CK-TH-2000-002 hep-th/0001217 January 2000 String Loop Threshold Corrections for N=1 Generalized Coxeter Orbifolds 0 0 0 2 C. Kokorelis n a CITY University Business School J 1 Department of Investment, Risk Management and Insurance 3 Frobischer Crescent, Barbican Centre, London, EC2Y 8HB,U.K 1 v 7 1 2 1 0 0 0 ABSTRACT / h t - p We discussthecalculation of thresholdcorrections togauge couplingconstants for e the, only, non-decomposable class of abelian (2,2) symmetric N=1 four dimendional h v: heterotic orbifold models, wheretheinternal twistis realized as ageneralized Coxeter i X automorphism. The latter orbifold was singled out by earlier work as the only N=1 r heteroticZ orbifoldthatsatisfythephenonelogical criteriaofcorrectminimalgauge N a coupling unification and cancellation of target space modular anomalies. The purpose of this paper is to examine the appearance of one-loop string threshold corrections in the gauge couplings of the four dimensional generalized non-decomposable N = 1 orbifolds of the heterotic string. In 4 N = 1 orbifold compactifications the D process of integrating out massive string modes, causes the perturbative one-loop threshold corrections1, to receive non-zero corrections in the form of automorphic functions of the target space modular group. At special points in the moduli space previously massive states become massless and contribute to gauge symmetry enhancement. As a result the appearance of massless states in the running coupling constants appears in the form of a dominant logarithmic term [1, 2]. The moduli dependent threshold corrections of the N = 1 4D orbifolds receive non-zero one loop corrections from orbifold sectors for which there is a complex plane of the torus T6 left fixed by the orbifold twist Θ. When the T6 can be decomposed into the direct sum T2 T4, the one-loop moduli dependent threshold corrections (MDGTC) are invariant ⊕ under the SL(2,Z) modular group and are classified as decomposable. Otherwise, when the action of the lattice twist on the T torus does not decompose into the orthogonal 6 sum T = T T with the fixed plane lying on the T torus, MDGTC are invariant 6 2 4 2 ⊕ under subgroups of SL(2,Z) and the associated orbifolds are called non-decomposable. The N = 1 perturbative decomposable MDGTC have been calculated, with the use of string amplitudes, in [3]. The one-loop MDGTC integration technique of [3] was extended to non-decomposable orbifolds in [4]. Further calculations of non-decomposable orbifolds involved in the classification list of N = 1 orbifolds of [5] have been performed in [6]. Here we will perform the calculation of one-loop threshold corrections for the class of Z orbifolds, that can be found in the classification list of [5], defined by the Coxeter twist, 8 Θ = exp[2πi(1, 3,2)] on the root lattice of A A . This orbifold was missing from the 8 − 3 × 3 list of calculations of MDGTG of non-decomposable orbifolds of [4, 6] and consequently its one-loop moduli dependent gauge coupling threshold corrections were not calculated in [4, 6]. In [7, 8] we found that this orbifold is non-decomposable and it is the only one that poccesses this property from the list of generalized Coxeter orbifolds given in [5]. Its twist can be equivalently realized through the generalized Coxeter automorphism S S S P P P on the root lattice. 1 2 3 35 36 45 Moreover in [8], where a classification list of the non-perturbative gaugino condensation 1which receive non-zero moduli dependent corrections from the N =2 unrotated sectors. 1 generated superpotentials and µ-termsof allthe N = 1 four dimensional non-decomposable heterotic orbifolds was calculated, its non-perturbative gaugino condensation generated su- perpotentialwasgiven. InthisworkwewillcalculateitsMDGTCfollowingthetechniqueof [4]. Our calculation completes the calculation of the threshold corrections for the classifica- tion list of four dimensional Coxeter orbifold compactifications with N = 1 supersymmetry of [5]. The generalized Coxeter automorphism is defined as a product of the Weyl reflections2 S of the simple roots and the outer3 automorphisms, the latter represented by the transpo- i sition of the roots. An outer automorphism represented by a transposition which exchanges the roots i j, is denoted by P and is a symmetry of the Dynkin diagram. ij ↔ In string theories the one-loop gauge couplings below the string scale evolves according to the RG equation 1 k b M2 1 a a string = + ln + , (2) g2(p2) g2 16π2 p2 16π2△a a Mstring where M the string scale and b the β-function coefficient of all orbifold sectors. For string a decomposable orbifolds [3] the MDGTC associated with the gauge couplings g−2 corre- △a a sponding to the gauge group G , are determined in terms of the N = 2 sectors, fixed under a both (g, h) boundary conditions, of the orbifold, namely d2τ d2τ = b(h,g) (τ,τ¯) bN=2 . (3) △ τ a Z(h,g) − a τ ZF 2 (g,h) ZF 2 X Here, bN=2 is the β-function coefficient of all the N = 2 sectors of the orbifold, b(h,g), a a Z the β-function coefficient of the N = 2 sector untwisted under (g,h) and its partition (h,g) function (PF) respectively. The integration is over the fundamental domain of the F PSL(2,Z). For the case of non-decomposable orbifolds the situation is slightly different, namely d2τ d2τ = b(h0,g0) (τ,τ¯) bN=2 . (4) △ τ a Z(h0,g0) − a τ ZF˜ 2 (g0X,h0)∈O ZF 2 2The Weyl reflection Si is defined as a reflection <x,ei > Si(x)=x 2 ei, (1) − <ei,ei > with respect to the hyperlane perpendicular to the simple root. 3 an automorphism is called outer if it cannot be generated by a Weyl reflection. 2 The difference with the decomposable case now is that the sum, in the first integral of (4) is over those N = 2 sectors that belong to the N = 2 fundamental orbit and the O integration is not over but over the fundamental domain ˜. Because the PF , for F F Z(g0,h0) non-decomposable orbifolds, is invariant under subgroups of the modular group i.e Γ˜, the ˜ ˜ domain is generatedby theactionofthose modulartransformationsthatgenerate Γfrom F Γ. In the example that we examine later in this work, Γ˜ = Γ (2) and ˜ = 1,S,ST . In o F { }F ˜ turn the fundamental orbit is generated by the action of on the fundamental element O F of this orbit. For the orbifold Z there are four complex moduli fields. There are three (1,1) moduli 8 due to the three untwisted generations 27 and one (2,1)-modulus due to the one untwisted generation ¯27. The realization of the point group is generated by 0 0 0 0 0 1 − 1 0 0 0 0 0   0 1 0 0 0 1 Q =  − . (5)  0 0 1 0 0 0       0 0 0 1 0 1   −   0 0 0 0 1 0      If the action of the generator of the point group leaves some complex plane invariant then the corresponding threshold corrections have to depend on the associated moduli of the unrotated complex plane. There are three complex untwisted moduli: three (1,1)–moduli andno(2,1)-modulusduetothethreeuntwisted 27generationsandnon-existent untwisted ¯27 generation. The metric g (defined by g =< e e >) has three and the antisymmetric ij i j | tensor field B an other three real deformations. The equations gQ = Q∗g and4 bQ = Q∗b determine the background fields in terms of the independent deformation parameters. Solving the background field equations one obtains for the metric R2 u v u 2v R2 u − − − − u R2 u v u 2v R2   − − − v u R2 u v u G =  − , (6)  u v u R2 u v   −     2v R2 u v u R2 u   − − −   u 2v R2 u v u R2   − − − −    4 ∗ T −1 By definition () mean (() ) . 3 with R,u,v and the antisymmetric tensor field : ∈ ℜ 0 x z y 0 y − x 0 x z y 0   − z x 0 x z y B =  − − , (7)  y z x 0 x z   − − −     0 y z x 0 x   − − −   y 0 y z x 0   − − −    with x,y,z . ∈ ℜ The N=2 orbit is given by these sectors which contain completely unrotated planes, = (1,Θ4),(Θ4,1),(Θ4,Θ4). O The element (Θ4,1) can be obtained from the fundamental element (1,Θ4) by an S– transformation on τ and similarly (Θ4,Θ4) by an ST–transformation. The partition func- tion for the zero mode parts Ztorus of the fixed plane takes the following form[5] (g,h) Z(t1o,rΘu4s)(τ,τ¯,G,B) = q12PL2q¯PR2, P∈X(ΛN⊥) 1 Z(tΘor4u,1s)(τ,τ¯,G,B) = V q21PL2q¯21PR2, Λ⊥N P∈X(Λ⊥N)∗ 1 Z(tΘor4u,Θs4)(τ,τ¯,G,B) = V q21PL2q¯21PR2qiπ(PL2−PR2), (8) Λ⊥N P∈X(Λ⊥N)∗ where with Λ⊥ we denote the Narain lattice of A A which has momentum vectors N 3 × 3 p p P = +(G B)w , P = (G+B)w (9) L R 2 − 2 − and Λ is that part of the lattice which remains fixed under Q4 and V its volume. N⊥ Λ⊥ N The lattice in our case is not self dual in contrast with the case of partition functions Ztorus(τ,τ¯,g,b)of[3]. Stateddifferentlythegeneralresultis-forthecaseofnon-decomposa (g,h) bleorbifolds-thatthemodularsymmetrygroupissomesubgroupofΓandasaconsequence the partition function τ Ztorus(τ,τ¯,g,b) is invariant under the same subgroup of Γ. 2 (g,h) The subspace corresponding to the lattice Λ⊥ can be described by the following winding N 4 and momentum vectors, respectively: n1 m 1 n2 m    2  0 m w =   , n1,n2 Z and p =  − 1  ,m ,m Z. (10) 1 2  0  ∈  m  ∈    − 2   n1   m1       n2   m     2      They are determined by the equations Q4w = w and Q∗4p = p . The partition function τ Ztorus (τ,τ¯,g,b) is invariant under the group Γ (2), congruence subgroup of Γ. Before we 2 (1,Θ4) 0 discuss the calculation of threshold corrections let us give some details about congruence subgroups. The homogeneous modular group Γ′ SL(2,Z) is defined as the group of two ≡ by two matrices whose entries are all integers and the determinant is one. It is called the ”full modular group and we symbolize it by Γ′. If the above action is accompanied with the quotient Γ PSL(2,Z) Γ′/ 1 then this is called the ’inhomogeneous modular group’ ≡ ≡ {± } andwesymbolize itbyΓ. The fundamentaldomainofΓisdefinedastheset ofpointswhich are related through linear transformations τ aτ+b. If we denote τ = τ + τ then the → cτ+d 1 2 fundamental domain of Γ is defined through the relation = τ C τ > 0, τ 1, τ F { ∈ | 2 | 1| ≤ 2 | | ≥ 1 . One of the congruence subgroup of the modular group Γ is the group Γ (n). The group 0 } Γ (2) can be represented by the following set of matrices acting on τ as τ aτ+b: 0 → cτ+d a b Γ (2) = ad bc = 1,(c = 0 mod 2) (11) 0 { c d ! | − } It is generated by the elements T and ST2S of Γ. Its fundamental domain is different from the group Γ and is represented from the coset decomposition ˜ = 1,S,ST . In addition F { }F the group has cusps at the set of points ,0 . Note that the subgroup Γ0(2) of SL(2,Z) {∞ } is defined as with b = 0 mod 2. The integration of the contribution of the various sectors (g,h) is over the fundamental domain for the group Γ (2) which is a three fold covering of the upper complex plane. By 0 taking into account the values of the momentum and winding vectors in the fixed directions 5 we get for Ztorus (1,Θ4) Z(t1o,rΘu4s)(τ,τ¯,g,b) = q21PLtG−1PLq12PRtG−1PR (PL,XPR)∈Λ⊥N = e2πiτptwe−πτ2(12ptG−1p−2ptG−1Bw+2wtGw−2wtBG−1Bw−2ptw). (12) p,w X Consider now thethe following parametrizationof the torusT2, namely define the the (1,1) T modulus and the (2,1) U modulus as: T = T +iT = 2(b+i√detg ), 1 2 ⊥ (13) U = U +iU = 1 (G +√detG ), 1 2 G⊥11 ⊥12 ⊥ where g is uniquely determined by wtGw = (n1n2)G (n1n2) and b the value of the B ⊥ ⊥ 12 element of the two-dimensional matrix B of the antisymmetric field. This way one gets T = 4(x y)+i 8 v, (14) − U = i. (15) Even if we have said that we expect that this Z orbifold doe not have a h(2,1) U-modulus 8 field, the T2 torus has a U-modulus. However its value for the Z orbifold is fixed. The 8 partition function Ztorus (τ,τ¯,g,b) takes now the form (1,Θ4) Z(t1o,rΘu4s)(τ,τ¯,T,U) = e2πiτ(m1n1+m2n2)e−T2πUτ22|TUn2+Tn1−Um1+m2|2 . (16) m1,m2∈X2Z;n1,n2∈Z By Poisson resummation on m and m , using the identity: 1 2 1 e[−π(p+δ)tC(p+δ)]+2πiptφ] = V−1 e[−π(l+φ)tC−1(l+φ)−2πiδt(l+φ)], (17) Λ √detC l∈Λ p∈Λ∗ X X we conclude τ2 Z(t1o,rΘu4s)(τ,τ¯,T,U) = 14 A∈Me−2πiTdetA T2e−τ2πUT22|(1,U)A(τ1)|2 , (18) P where n 1l = 1 2 1 (19) M n 1l ! 2 2 2 6 and n ,n ,l ,l Z. 1 2 1 2 ∈ From (18) one can obtain τ Ztorus (τ,τ¯) by an S–transformation on τ. After exchanging 2 (Θ4,1) n and l and performing again a Poisson resummation on l one obtains i i i Z(tΘor4u,1s)(τ,τ¯,T,U) = 14 e2πiτ(m1n21+m2n22)e−T2πUτ22|TUn22+Tn21−Um1+m2|2. (20) m1X,m2∈Z n1,n2∈Z The factor 4 is identified with the volume of the invariant sublattice in(20). The expression τ Ztorus (τ,τ¯,T,U) is invariant under Γ0(2) acting on τ and is identical to that for the 2 (Θ4,1) (Θ4,Θ4) sector. Thus the contribution of the two sectors (Θ4,1) and (Θ4,Θ4) to the coefficient bN=2 of a the β–function is one fourth of that of the sector (1,Θ4), thus 3 bN=2 = b(1,Θ4). (21) a 2 a The coefficient bN=2 is the contribution to the β functions of the N = 2 orbit. Including a the moduli dependence of the different sectors, we conclude that the final result for the threshold correction to the inverse gauge coupling reads 8πe1−γE T (T,T¯,U,U¯) = bN=2 ln T η 4U η((U) 4 . (22) △a − a · | 3√3 2| 2 | 2| | | (cid:18) (cid:19) The value of U is fixed and equal to one as can be easily seen from eqn.(15). In general for 2 Z orbifolds with N 2 the value of the U modulus is fixed. The final duality symmetry N ≥ of (22) is Γ0(2) with the value of U replaced with the constant value i. T Let us use (4), (22) to deduce some information about the phenomenology of the Z 8 orbifold considered in this work. We want to calculate the one-loop corrected string mass unification scale M , that is when two gauge group coupling constants become equal, i.e. X 1 = 1 . We further assume that the gauge group of our theory at the string unification kaga2 kbgb2 scale if given by G = G , where G a gauge group factor. Taking into account (22) we get i i ⊕ MX = Mstring[T2 η(T) 4U2 η(U) 4](bNb2=(b2akkab−−bbNabk=a2)kb), | 2 | | | M 0.7g 1018 Gev. (23) string string ≈ where k the Kac-Moody level associated to the gauge group factor G . i i 7 We will give now some details about the integration in (4) of the integral that we used so far to derive (22). The integration of eqn. (16) is over a Γ (2) subgroup of the 0 modular group Γ since (16) is invariant under a Γ (2) transformation τ aτ+b (with 0 → cτ+d ad bc = 1,c = 0 mod 2). Under a Γ (2) transformation (16) remains invariant if at the 0 − same time we redefine our integers n ,n ,l and l as follows: 1 2 1 2 n′ n′ a c/2 n n 1 2 = 1 2 (24) l1′ l2′ ! 2b d ! l1 l2 ! The integral can be calculated based on the method of decomposition into modular orbits. There are three set’s of inequivalent orbits under the Γ (2), namely : 0 a.) The degenerate orbit of zero matrices, where after integration over ˜ = 1,S,ST F { }F gives as a total contribution I = πT /4. 0 2 b.) The orbit of matrices with non-zero determinants. The following representatives give a non-zero contribution I to the integral: 1 k j 0 p 0 p , − , − , 0 j < k, p = 0 , (25) 0 p ! k j ! k j +p ! ≤ 6 where I +I = (3/2) 4Relnη(T) o 1 − · 2 c.) The orbits of matrices with zero determinant, 0 0 j p , , j,p Z , (j,p) = (0,0) . (26) j p ! 0 0 ! ∈ 6 The first matrix in (26) has to be integrated over the half–band τ C τ > 0 , τ < h 2 1 { ∈ | | } while the second matrix has to be integrated over a half–band with the double width in τ . 1 The total contribution from the modular orbit I gives, 3 8π I = 4 lnη(U) ln(T U )+ γ 1 ln 3 2 2 E − ℜ − − − 3√3 (cid:18) (cid:19) 1 1 1 8π 4Relnη(U) ln(T U )+ (γ 1 ln ). 2 2 E −2 × − 2 × 2 × − − 3√3 Putting I , I , I together we get (22). o I 2 All N = 1 four dimensional orbifolds have been tested in [9] as to whether they satisfy several phenomenological criteria, involving 8 a) correct unification of the three gauge coupling constants at a scale M 1016 Gev, as- X ≈ suming the minimal supersymmetric, Standard Model gauge group G = SU(3) SU(2) × × U(1), particle spectrum with a SUSY threshold close to weak scale, in the two cases of i) a single overall modulus inthe three complex planes T = T = T = T and ii) the anisotropic 1 2 3 squeezing case T >> T ,T , 1 2 3 b) anomaly cancellation with respect to duality transformations of the moduli in the planes rotated by all the orbifold twists. 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