in eq.(14).In fact, as we will see below, the on-shell contribution of KK scalars is canceled out uponintegration over mass, and the heaviest mode of the KK graviton, whose mass is αM f , givesthe dominant contribution. In the next subsection, to grasp the details of these problems,we will discuss the narrow-width approximation for the two-body scattering process via theKK scalars. The leading terms in the two methods correspond the result assuming √ s and Γ( m ) in the denominatorin eq.(9) to be zero. . Narrow width approximation To discuss whether the two features of the cross section are physical or unphysical, letus next see whether the narrow width approximation is satisfied in the virtual KK scalarexchange process. The narrow width approximation eq.(2) is expected to be satisfied forcross sections of production processes of decaying particles, in which a particle X is inintermediate state, if (i) X can be on-shell in intermediate state, and (ii) the decay widthΓ X of the particle is small enough compared to its mass m X (Γ X /m X ≪ √ s > /R , and (ii) the decaywidth of KK scalars is small enough (Γ( m ) /m ∼ m / ¯ M p ≪ .In the toy model, the inclusive cross section of KK scalars production is given by σ ( χχ → φ ) = g πS δ − √ s δ M δf . (14)If δ ≤
6, the cross section σ Γ ( χχ → χ ′ χ ′ ) of two-body scattering process in eq.(13) is muchsmaller than the cross section σ ( χχ → φ ) in eq.(14) , because the ratio of the two crosssections is extremely smaller than one: σ Γ ( χχ → χ ′ χ ′ ) σ ( χχ → φ ) ∼ g α δ − (cid:18) √ sM f (cid:19) − δ ≪ , ( δ ≤ , (15)or σ Γ ( χχ → χ ′ χ ′ ) ≪ σ ( χχ → φ ) × Br ( φ → χ ′ χ ′ ) , (16)where the branching ratio Br ( φ → χ ′ χ ′ ) = 1 /
2. Even though the two-body scattering pro-cess includes the real KK scalar production process, the cross section of two-body scatteringis much smaller than that of the real KK scalar production process, and the narrow widthapproximation is not valid.
C. Over counting problem
In the previous subsection, we saw that the narrow width approximation is not satisfiedfor the virtual KK graviton exchanges. The reason why the narrow width approximation fails Note that in case of the SM, we can not divide cross sections of two-body scattering process into the twoparts: a real particle production from two bodys and its decay. The production cross sections are zeroat almost all the values of √ s , because the phase space of a particle production from two bodys in thecenter of mass frame is just a point. On the other hand, in case of the ADD model, the phase spaceof one KK graviton production from two SM particles is also a point, but the inclusive production crosssection is non-zero at any value of √ s >
0. From the higher dimensional point of view, the phase space ofthe higher dimensional graviton production from two SM particles is not a point. Lack of the momentumconservation law in extra dimensions makes the volume of phase space finite.
7s because of cancellation of the on-shell contribution from KK gravitons in the intermediatestate. The integrand ρ ( m ) 1 s − m + im Γ( m ) (17)of the integration over mass in eq.(10) has a sharp peak at m = √ s , and the value of thisfunction is switch from the plus peak to the minus peak at m = √ s . Therefore, the on-shellcontribution from the peak at m = √ s is canceled out upon integration over mass, and weget D ( √ s ) ≃ Z αM f dm ρ ( m ) 1 − m = − ¯ M p M δf ( αM f ) δ − δ − ∝ ρ ( m = αM f ) 1 − ( αM f ) . (18)The cross section of the two-body scattering process is then σ Γ ( χχ → χ ′ χ ′ ) ∝ (cid:12)(cid:12)(cid:12)(cid:12) ρ ( m = αM f ) 1 − ( αM f ) (cid:12)(cid:12)(cid:12)(cid:12) . (19)This result implies that the heaviest mode of the KK graviton, whose mass is αM f , gives thedominant contribution to the two-body scattering, no matter whatever value √ s ( > /R )may have. Thus the narrow width approximation is not valid, and the cross section dependsstrongly on the cutoff parameter α .We can regard this problem as an over counting problem in the following way. The crosssection of the virtual KK graviton exchanges σ Γ is proportional to the integration σ Γ ∝ Z αM f dm Z αM f dm ρ ( m ) ρ ( m ) M Γ ( m ) M † Γ ( m ) , (20)where the M Γ ( m ) is an amplitude of the exchange process of the virtual KK graviton withmass m . Since the integration over mass corresponds to the phase space integration in theextra dimensions, the double integration in eq.(20) includes a double counting of the phasespace in the extra dimensions. If there were no double counting (or no interference termsbetween KK gravitons), the problem would not appear in the virtual KK graviton exchangeprocess. In fact, such double counting of the phase space never appears in any process ofthe SM because of the conservation law of the 4-momentum. D. Case of the process with more than two final states
In the previous subsections, we discussed only the two-body scattering processes, but inother process of the virtual KK graviton exchanges, for example pp → jet + G ⌊−−−−→ l ¯ l
8e can see the same problem. In case that the number of final states is three, it can beshown that a fake enhancement of the on-shell contribution occurs under the integrationover mass, and the cross section becomes unphysically huge compared to the cross sectionof that of the real KK graviton production process, for example pp → jet + missing.The cross section σ Γ of the virtual KK graviton exchange processes with the three finalstates, like pp → (jet + G ) → jet + l ¯ l , is expressed as σ Γ = X | ~n | , | ~n | < αM f R σ Γ ( ~n , ~n ) ≈ Z αM f dm Z αM f dm ρ ( m ) ρ ( m ) σ Γ ( m , m ) . (21)In eq.(21), σ Γ ( m , m ) stands for the value of the three-dimensional phase space integralof square of the absolute value of the amplitude of the process. On the other hand, thecross section σ real of the real KK graviton production processes which are included in sucha virtual KK graviton exchange process, like pp → jet + missing, is expressed as σ real = X | ~n | < √ sR σ real ( ~n ) ≈ Z √ s dm ρ ( m ) σ real ( m ) , (22)where σ real ( m ) stands for the production cross section of the one KK graviton with mass m .We can separate σ Γ into the two parts σ Γ = σ m =m Γ + σ m =m Γ , (23)where σ m =m Γ and σ m =m Γ are given as follows: σ m =m Γ = Z αM f dm Z m dm ρ ( m ) ρ ( m ) σ Γ ( m , m ) , (24) σ m =m Γ = Z αM f dm Z αM f dm ρ ( m ) ρ ( m ) σ Γ ( m , m ) δ ( | ~n | − | ~n | )= Z αM f dm ρ ( m ) × R σ Γ ( m, m ) . (25)In eq.(23), σ m =m Γ represents the contribution from the interference terms between KKgravitons whose mass are different and σ m =m Γ is composed of the two contributions: onefrom the real KK gravitations propagation and from the interference terms between the KKgravitons whose mass are the same. The σ m =m Γ gives dominant contribution for the σ Γ .Changing the integration variable m to the absolute value | ~n | of the KK index, σ m =m Γ and σ real are expressed as σ real = Z √ sR d | ~n | ρ ( | ~n | ) σ real ( m ) (26) σ m =m Γ = Z αM f R d | ~n | ρ ( | ~n | ) σ Γ ( m, m ) , (27)9here ρ ( | ~n | ) = S δ m δ − R δ − (28)is the multiplicity of the KK gravitons with mass m = | ~n | /R , and the integrand in eq.(27)is proportional to the factor ρ ( | ~n | ), because all the KK gravitons with the same mass | ~n | equally contribute,Furthermore, σ m =m Γ is decomposed as the two contributions, one from the KK gravi-ton propagations (the diagonal part) and one from the interference terms between the KKgravitons whose mass are the same (the off-diagonal part): σ m =m Γ = σ diagΓ + σ off − diagΓ , (29)where σ diagΓ and σ off − diagΓ , the contributions from the KK gravitons propagations and fromthe interference terms, are given by σ diagΓ = X | ~n | , | ~n | < αM f R σ Γ ( ~n , ~n ) δ ~n , ~n ≈ Z αM f dm ρ ( m ) σ Γ ( m, m ) , (30) σ off − diagΓ ≈ Z αM f dm (cid:0) ρ ( m ) − ρ ( m ) (cid:1) σ Γ ( m, m ) . (31)We can regard σ Γ ( m, m ) as the production cross section of one KK graviton with mass m because σ Γ ( m, m ) = σ real ( m ) × h Br (KK decay) + O (cid:16) Γ( m ) m (cid:17)i (if m ≤ √ s ) σ real ( m ) × O (cid:16) Γ( m ) m (cid:17) (if m > √ s ) , (32)where Br (KK decay) is a branching ratio of the decay channel of the KK gravitons whichis included in the full process. The first equation in eq.(32) is exactly the expression ofthe narrow width approximation for the one KK graviton exchange process. Substitutingeq.(32) into eq.(30), we get σ diagΓ ≃ Z √ s dm ρ ( m ) σ real ( m ) × Br (KK decay) . (33)If this were the whole contribution, then the narrow width approximation would be satisfied.However, it can be shown that σ Γ is dominated by the contribution from the interferenceterms: σ Γ ≃ σ off − diagΓ ≃ Z √ sR d | ~n | (cid:0) ρ ( | ~n | ) − ρ ( | ~n | ) (cid:1) σ real ( m ) × Br (KK decay) , (34)and we get the ratio of the cross section of σ Γ to σ real σ Γ σ real ∼ ρ ( | ~n | = √ sR )= O ¯ M − δ p M (2+ δ ) ( − δ ) f √ s δ − , (35)10here ρ ( | ~n | = √ sR ) is the number of the KK gravitons with mass m = √ s .Thus we find that the narrow width approximation is broken. This is because the inter-ferences between the KK gravitons whose mass are same are mistaken for the propagationsof the real KK gravitons, and the cross section of the virtual KK graviton exchange σ Γ getto be huge compared to the KK graviton production cross section σ real .As compared with the case of the two-body scattering, there is a difference in the rep-resentation of the over counting problem. In case the virtual KK graviton exchanges withthe three final states, the over counting appears not only in the virtual KK gravitons butalso in the contribution from the on-shell KK gravitons production, although in case of thetwo-body scattering, it appears in the most heaviest KK gravitons production and on-shellcontribution of the KK gravitons is canceled out. This is because, in the case with thethree final states, the pole of the propagators of the KK gravitons are taken care of with thethree-dimensional phase space integral (not the mass integration). In the case with morethan three final state, first we can take care of the on-shell pole of the KK gravitons withthe three-dimensional phase space integral, and calculate the on-shell contribution and theoff-shell contribution of the KK gravitons correctly. Therefore, the on-shell contribution ofthe KK gravitons are not canceled, and the over counting also appears in the contributionfrom the on-shell KK graviton productions.In case that the number of final states is more than three, we find a similar fake enhance-ment of the on-shell contribution and the cross section again becomes unphysically hugecompared to the cross section of that of the real KK graviton production process. IV. SUMMARY AND CONCLUSION
In this paper, we have pointed out that the the cross sections of the virtual KK gravitonexchanges in the ADD model have pathological behaviors. We have shown that the narrowwidth approximation is not valid for the cross section of the two-body scattering process viathe KK gravitons. The cross section of the two-body scattering process get to be unphysicallysmall compared to that of the real KK graviton production process which is included in sucha process, and strongly depends on the cutoff parameter α .The cause of this problem lies in the over counting of the phase space in the extradimensions due to the violation of the momentum conservation in the extra dimensions. Ifthere were no double counting, the problem would not appear in the virtual KK gravitonexchange process.To search for the ADD model (or the large extra dimensions) with collider experiments,we need more care in evaluation of the virtual KK graviton exchange process. Acknowledgments
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