Can effective four-dimensional scalar theory be asymptotically free in a spacetime with extra dimensions?
aa r X i v : . [ h e p - t h ] F e b Can effective four-dimensional scalartheory be asymptotically free in aspacetime with extra dimensions?
A.V. Kisselev ∗ and V.A. Petrov † Division of Theoretical Physics, A.A. Logunov Institute for High Energy Physics,NRC “Kurchatov Institute”, 142281, Protvino, Russia
Abstract
We trace what happens with asymptotically free behavior of therunning coupling in φ theory in six-dimensional spacetime, if to com-pactify two spatial dimensions on a 2D closed manifold. The resultcan be considered as an effective 4D theory of infinitely many KK-type scalar fields with triple interactions. The effective dimensional coupling constant inherits running to zero at high mass scales in amodified form depending on the size of the compact manifold. Somephysical implications are discussed. Freedom is not worth having ifit does not include the freedomto make mistakes.
Mahatma Gandhi.
Asymptotic freedom in QCD was discovered by David Gross and FrankWilczek [1], and independently by David Politzer [2] in 1973. The asymptot-ically free renormalizable field theory in four dimensions necessarily involves ∗ Electronic address: [email protected] † Electronic address: [email protected] D = 4. The striking examples are the 2D Gross-Neneumodel [5] and 2D nonlinear sigma model [6]. All these theories are renor-malizable and asymptotically free. A special case is the 4D φ theory with anegative coupling constant [7]. It is a common belief that its spectrum canbe shown to be unbounded from below. Nevertheless, as was shown in [8],this theory may be consistent. Especially note the 6D scalar φ theory whichalso exhibits the property of asymptotic freedom [9]. One may ask is thereany 4D effective asymptotically free theory without gauge fields? By effectivetheory we mean a reduced theory obtained from a higher dimensional theoryafter “integrating out” extra spatial coordinates. To answer our question,one has to consider theories in a spacetime with extra dimensions (EDs).Effective field theories with one or more compact EDs are of considerableinterest during last years. In particular, in [10] the total cross section of thescattering of two light particle was calculated in the φ scalar model witha spherical compactification. In [11] one-loop order contributions from onecompact universal ED to the self-energy and four-point vertex functions ina φ scalar theory are given. The one-loop low-energy effective action inthe φ scalar theory and scalar QED with spacetime topology R , ⊗ S iscalculated in [12]. The decoupling of heavy KK modes in Abelian Higgsmodel with spacetime topologies R , ⊗ S and R , ⊗ S / Z is examinedin [13]. The photon self-energy, the fermion self-energy, and fermion vertexfunction in the one-loop approximation in the context of QED with one EDare presented in [14]. In [15] D + 1 dimensional φ model with an arbitrary D and one compact manifold is studied. The renormalizable compactificationmodels, when a size of compact dimensions is of the order of cutoff scale,are examined in [16]. The universal extra dimensional models defined onthe six-dimensional spacetime with two spatial dimensions compactified toa two-sphere orbifold S /Z were studied in [17]-[20]. In [21] T /Z , S /Z ,and other orbifolds were examined.The goal of our study is to derive an effective four-dimensional φ scalarfield theory in a spacetime with two compact EDs and calculate a runningcoupling constant in the one-loop approximation. There are three possibil-ities to realize a scalar theory with a power interaction gφ n which has a dimensionless coupling constant g , see Tab. 1. Among them only the scalar gφ theory in six dimensions is known to be asymptotically free [9] (see also[22]). That is why, we will start from this theory.The paper is organized as follows. In Section 2 we briefly remind a renor-2 n n on a number of spacetimedimensions D in scalar theories with an interaction gφ n ( x ) and dimensionless coupling constant g .malization of the φ theory in six infinite dimensions (denoted hereafter as φ , with the subscript 6 indicating the spacetime dimensionality). In the nextsection we examine an effective φ theory in the spacetime with four infiniteand two compact dimensions (referred below as φ ) and calculate a runningcoupling constant. In Section 4 we examine a dependence of our results on atopology of the compact dimensions. Finally, in Section 5 a scale dependenceof physical observables is analyzed. Some properties of two-dimensional in-homogeneous Epstein zeta function and truncated Epstein-like zeta functionare collected in Appendix A. φ theory in six infinite dimensions The classical Lagrangian for the φ theory in terms of bare parameters lookslike L = 12 [ ∂ µ φ ( x )] − m φ ( x ) − g φ ( x ) , (1)where the bare coupling constant g has a dimensionality of mass. On aclassical level a cubic potential of the φ theory is not bounded below. As aconsequence, there cannot be a stable ground state. However, it is not thecase, if one consider the theory on a quantum level and takes into accounta kinetic term in a Hamiltonian, along with the cubic and quadratic ones[23]. In terms of the renormalized (R) field φ R , mass m and coupling g theLagragian is given by L = L R + L CT , (2)where L R = 12 [ ∂ µ φ R ( x )] − m φ ( x ) − g φ ( x ) (3)is its renormalized part, and the counterterm part of (2) is of the form L CT = 12 ( Z φ − ∂ µ φ R ( x )] − δm φ ( x ) − ( Z Γ − g φ ( x ) . (4)3he Feynman rules are i/ ( p − m ) for a scalar propagator, and ( − ig ) fora three-particle vertex. Let Γ ( n ) ( p , p , . . . p n − ) be one-particle irreducible(OPI) Green’s function. The inverse propagator is given by S − ( p ) = − i [ p − m + Σ( p )] = − i Γ (2) ( p ) , (5)where Σ( p ) is a self-energy. Γ (3) ( p, q ) is a three-particle vertex with “ampu-tated” external legs.The renormalized quantities ( φ R , g, m ) are related with the bare quantities( φ, g , m ) through renormalization constants (see, for instance, [22]). Inparticular, the scalar field is renormalized as φ R = Z − / φ φ . (6)The mass renormalization looks like m = Z − m m . (7)The renormalization of the coupling constant is given by g = Z / φ Z − g . (8)If we express in (2) all the parameters in terms of the bare quantities usingeqs. (6)-(8), we come to (1).In our study, we use the dimensional regularization [24] for Feynmanintegrals, and the MOM scheme with the Euclidean normalization point − µ ( µ >
0) for the renormalization procedure. Usually an on-shell conditionis imposed on propagators and vertices of scalar fields. In massive theorieswhere the zero momentum lies in the analyticity domain, a subtraction point p = 0 is used [22]. Nevertheless, it is more appropriate for us to normalizeOPI Green’s functions at some Euclidean point, as it is done in QCD [1,25], where quarks and gluons are confined, and, consequently, have no polemasses.The beta function of the φ theory, β [ g ( µ )] = µ dg ( µ ) dµ , (9)is known to be [9, 22, 27] β ( g ) = − β g + O( g ) , (10)4here β = 34(4 π ) . (11)It is calculated up to five loops [26]. All known terms in an expansion of β ( g )are negative. Since β >
0, there is the asymptotic freedom in φ theory, and α ( µ ) = α ( µ )1 + 34 α ( µ ) ln( µ /µ ) , (12)where α = g (4 π ) . (13)Note that, instead of using eq. (9), the β -function can be alternatively definedas β [ g (¯ µ )] = − ¯ µ dg (¯ µ ) d ¯ µ , (14)where ¯ µ is a scale needed to preserve the canonical dimension of the couplingconstant in the dimensional regularization. The reason is that the renormal-ization constants Z φ and Z Γ depend on the ratio µ/ ¯ µ . φ theory in spacetime with two extra com-pact dimensions Let us consider φ theory in a spacetime with two extra coordinates y , y ,and metric tensor G MN = (1 , − , − , − , η mn ) = ( γ µν , η mn ) , (15)where M, N = ( µ, m ), µ = 0 , , , m = 1 ,
2, and η mn stands for the metrictensor of a 2D compact manifold. The scalar field φ ( x, y ) is assumed tobe defined on a manifold M ⊗ T /Z with equal compactification radii R c .Thus, the field fulfills the periodicity and parity conditions φ ( x, y ) = φ ( x, y + 2 πR c ) ,φ ( x, y ) = φ ( x, − y ) , (16)where y = ( y , y ). A manifold with another topology will be considered inSection 4. 5he action in six dimensions with two compact dimensions is given bythe following expression S = Z d x πR c Z − πR c dy πR c Z − πR c dy √− G h ∂ M φ ( x, y ) ∂ M φ ( x, y ) − m φ ( x, y ) − g φ ( x, y ) i , (17)where G = det( G MN ). The canonical dimension of φ ( x, y ) is equal to 2. Thecoupling constant g is dimensionless. It is clear that in the limit R c → ∞ the action (17) becomes an 6D action of a scalar field with interaction gφ in six infinite spacetime dimensions (see the previous section).We can use the following Fourier expansion of the field φ ( x, y ) = 12 πR c ∞ X n = −∞ ∞ X n = −∞ e i ( n y + n y ) /R c φ n ( x ) , (18)where n = ( n , n ). Correspondingly, we have φ n ( x ) = 12 πR c πR c Z − πR c dy πR c Z − πR c dy e − i ( n y + n y ) /R c φ ( x, y ) . (19)Note that every KK mode has canonical dimension 1.If we require that the Kaluza-Klein (KK) modes φ n ( x ) are normalized, Z d xφ n ( x ) φ n ′ ( x ) = δ n,n ′ , (20)then Z d x Z d y φ n ( x, y ) φ ∗ n ′ ( x, y ) = δ n,n ′ . (21)The masses of the KK excitations are m n = m + n R c , (22)6here n = n + n , and m means zero mode mass. Thus, the effective S = Z d x √− γ n ∂ µ φ ( x ) ∂ µ φ ( x ) − m φ ( x ) − X n =0 h ∂ µ φ n ( x ) ∂ µ φ n ( x ) − m n φ n ( x ) i − g h φ ( x ) + φ ( x ) X n =0 φ n ( x ) φ − n ( x )+ X n,m,k =0 φ n ( x ) φ m ( x ) φ k ( x ) δ n + m + k, io , (23)where γ = det( γ µν ). Here g = g πR c (24)is an effective four-dimensional coupling constant. Thus, it is the inversecompactification scale R − c that makes g a quantity with the dimension ofmass. One of our main goals is a calculation of a scale dependence of the couplingconstant g (24). As one can see from (23), it is the same for zero modeinteractions, interactions between zero and KK modes, and non-zero modeinteractions. That is why, here and in what follows it is assumed that allexternal particles have zero KK numbers. From the very beginning, we put m = 0.The four-dimensional self-energy of the scalar field at order O( g ) is givenby the diagram in Fig. 1. It can be divided into two partsΣ( p ) = Σ ( p ) + Σ KK ( p ) , (25)where Σ ( p ) = − i g ¯ µ ǫ Z dx Z d D k (2 π ) D k + p x (1 − x )] (26)7 pk Figure 1: The self-energy diagram for the scalar field in the φ theory in theone-loop approximation.is the contribution from zero mode, andΣ KK ( p ) = − i g ¯ µ ǫ X n =0 1 Z dx Z d D k (2 π ) D k + p x (1 − x ) − m n ] (27)is the contribution from KK massive modes. It is assumed that p <
0. Wedefine D = 4 − ε . (28)We find that Σ ( p ) = g π ) − ǫ Γ( ǫ ) (cid:18) ¯ µ − p (cid:19) ε Z dx [ x (1 − x )] − ε = α π R − c (cid:20) N ε − ln − p ¯ µ + 2 (cid:21) + O( ε ) , (29)where N ε = 1 ε − γ E + ln 4 π . (30)Thus, zero mode contributes to the mass renormalization only.Now we consider the contribution from the massive modesΣ KK ( p ) = − i g (¯ µR c ) ǫ R − c X n ,n =0 1 Z dx Z d D l (2 π ) D × l + p R c x (1 − x ) − n − n ] , (31)8here l = kR c . Since Z d D l (2 π ) D l + p R c x (1 − x ) − n − n ] = i (4 π ) − ε Γ( ε )[ − p x (1 − x ) + n + n ] − ε , (32)we obtain Σ KK ( p ) = α π Γ( ε )(4 π ) ε (¯ µR c ) ǫ R − c × X n ,n =0 1 Z dx [ − p R c x (1 − x ) + n + n ] − ε . (33)The series in (33) converges absolutely for Re ε >
1. To define this series forother values of ε , we require its analytic continuation using two-dimensionalinhomogeneous Epstein zeta function Z a ( s ) [28] Z a ( s ) = X n ,n ∈ Z ′ n + n + a ) s , (34)with a > n = 0 is to be excludedfrom the sum). The zeta function regularization method for the quantumphysical systems was proposed for the first time in [29, 30]. The Riemannzeta function ζ ( s ) was used in fixing a critical spacetime dimension of thestring theory (see, for instance, [31]). Recently, one-dimensional inhomoge-neous Epstein zeta function Z a ( s ) was applied to quantify the UV divergencesinduced by the KK fields [11]-[13]. In [14] both Z a ( s ) and n -dimensional in-homogeneous function Z an ( s ) were used.In Appendix A formula (A.1) is presented which gives an analytical con-tinuation for the function Z a ( s ). It is defined on the complex plane of s . Ithas an infinite number of simple poles, but converges both in the s → a = 0. These results is a consequence of the analytical propertiesof the inhomogeneous Epstein zeta function.Let us define c = − p R c x (1 − x ) . (35)Note that c >
0, except for two points x = 0 ,
1. We obtain from (33)-(35)Σ KK ( p ) = α π Γ( ε )(4 π ) ε (¯ µR c ) ǫ R − c Z dxZ c ( ε ) , (36)9here Z c ( ε ) = − c − ε − πc − ε − ε + A ( ε ; c )Γ( ε ) . (37)Since Z c ( a ) is finite for c = 0, see (A.3), we can take c > x ∈ [0 , A ( s ; a ) in (37) is given by eq. (A.2). The function A ( ε ; c ) converges, as ε →
0, and, consequently, Z c (0) = − (1 + πc ). The KKdivergence (the first term in (37)) exactly cancels the zero mode divergence(29). A similar effect was seen in the context of quantum electrodynamicswith one ED [12]. Since A ( ε ; c ) decreases exponentially as c → ∞ , we findfor large R c Σ( p ) = p α ε )(4 π ) ε (cid:18) ¯ µ − p (cid:19) ε Z dx [ x (1 − x )] − ε = p α (cid:18) N ε − ln − p ¯ µ + 53 (cid:19) + O( ε ) . (38)As a result, for µR c ≫
1, the field renormalization constant is equal to Z φ = 1 − α (cid:18) N ε − ln µ ¯ µ + 53 (cid:19) . (39)It differs from the field renormalization in the φ theory by a constant termonly. Note, there is no dependence on R c in (39). Since Σ( p ) ∼ p , the renor-malized theory remains massless in the one-loop approximation (no massrenormalization holds). The effective four-dimensional three-point vertex Γ (3) is defined by the dia-gram presented in Fig. 2. It is a sum of two terms,Γ (3) ( p, q ) = Γ (3)0 ( p, q ) + Γ (3)KK ( p, q ) , (40)where Γ (3)0 ( p, q ) = 2 g ¯ µ ǫ Z dx x Z dy Z d D k (2 π ) D k − M ) , (41)10 p + qp q Figure 2: Three-particle vertex in the scalar φ theory in the one-loop ap-proximation.andΓ (3)KK ( p, q ) = 2 g ¯ µ ǫ X n ,n =0 1 Z dx x Z dy Z d D k (2 π ) D k − M − m n ) = 2 g ¯ µ ǫ R εc × X n ,n =0 1 Z dx x Z dy Z d D l (2 π ) D l − M R c − n − n ) . (42)Here a notation M = − x [ p xy (1 − y ) + q y (1 − x ) + ( p + q ) (1 − x )(1 − y )] (43)is introduced. We assume that p , q , ( p + q ) <
0. It means that M > x, y ) = (1 , , (1 , ε →
0, and we obtainΓ (3)0 ( p, q ) = ( − ig ) απ Γ(1 + ε )(4 π ) ε ¯ µ ǫ R − c Z dx x Z dy ( M ) − − ε . (44)In particular, we find for ε = 0Γ (3)0 ( p, q ) | p = q =( p + q ) = − µ = ( − ig ) α π B ( µR c ) − , (45)11here B = 2 Z dx Z dy [1 − x + xy (1 − y )] − = 127 (cid:20) ψ (cid:18) (cid:19) + ψ (cid:18) (cid:19) − ψ (cid:18) (cid:19) − ψ (cid:18) (cid:19)(cid:21) , (46) ψ ( z ) = ( d /dz ) ln Γ( z ) being the trigamma function [32].The integral on the right-hand side of eq. (42) is equal to Z d D l (2 π ) D l − M R c − n − n ) = − i π ) − ε Γ(1 + ε )( M R c + n + n ) − − ε , (47)that results inΓ (3)KK ( p, q ) = ( − ig ) απ Γ(1 + ε )(4 π ) ε (¯ µR c ) ǫ Z dx x Z dyZ M R c (1 + ε ) . (48)Thus, the infinite number of UV divergences results in the two-dimensionalinhomogeneous Epstein zeta function. We find from eqs. (A.1), (A.2) Z M R c (1 + ε ) = − ( M R c ) − − ε + π ( M R c ) − ε ε + A (1 + ε ; M R c )Γ(1 + ε ) , (49)with A (1; c ) being a finite quantity. As one can see from (49), Z M R c (1 + ε )has a simple pole at ε = 0. It can be easily shown that in the limit ε → (3) ( p, q ) = ( − ig ) απ Γ(1 + ε )(4 π ) ε (¯ µR c ) ǫ × Z dx x Z dy " π ( M R c ) − ε ε + A (1 + ε ; M R c )Γ(1 + ε ) . (50)Thus, for ε → Z Γ = 1 − απ Z dx x Z dy " Γ( ε ) π (4 π ) ε (cid:18) ¯ µ M µ (cid:19) ε + A (1; M µ R c ) . (51)12here M µ = µ x [1 − x + xy (1 − y )] , (52) − µ being the renormalization point. As one can see from (51), the vertexrenormalization constant Z − depends both on the ratio µ/ ¯ µ and on thecompactification radius via dimensionless parameter µR c . The vertex hasa divergence related with a summation over KK number, while Feynmanintegral is finite.However, for µR c ≫ M µ R c ≫ A (1; M µ R c ) decreases exponentially (see eq. (A.2)), and we obtain Z − = 1 + α (cid:20) N ε − ln (cid:18) µ ¯ µ (cid:19) − C (cid:21) , (53)where C = 2 Z xdx Z dy { ln x + ln[1 − x + xy (1 − y )] } = 2 B − . (54)As we can see, if the compactification radius exceeds the physical scale, R c ≫ µ − , it disappears from the renormalization constants (39) and (51).The renormalized effective four-dimensional vertex is proportional to g (24). The fact that the coupling of the four-dimensional fields becomessmaller at larger R c can be easily understood. As it follows from (18), thewave function of the field φ n ( x ) in the y -space is given by ψ n ( y ) = 12 πR c e iny/R c . (55)The coupling constant of three fields φ n ( x ), φ m ( x ), φ k ( x ) is defined by over-lapping of their wave functions g πR c Z − πR c dy πR c Z − πR c dy ψ n ( y ) ψ m ( y ) ψ k ( y ) = g πR c δ n + m + k, . (56)It tends to zero as R c grows. Thus, in the limit R c → ∞ (all six dimensionsare infinite), the φ theory becomes a theory of a free scalar field, whosepropagator is equal to that of the φ theory.13 .3 Running coupling constant Let us consider large values of the mass scale µ , namely, µ ≫ R − c . It followsfrom eqs. (8), (39) and (53) that in the one-loop approximation the betafunction is equal to β ( g ) = − R c π g , (57)and, correspondingly, µ ∂α ( µ ) ∂µ = − R c α ( µ ) , (58)where α = g π . (59)Let us note, it is the dimensional variable R c ln( µ /µ ), not the dimensionlessquantity ln( µ /µ ), which should be regarded as an evolution parameter forthe coupling constant α ( µ ). It is to be expected, since the coupling α hasdimension −
2. As a result, we obtain α ( µ ) = α ( µ )1 + 316 α ( µ ) R c ln( µ /µ ) = 163 R c ln( µ / Λ ) , (60)where Λ = µ exp (cid:2) − / (3 α ( µ ) R c ) (cid:3) = µ exp (cid:2) − / (3 α ( µ ) (cid:3) . (61)We remind that eqs. (60) and (61) hold in the one-loop approximation andat µ ≫ Λ. Ghost pole at µ = Λ is safely eliminated if to respect the causality[33].Thus, the effective four-dimensional scalar φ theory in the flat spacetimewith two compact EDs exhibits the property of asymptotic freedom . Namely,its effective coupling constant α ( µ ) tends logarithmically to zero, as the massscale µ grows. One can say that four-dimensional theory does not forget itshigher dimensional origin.All this can be understood as follows. The renormalization of the couplingconstant is defined by the UV divergences and renormalization scale µ , and“it is not aware” of the scale R − c , provided µ ≫ R − c . In other words,the large scale R c is irrelevant to a small-distance physics. As a result, the14ffective four-dimensional coupling constant g exhibits a large-scale behaviorof the coupling constant in the φ theory. For a detailed discussion of thisphenomenon, see Section 5.It is interesting to compare our prediction (57) with the results obtainedfor an effective 4D λφ theory in a spacetime with one compact ED [12].Is has been found that in such a theory an effective coupling constant inone-loop approximation is renormalized by the constant Z / φ Z − = 1 + 3 λ π (cid:20) ε + ln( µR c ) (cid:21) . (62)Note that λ = ¯ λ/ (2 πR c ), where ¯ λ is the coupling constant in a 5D actionwith dimension −
1. Thus, one can not obtain a RG-like equation for λ withrespect to the scale ¯ M = R − c , as it is erroneously stated in [12] (see also[15]), except when ¯ λ = ¯ λ ( R c ) = constant × R c . For instance, if we assumethat this relation takes place for small R c , then we come to the equation withrespect to the intrinsic scale of the spacetime topology, dλd ln ¯ M = − λ π , (63)valid for large ¯ M .As for the case µ ≪ R − c , it can be shown that α ( µ ) tends to a constantvalue, as µ grows (while being less than R − c ). As one see from (36), thetotal divergence in ε = (4 − D ) / Z c ( ε ) is finite,as ε → Z c (1 + ε ) ∼ ε − , as ε → µ ≫ R − c , infinite and µ -dependentparts of the counterterms of the origin, six-dimensional, theory and thoseof the reduced theory coincide. However, our calculations have shown that µ -dependent parts of the renormalization constants differ for µ ≪ R − c , and anontrivial dependence on R c occurs. Let us note that the divergent ε − termsremain the same regardless of a value of R c , in a full accordance with theresults of [34]. It is to be expected, since the compactification is an infraredprocess which can not change the UV properties of the theory.15 Compactification on orbifold S /Z M ⊗ T /Z was studied. In this section we examinethe case when the six-dimensional scalar field φ is defined on a manifold M ⊗ S /Z , with a radius of two-dimensional sphere S to be R c . It isappropriate to introduce spherical coordinates θ, φ , and use the followingexpansion φ ( x, θ, φ ) = 1 R c ∞ X l =0 l X m = − l Y ml ( θ, φ ) φ lm ( x ) , (64)where Y ml ( θ, φ ) ( m = − l, − l + 1 , . . . , l − , l ) are spherical harmonics [35].They obey the orthogonality condition π Z dφ π Z sin θdθ Y ml ( θ, φ )[ Y m ′ l ′ ( θ, φ )] ∗ = δ ll ′ δ mm ′ . (65)Using formula π Z dφ π Z sin θdθ Y ( θ, φ ) Y ml ( θ, φ ) Y − ml ( θ, φ ) = ( − m √ π , (66)one can show that an effective four-dimensional coupling constant is¯ g = g √ πR c , (67)for zero mode interaction. For interactions between zero mode and KKmodes, a coupling constant is equal to ( − m ¯ g . The masses of the KKexcitations are known to be numerated by an integer l = 0 , , , . . . [17, 18], m l = m + l ( l + 1) R c . (68)Let us consider zero-mode self-energy Σ( p ) in the one-loop approxima-tion (Fig. 1). It is given byΣ( p ) = α ε )(4 π ) ε (¯ µR c ) ǫ R − c ∞ X l =0 l X m = − l Z dx [ l ( l + 1) + c ] − ε = α ε )(4 π ) ε (¯ µR c ) ǫ R − c Z dx ∞ X l =0 l + 1[ l ( l + 1) + c ] ε , (69)16here c is defined by eq. (35). The series on the right-hand side can berepresented as ζ t ( s ; c ) = ∞ X l =0 l + 1[ l ( l + 1) + c ] s = 11 − s ddα ∞ X l =0 l ( l + 1) + α (2 l + 1) + c ] s − (cid:12)(cid:12)(cid:12) α =0 . (70)We have ∞ X l =0 l ( l + 1) + α (2 l + 1) + c ] s − = ∞ X l =0 l + a ) + q ] s − = ζ t ( s ; a, q ) , (71)where a = 12 + α , q = c − − α , (72)and an analytic expression for ζ t ( s ; a, q ) is given by eq. (A.6). Note that[ dq/dα ] | α =0 = 0. For c ≫
1, we obtain form (70)-(72), and (A.6) that ζ t ( ε ; c ) = − c − ε (cid:20) − c (cid:21) + O( c − ) , (73)as ε →
0. As a result, we come to expression (39) (up to unimportant finiteconstant).The above consideration can be also applied to a calculation of the ef-fective four-dimensional vertex for zero mode interaction in the one-loopapproximation (Fig. 2). Taking into account that ζ t (1 + ε ; c ) = c − ε (cid:20) ε + 112 c (cid:21) + O( c − ) , (74)as ε →
0, we reproduce formula (53) (up to a constant factor). All saidabove allows us to conclude that in the large R c region our main results donot depend on a topology of the two-dimensional compact manifold. Ultimately the circumference of an infinitecircle and a straight line are the same thing.
Galileo Galilei, “Dialogue concerning the twochief world systems: Ptolemaic and Coperni-can. The third day.”17s we already mentioned in Section 3, a nontrivial dependence of physicalquantities on the compactification radius appears when the physical scale( µ − , in our case) becomes much larger than R c . In the opposite case, µ − . R c , when a physical process goes “inside a sphere of the radius R c ”, such adependence disappears.Some other physical examples can be given which illustrates these state-ments. In [36] a generalization of the Froissart-Martin bound for scatteringin D -dimensional spacetime with one compact dimension has been derived.The upper bound for the imaginary part of the hadronic scattering amplitude T D ( s, t ) was found to beIm T D ( s, s R D − ( s ) Φ (cid:18) R R c , D (cid:19) . (75)In (75) the “transverse radius” is given by R ( s ) ∼ t − / ln s , where t denotesthe nearest singularity in the t -channel. R c is the compactification radius ofthe ED, and Φ( R /R c , D ) is a known function. At R c ≫ R ( s ), the inequality(75) reproduces the Froissart-Martin bound in a flat spacetime with arbitrary D dimensions [37] σ D tot ( s ) const( D ) R D − ( s ) , (76)while in the opposite limit R c ≪ R ( s ) it results in the inequality [36]Im T D ( s, const( D ) s R D − ( s ) R c . (77)In [38] an analogous result has been obtained for the scattering of two SMparticles on a 3D brane embedded into a flat spacetime with n compact EDs( D = 4 + n ). The inelastic cross section σ D in ( s ) was calculated in the trans-planckian region √ s ≫ M D , | t | , where t is a momentum transfer squared,and M D is a fundamental Planck scale in D dimensions. The result of thecalculations is the following σ D inel ( s ) ≃ const( D ) × ( R n ( s ) , R c ≫ ¯ R ( s ) ,R ( s ) R nc , R c ≪ ¯ R ( s ) , (78)where ¯ R ( s ) = 2 R g ( s ) p ln( s/M D ), R g ( s ) being the “Regge gravitational ra-dius” (for more details, see [38]).To summarize, we can say that the dependence of physical observableson the compactification radius of the ED(s) arises only when the physical18cale R phys of the process becomes larger than (comparable with) R c . Onthe contrary, if R phys ≪ R c , this dependence disappears (a physical processoccurs on distances ∼ R phys , and it does not “feel” the large scale R c at all). We have considered compactification of the asymptotically free φ D =6 theoryto manifolds M ⊗ T /Z and M ⊗ S /Z . The asymptotically free behaviorof the dimensionless triple coupling in M is being inherited by dimensionaltriple couplings of the light modes in both cases of compactification, withdetails depending of the shape of compactification. We also have consid-ered the physical implications for high-energy scattering which has the sameenergy dependence as in simple four-dimensional case but retaining the com-pactification radius as a parameter, when the interaction radius exceeds thecompactification size, while the “memory” of the latter disappears at short-distance interactions which has now a different energy dependence. Appendix A
We give some useful properties of the two-dimensional inhomogeneous Ep-stein zeta function Z a ( s ) (34), with a >
0. In [39] the following explicitexpression was derived Z a ( s ) = − a − s + πa − s s − A ( s ; a )Γ( s ) , (A.1)where A ( s ; a ) = 4 " a / (cid:18) π √ a (cid:19) s ∞ X n =1 n s − / K s − / (2 πn √ a )+ a / (cid:18) π √ a (cid:19) s ∞ X n =1 n s − K s − (2 πn √ a )+ √ π ) s ∞ X n =1 n s − / X d k n d − s (cid:18) ad (cid:19) / − s/ × K s − / πn r ad ! . (A.2)19ere K ν ( z ) is the modified Bessel function of the second kind, d k n is thedivision of n . As one can see from (A.2), A ( s ; a ) decreases exponentially, as a → ∞ . In the limit a → Z ( s ) = 2 ζ (2 s ) + 2 √ π Γ( s − / s ) ζ (2 s − π s Γ( s ) ∞ X n =1 n s − / X d k n d − s K s − / (2 πn ) , (A.3)where Z ( s ) ≡ Z ( s ) = X n ,n ∈ Z ′ n + n ) s (A.4)is the two-dimensional Epstein zeta function [41], and ζ ( s ) is the Riemannzeta function [32]. Note that all (multi)series in (A.2), (A.3) are exponentiallyconvergent.The above formulas are valid over the whole complex plane. The inho-mogeneous Epstein function (A.1) exibits an infinite number of simple polesat s = 1 , / , − / , − / , . . . , while the homogeneous Epstein function (A.3)has only two simple poles at s = 1 and s = 1 /
2, with the residues π and − /
2, respectively [28]. Note that both functions are regular at s = 0. Theformula Z a ( s ) (cid:12)(cid:12) s → = − a − s + πa − s s − s ) ∞ X n =1 n (cid:20) e − πn √ a + √ a K (2 πn √ a )+ X d k n d e − πn √ a/d (cid:21) + O( s ) (A.5)gives an expansion of two-dimensional inhomogeneous Epstein function aroundthe point s = 0.The truncated Epstein-like zeta function is given by the expression [39] ζ t ( s ; a, q ) = ∞ X n =0 n + a ) + q ] s = (cid:18) − a (cid:19) q − s + q − s Γ( s ) ∞ X m =1 ( − m Γ( m + s ) m ! × ζ H ( − m ; a ) q − m + √ π Γ( s − / s ) q / − s + 2 π s Γ( s ) q / − s/ ∞ X n =1 n s − / cos(2 πna ) K s − / (2 πn √ q ) , (A.6)20ith q >
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