Gauge/Gravity Duality for Interactions of Spherical Membranes in 11-Dimensional Pp-Wave

Nuclear Physics B 728 (2005) 1–30

Gauge/gravity duality for interactions of spherical membranes in 11-dimensional pp-wave

Hok Kong Lee a, Tristan McLoughlin a, Xinkai Wu b

a California Institute of Technology, Pasadena, CA 91125, USA b Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA Received 3 March 2005; received in revised form 19 July 2005; accepted 5 September 2005 Available online 12 September 2005

Abstract We investigate the gauge/gravity duality in the interaction between two spherical membranes in the 11- dimensional pp-wave background. On the supergravity side, we ﬁnd the solution to the ﬁeld equations at locations close to a spherical source membrane, and use it to obtain the light-cone Lagrangian of a spherical probe membrane very close to the source, i.e., with their separation much smaller than their radii. On the gauge theory side, using the BMN matrix model, we compute the one-loop effective potential between two membrane fuzzy spheres. Perfect agreement is found between the two sides. Moreover, the one-loop effective potential we obtain on the gauge theory side is valid beyond the small-separation approximation, giving the full interpolation between interactions of membrane-like objects and that of graviton-like objects.  2005 Elsevier B.V. All rights reserved.

1. Introduction

In this paper we investigate the gauge/gravity duality in the maximally supersymmetric eleven-dimensional pp-wave background. The gauge theory side is represented by a one- dimensional matrix theory ﬁrst proposed in [1] while the gravity side is represented by eleven- dimensional supergravity. Similar investigations have been performed in ﬂat space for various M-theory objects [3–9]. In particular, we will be interested in the two-body interactions of point like gravitons and spherical M2-branes.

E-mail addresses: [email protected] (H.K. Lee), [email protected] (T. McLoughlin), [email protected] (X. Wu).

In pp-wave there is an interesting new connection we can make between a graviton and an M2-brane. Under the influence of the 3-form background whose strength is proportional to a parameter µ, each stable M2-brane curls up into a sphere, with its radius r0 proportional to µ and − its total momentum in the x direction, i.e., we have r0 ∼ µP−. If one takes the limit of µ → 0 while keeping the total momentum P− fixed, then the radius of the sphere goes to zero and we get a point like graviton. If on the other hand, we increase P− simultaneously such that the radius goes to infinity, then the end product will be a flat membrane instead. Thus the gravitons and the flat membranes of flat space can both be regarded as different limits of spherical membranes in pp-wave. One could make another observation by considering two spherical membranes separated by a distance z.Ifz r0, then one expects the membranes to interact like two point-like gravitons. If z r0 then the interaction should be akin to that between flat membranes. There- fore by computing the interactions between membranes of arbitrary radii and separation, we can take different limits to understand the interactions of both gravitons and membranes. In this paper we will compare the light-cone Lagrangian of supergravity with the effective potential of matrix theory. With a slight abuse in terminology, we will sometimes refer to both as the effective potential. On the supergravity side, we will use a linear approximation when solving the field equations. This means all the metric components computed this way will be 2 proportional to no higher than the first power of κ11, higher order effects such as recoiling and other back reactions could then be neglected. On the matrix theory side, the interaction between two spherical membranes begins at one loop, and for the purpose of comparing to linearized supergravity, only the one-loop effective potential is needed. A more detailed discussion can be found in [2]. Due to the complexity of the field equations of 11D supergravity, we will only compute the effective potential of the supergravity side in the near membrane limit (z r0). The graviton limit (z r0) was already computed in a previous paper [2]. On the matrix theory side, however, it is possible to compute the expression for general z and r0, and by taking the appropriate limits, we are able to find perfect agreement with supergravity. In other words, the matrix theory result provides a smooth interpolations between the near membrane limit and the graviton limit. We also compare our results with the those of Shin and Yoshida [10,11] and where there is overlap we again have perfect agreement. This paper is organized as follows. We begin in Section 2 with a brief discussion of the theories on both sides of the duality. The relations between the parameters on the two sides as well as a discussion on the special limits of interests can be found here. In Section 3 we first define the probe membrane’s light-cone Lagrangian on the supergravity side for a general background, and then consider it in the pp-wave background perturbed by a source. Next in Section 4 the linearized field equations are first diagonalized for an arbitrary static source, and then solved in the special case of a spherical membrane source in the near membrane limit (z r0). The metric and the three-form potential are then used to compute the supergravity light-cone Lagrangian. Section 5 is devoted to computation on the matrix theory side. The one-loop effective potential is found by integrating over all the fluctuating fields. The near membrane limit of the resulting potential is compared with the supergravity light-cone Lagrangian and agreement is found. In Section 6 we compute the one-loop interpolating effective potential on the matrix theory side for arbitrary separation that takes us between the membrane limit and the graviton limit. The result is compared with our earlier work [2] as well as with that of Shin and Yoshida [10,11].Thisis followed by a discussion in Section 7. H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30 3

2. The two sides of the duality

2.1. The spherical membranes

The nonzero components of the maximally supersymmetric eleven-dimensional pp-wave metric and the four-form field strength are given by: 3 9 2 1 i 2 1 a 2 g+− = 1,g++ =−µ x + x ,g= δ , (1) 9 36 AB AB i=1 a=4 F123+ = µ. (2) The index convention throughout this paper is: µ,ν,ρ,... take the values +, −, 1,...,9; A,B,C,... take the values 1,...,9; i,j,k,... take the values 1,...,3; and a,b,c,... take the values 4,...,9. The matrix theory action in such a background is known [1]: 9 1 (M3R)2 9 S = dt Tr D XA 2 + iψT D ψ + XA,XB 2 2R t t 4R A=1 A,B=1 9 1 µ 2 3 µ 2 9 − M3R ψT γ A ψ,XA + − Xi 2 − Xa 2 = 2R 3 = 6 = A 1 i 1 a 4 µ (M3R)µ 3 − i ψT γ ψ − i XiXj Xk , (3) 4 123 3R ij k i,j,k=1 I I where Dt X = ∂t X − i[X0,X ]. The constant M in the action above is the eleven-dimensional Planck mass, and R is the compactification radius in the X− light-like direction in the DLCQ formalism. The supersymmetric configurations of this action have been well studied, see for example [12]. Although in the end we will add a perturbation in the X4 to X9 directions to make it nonsupersymmetric, we begin with the following configuration: µ Xi = Ji,Xa = 0, (4) 3MR3 where [Ji,Jj ]=iij k Jk. If Ji is an N × N irreducible matrix, then it represents a single sphere with radius r0 given by:

1 µ µN r = Tr X2 = N 2 − 1 ≈ . (5) 0 N i 6M3R 6M3R The last approximation is taken with N large, such that the matrix theory correction to the radius (denoted as δr0) is negligible compared to r0. In other words, we choose N large enough that δr0 ∼ 1 → 0. The reader is reminded that the purpose of this paper is to compare matrix theory r0 N2 to the predictions of supergravity, so we are not interested in any ﬁnite N effects which are related to matrix theory corrections to supergravity. If Ji is reducible it represents multiple concentric spherical membranes, and the radius of each irreducible component can be found in the same way as for the case above. 4 H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30

The conﬁguration we are going to use is one where Ji is the direct sum of two irreducible components. This represents two membranes of different radius. In Section 5 we will put in nontrivial Xa to break the supersymmetry, which physically corresponds to placing one of the membranes away from the origin. On the supergravity side, the bosonic part of the Lagrangian of a probe membrane in a general background is 1 L Xµ,∂ Xµ =−T − det(g ) − ij k A ∂ Xµ∂ Xν∂ Xρ , (6) i ij 6 µνρ i j k µ ν with T being the membrane tension, and gij ≡ Gµν∂iX ∂j X being the pullback metric. − One can deﬁne the light-cone Lagrangian Llc via a Legendre transformation in the X direction (see Section 3). In the unperturbed pp-wave background, we have 1 A A Π− 2 1 i 2 1 a 2 (L ) = Π−∂ X ∂ X − µ X + X lc pp 2 0 0 2 9 36 2 T A B B A 2 µ i j k − ∂1X ∂2X − ∂1X ∂2X + T ij k ∂1X ∂2X X , (7) 4Π− 3 − with Π− being the momentum density in the X direction. Looking at (Llc)pp, above, we see that one solution to the equations of motion in the unperturbed pp-wave is a spherical membrane i i = 2 a = + = − = = µΠ− at rest with X X r0 , X 0, X t, X 0, with its radius being r0 3T sin θ (note that we 0,1,2 Π− take the worldvolume coordinates σ to be t,θ,φ; also recall that it is sin θ that is a constant on the worldvolume). In terms of the total momentum P− = dθ dφΠ−, this radius is

µP− r = . (8) 0 12πT Later in this paper, we will take the background to be the pp-wave perturbed by a source membrane: Gµν = (Gµν)pp + hµν , and Aµνρ = (Aµνρ)pp + aµνρ . Then Llc = (Llc)pp + δLlc, and we shall compare δLlc with the one-loop effective potential on the matrix theory side (while (Llc)pp agrees with the tree level Lagrangian on the matrix theory side, of course). Identifying the radius calculated on the two sides and P− with N/R, we get the following relation between the tension of the membrane and the Planck mass: 2πT = M3. (9) 5 Another relation we will use to compare the two side is the identiﬁcation κ2 = 16π [2,13], 11 M9 and for convenience we will deﬁne the parameter α = 1 that will appear often on the matrix M3R theory side.

2.2. The effective potential

As already stated, on the supergravity side the quantity of interest is the probe light-cone Lagrangian. On the matrix theory side, the one-loop effective potential is the relevant quantity, and is defined through the Euclideanized effective action with all quadratic fluctuations integrated out. In the following, the subscripts M and E will be used to denote Minkowski and Euclidean signature, respectively. Euclideanization is carried out by defining the following Euclideanized quantities:

τE = iτM , SE =−iSM . (10) H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30 5

The Euclideanized action SE will then be expanded about a certain background (to be specified in Section 5) and the quadratic fluctuations integrated out to produce the one-loop effective action Γ . The effective action on the matrix theory side is then defined via the relation: E

ΓE =− dτE Veff. (11)

The minus sign in front of the integral is slightly unconventional, but it is put there for the convenience of comparison with supergravity. It is chosen such that the tree level part of the effective potential is simply the light-cone Lagrangian (Llc)pp rather than −(Llc)pp.AfterVeff is computed, the result could then be analytically continued back into Minkowski signature by replacing vE → ivM . In order to facilitate the comparison of the two sides, it is useful to ﬁrst examine the form of the action. The supergravity effective action is given in Eq. (95). For the simple case when the two membranes have the same radius and in the limit where their separation in the X4 to X9 direction, z, is small, i.e., z r0, the supergravity effective action is given in Eq. (95). Putting w, the difference of the two radii to zero, and not keeping track of the exact coefﬁcients, we could rewrite the supergravity result in term of matrix theory parameters:

v4 v2 µ2 V = α + + . (12) eff µ2z5 z3 z Hence we see that a comparison with supergravity means looking at order (α)1 on the matrix theory side.

2.3. The membrane limit and the graviton limit

We will ﬁrst state the two limits we are interested in:

• membrane limit: z 1, (13) αµ z N, (14) αµ • graviton limit: z 1, (15) αµ z N, (16) αµ whereweusedz to denote the separation of the two spherical membranes in the X4 to X9 directions, and recall α = 1 . M3R The membrane limit is derived from the condition: 1 z 1. (17) N r0 The ﬁrst inequality ensures that the effect of nonzero z is greater than any matrix theory corrections to supergravity, which we are not interested in. The second inequality ensures we are at the = αµN near membrane limit. Using r0 6 (see Eq. (5)), we arrive at the limit as stated. 6 H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30

The graviton limit is when z is much greater than r0, so that approximately the two spheres interact like two point like gravitons. We still enforce the condition 1 z for comparison with N r0 supergravity but reverse the second inequality in the membrane limit to z 1 to arrive at the r0 graviton limit stated above. In this paper, on the supergravity side we will calculate the light-cone Lagrangian only in the membrane limit. The light-cone Lagrangian in the graviton limit was already computed in an earlier work [2]. On the matrix theory side, the one-loop effective potential can be computed for general z. The two limits of this potential are then compared with the supergravity side and we will ﬁnd perfect agreement. Later in the paper we will use matrix theory to ﬁnd explicitly the potential that interpolates between the two limits.

3. The supergravity light-cone Lagrangian

The light-cone Lagrangian Llc of the probe is basically its Lagrangian Legendre transformed − in the x degree of freedom. Here we brieﬂy derive the Llc for a probe membrane in a pp-wave background perturbed by some source. (The reader is referred to Section 4.1 of [14] for the detailed discussion of the light-cone Lagrangian of point particle probes and membrane probes in terms of Hamiltonian systems with constraints.) Denote the background metric and three-form as Gµν(x), Aµνρ(x), respectively, and the membrane embedding coordinates as Xµ(σ i), with σ i,i= 0, 1, 2 being the world-volume coordinates. The bosonic part of the membrane Lagrangian density is given by 1 L Xµ,∂ Xµ =−T − det(g ) − ij k A ∂ Xµ∂ Xν∂ Xρ , (18) i ij 6 µνρ i j k µ ν with T being the membrane tension, and gij ≡ Gµν∂iX ∂j X being the pullback metric. The momentum density is L ≡ ∂ =− − 0k µ − ν ρ Πλ λ T det(gij )g ∂kX Gλµ Aλνρ∂1X ∂2X . (19) ∂(∂0X ) Deﬁne ˜ ν ρ 0k µ Πλ ≡ Πλ − TAλνρ∂1X ∂2X =−T − det(gij )g ∂kX Gλµ. (20)

There are the following three primary constraints

λξ ˜ ˜ 2 λ φ0 ≡ G ΠλΠξ + T det(grs) = 0,φr ≡ Πλ∂r X , (21) where r, s = 1, 2 label the spatial world-volume coordinates. Now use the light-cone coordinates {x+,x−,xA}, and assume static, i.e., x+-independent, − A − A + 0 background Gµν(x ,x ) and Aµνρ(x ,x ). Use the light-cone gauge X = σ . Then the equa- + 0 tion of motion for X ﬁxes the Lagrange multiplier c for the constraint φ0 1 c0 = . (22) +ξ ˜ 2G Πξ

+ 0 The constraint φ0 = 0 (in which X is set to σ ) can be used to solve for Π+ as a function of − A − A (X ,X ,∂r X ,∂r X ,Π−,ΠA). H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30 7

− A − A A We shall also express ΠA as a function of (X ,X ,∂r X ,∂r X ,Π−,∂0X ) by inverting the equations of motion for XA: ∂XA GAξ Π˜ ∂XA = ξ + cr , (23) 0 +ν ˜ r ∂σ G Πν ∂σ where we have used the expression for c0 given above and the cr ’s are the Lagrange multipliers for the φr ’s. − A − A We can first define the light-cone Hamiltonian Hlc(X ,X ,∂r X ,∂r X ,Π−,ΠA) ≡−Π+, and then define the light-cone Lagrangian − A − A A A A Llc X ,X ,∂r X ,∂r X ,Π−,∂0X ≡ ΠA∂0X − Hlc = ΠA∂0X + Π+. (24) Now the background in which the probe membrane moves is the pp-wave perturbed by some source: Gµν = (Gµν)pp + hµν , and Aµνρ = (Aµνρ)pp + aµνρ , where the quantities with subscript pp are those of the unperturbed pp-wave background, and hµν,aµνρ are the perturbations caused by the source. We only need the light-cone Lagrangian to linear order in the perturbation. Solving φ0 = 0gives − A − A Π+ X ,X ; ∂r X ,∂r X ; Π−,ΠA = (Π+)pp + δΠ+, (25) where − 2 1 2 T A B B A 2 (Π+)pp = −g++Π− + ΠAΠA + ∂1X ∂2X − ∂1X ∂2X 2Π− 2 µ + T ∂ Xi ∂ Xj Xk (26) 3 ij k 1 2 and − = 1 − ˜ 2 − − ˜ δΠ+ h−−(Π+)pp 2Π−(h+− g++h−−)(Π+)pp 2Π− ˜ ν ρ ν ρ ˜ − 2T(Π+) a− ∂ X ∂ X − 2TΠ−a+ ∂ X ∂ X − 2h− Π (Π+) pp νρ 1 2 νρ 1 2 A A pp 2 2 ν ρ − Π− h++ + (g++) h−− − 2g++h+− + 2TΠ−g++a−νρ ∂1X ∂2X ν ρ − h Π Π − 2Π−(h+ − g++h− )Π − 2TΠ a ∂ X ∂ X AB A B A A A A Aνρ 1 2 2 + T δ det(grs) , (27) where − 2 ˜ 1 2 T A B B A 2 (Π+)pp = −g++Π− + ΠAΠA + ∂1X ∂2X − ∂1X ∂2X . (28) 2Π− 2 The Lagrange multipliers cr ’s are r = r + r = r c c pp δc δc , (29) r where we have taken (c )pp to be zero. Inverting Eq. (23) gives − A − A A ΠA X ,X ; ∂r X ,∂r X ; Π−,∂0X = (ΠA)pp + δΠA, (30) A r where (ΠA)pp = Π−∂0X , and δΠA depends on δc . Now A A Llc = (ΠA)pp∂0X + δΠA∂0X + (Π+)pp + δΠ+. (31) 8 H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30

However, as can be seen easily, the second term in the above expression is canceled by the part containing δΠA in the third term. So Llc does not contain δΠA, thus being independent of the δcr ’s. Hence one ﬁnds

Llc = (Llc)pp + δLlc, (32) with 1 A A Π− 2 1 i 2 1 a 2 (L ) = Π−∂ X ∂ X − µ X + X lc pp 2 0 0 2 9 36 2 T A B B A 2 µ i j k − ∂1X ∂2X − ∂1X ∂2X + T ij k ∂1X ∂2X X (33) 4Π− 3 and δLlc = δΠ+, where δΠ+ is given by Eq. (27) with all the ΠA’s in it replaced by (ΠA)pp = A Π−∂0X (the reason being that δΠ+ contains hµν and aµνρ and is thus already first-order). We r would like to emphasize again that Llc is independent of the choice of the δc ’s. To use the general expression for Llc given in the previous paragraph in a specific problem, one just has to substitute in the hµν ’s and aµνρ ’s found by solving the supergravity field equations for the specific source, as well as the specific probe membrane embedding XA(σ i).

4. Supergravity computation

4.1. Diagonalizing the ﬁeld equations for arbitrary static source

In this subsection we present the diagonalization of the linearized supergravity equations of motion for arbitrary static sources (this subsection is basically Section 4.2 of [14]). There is, of course, no highbrow knowledge involved here: we are just solving the linearized Einstein equations and Maxwell equations, which are coupled; and by “diagonalization” we basically just mean the prescription by which we get a decoupled Laplace equation for each component of the metric and three-form perturbations. The unperturbed background is the 11D pp-wave, and we only consider static, i.e., x+-independent, field configurations, thanks to the fact that the sources considered are taken to be static, i.e., with x+-independent stress tensor and three-form current. Since we leave the source arbitrary, what we will present here are the left-hand side of the linearized equations. These are tensors whose computation is straightforward though a bit tedious: the reason we present them here is because they are necessary when solving the field equations, and to the best of our knowledge have not been explicitly given elsewhere. A somewhat related problem is the diagonalization of the equations of motion when the source is absent. This requires field configurations with x+-dependence. One work along this line is [15]. Roughly speaking, borrowing the language of electromagnetism, what is considered in [15] are electromagnetic waves in vacuum, while what we are considering here are electrostatics and magnetostatics for arbitrary static sources. The nonzero components up to (anti)symmetry of the Christoffel symbol, Riemann tensor, and Ricci tensor of the 11D pp-wave are

A 1 − 1 Γ++ =− ∂ g++,Γ= ∂ g++, 2 A +A 2 A 1 1 R+ + =− ∂ ∂ g++,R++ =− ∂ ∂ g++. (34) A B 2 A B 2 C C H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30 9

(We usually do not substitute the explicit expression of g++, unless that brings significant sim- plification to the resulting formula.) Now let us perturb the pp-wave background by adding a source. Denote the metric perturbation δgµν by hµν , and the gauge potential perturbation by δAµνρ = aµνρ . hµν,aµνρ are treated as rank-two and rank-three tensors, respectively, the covariant derivative ∇ acting on them is defined using the connection coefficient of the unperturbed pp-wave background, and indices are raised/lowered, traces taken using the background metric gµν . ¯ ≡ − 1 ≡ µν Let us deal with the Einstein equations first. Define hµν hµν 2 gµνh, where h g hµν . Without the source, the Einstein equation is 1 R − Rg − κ2 [T ] = 0 (35) µν 2 µν 11 µν A and we recall that the stress tensor of the gauge field is

1 1 [T ] = F F λξρ − g F ρσλξF . µν A 2 µλξρ ν µν ρσλξ (36) 12κ11 8 The source perturbs the Einstein equation to

1 δ R − Rg − κ2 δ[T ] = κ2 [T ] (37) µν 2 µν 11 µν A 11 µν S with [Tµν]S standing for the stress tensor of the source. As usual, it helps to proceed in an organized manner, grouping different terms in the above − 1 =−1 ∇σ ∇ ¯ + + perturbed Einstein equations. One ﬁnds, δ(Rµρ 2 Rgµρ ) 2 σ hµρ Kµρ Qµρ , and 2 [ ] = + ∇σ ∇ ¯ κ11δ Tµν A Nµν Lµν , where the explicit expressions of the symmetric tensors σ hµν , Kµν , Qµν , Nµν , and Lµν can be obtained after some work. Their deﬁnitions and components are given below1:

σ ¯ •∇∇σ hµν : σ ¯ µν ¯ ¯ 1 ¯ ∇ ∇ h++ = g ∂ ∂ h++ + −(∂ ∂ g++)h+− + (∂ g++∂ g++)h−− σ µ ν A A 2 A A ¯ ¯ + 2[∂Ag++∂−h+A − ∂Ag++∂Ah+−], (38)

σ ¯ µν ¯ 1 ¯ ¯ ∇ ∇ h+− = g ∂ ∂ h+− − (∂ ∂ g++)h−− + ∂ g++∂−h− σ µ ν 2 A A A A ¯ − ∂Ag++∂Ah−−, (39)

σ ¯ µν ¯ 1 ¯ ¯ ¯ ∇ ∇ h+ = g ∂ ∂ h+ − (∂ ∂ g++)h− + ∂ g++∂−h − ∂ g++∂−h+− σ C µ ν C 2 A A C A AC C ¯ − ∂Ag++∂Ah−C, (40) σ ¯ µν ¯ ∇ ∇σ h−− = g ∂µ∂νh−−, (41) σ ¯ µν ¯ ¯ ∇ ∇σ h−C = g ∂µ∂νh−C − ∂Cg++∂−h−−, (42) σ ¯ µν ¯ ¯ ¯ ∇ ∇σ hCD = g ∂µ∂νhCD − ∂Cg++∂−h−D − ∂Dg++∂−h−C. (43)

1 µν 2 Notice that ∂+ will never appear because we only consider the static case; also note g ∂µ∂ν =−g++∂− + ∂A∂A for static conﬁgurations. 10 H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30

• Kµν : Its deﬁnition is 1 1 1 K ≡ R ξ h¯ + R ξ h¯ + Rσ ξ h¯ + g Rξσh¯ − Rh¯ . (44) µρ 2 µ ξρ ρ ξµ µρ σξ 2 µρ ξσ 2 µρ Its components are given by

1 ¯ 1 ¯ 1 ¯ K++ = − ∂A∂Ag++ h+− + g++h−− + (∂A∂B g++)hAB , (45) 2 2 2 1 ¯ K+− = − ∂A∂Ag++ h−−, (46) 2 1 ¯ 1 ¯ K+ = − ∂ ∂ g++ h− + − ∂ ∂ g++ h− , (47) A 4 C C A 2 A B B K−− = 0, (48) = K−A 0, (49) 1 1 ¯ K = ∂ ∂ g++ − δ ∂ ∂ g++ h−−. (50) AB 2 A B 2 AB C C • ≡ 1 ∇ +∇ − 1 ∇α ≡∇β ¯ Qµν : Its definition is Qµρ 2 ( µqρ ρqµ) 2 gµρ qα, where qα hβα. As one can recognize, Qµρ contains the arbitrariness of making different gauge choices when solving the Einstein equation, where one makes a gauge choice by specifying the qµ’s. The components of Qµρ are 1 1 Q−− = ∂−q−,Q− = (∂−q + ∂ q−), Q−+ = (g++∂−q− − ∂ q ), A 2 A A 2 A A 1 1 Q = (∂ q + ∂ q ) − δ (∂−q+ − g++∂−q− + ∂ q ), AB 2 A B B A 2 AB A A 1 Q+ = ∂ q+ − (∂ g++)q− , A 2 A A 1 1 Q++ = (∂ g++)q − g++(∂−q+ − g++∂−q− + ∂ q ). (51) 2 A A 2 A A • 2 [ ] Nµν : It is defined to be the part of κ11δ Tµν A that contains only the metric perturbation, but not the three-form gauge potential perturbation. Its components are given by 3 9 2 1 ¯ 1 ¯ 1 ¯ 1 ¯ N++ = µ h+− + g++h−− − h + h , 3 12 3 ii 6 aa i=1 a=4 2 2 µ ¯ µ ¯ N+− = h−−,N+ = h− ,N+ = 0, 4 i 2 i b N−− = 0,N−i = 0,N−b = 0, 2 2 µ ¯ µ ¯ N =− δ h−−,N= 0,N= δ h−−. (52) ij 4 ij ib ab 4 ab • 2 [ ] Lµν : This is defined to be the part of κ11δ Tµν A that contains only the three-form perturbation, but not the metric perturbation. Its components are given by

1 L++ = µ δF + − g++δF − ,L+− = 0, 123 2 123 H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30 11

µ µ L+ = δF+ −,L+ = δF , i 4 ij k jk b 2 123b L−− = 0,L−i = 0,L−b = 0, µ µ µ L = δ δF −,L= δF −,L=− δ δF −. (53) ij 2 ij 123 ib 4 ij k bj k bd 2 bd 123 Next let us deal with the Maxwell equation. In the absence of the source, it is 1 √ η˜ µ1...µ11 √ ∂ −gF λµ1µ2µ3 − √ F F = 0, (54) −g λ 1152 −g µ4...µ7 µ8...µ11 where η˜ is either +1or−1 depending on the choice of convention, which one can fix later by requiring the consistency of the conventions for the equations and the solutions under consideration. When the source is present, we add its current J µ1µ2µ3 to the left-hand side of the above equation, and get 1 √ η˜ µ1...µ11 δ √ ∂ −gF λµ1µ2µ3 − √ F F = J µ1µ2µ3 . (55) −g λ 1152 −g µ4...µ7 µ8...µ11 We can write the left-hand side of the above equation as the sum of two totally antisymmetric tensors Zµ1µ2µ3 + Sµ1µ2µ3 , where Zµ1µ2µ3 is defined to be the part that contains the metric perturbation only, and Sµ1µ2µ3 is defined to be the part that contains the three-form perturbation only. One finds +−i ¯ +−b +ij ¯ ¯ Z = µij k ∂j h−k,Z= 0,Z= µij k (∂−h−k − ∂kh−−), + + Z ib = 0,Zbc = 0, 3 −ij 1 ¯ ¯ − ¯ ¯ Z = µ ∂ h − h− − h − ∂ h , ij k k 3 ii b kb i=1 − − Z ib = µ ∂ h¯ ,Zbc = 0, ij k j kb 3 ij k 1 ¯ ¯ − ¯ ¯ Z =−µ ∂− h − h− − h − ∂ h− , ij k 3 ii b b i=1 ij b ¯ ¯ Z = µij k (∂−hkb − ∂kh−b), Zibc = 0,Zbce = 0, (56) and +−A µν µ µ S = g ∂ ∂ a−+ + ∂ g++∂−a − − ∂− ∇ a + + ∂ ∇ a +− , (57) µ ν A B BA µ A A µ +AB µν µ µ µ S = g ∂µ∂νa−AB − ∂− ∇ aµAB + ∂A ∇ aµ−B − ∂B ∇ aµ−A , (58) −AB µν +AB S = g ∂ ∂ a+ − g++S µ ν AB µ + (∂Ag++)(∂−a−+B ) + ∂A ∇ aµ+B − ∂A(aEB−∂Eg++) −[A ↔ B] ˜ η −ABµ ...µ 123+ − (∂ g++)δF − − µ 4 7 δF , (59) D D AB 24 µ4...µ7 ABE µν S = g ∂ ∂ a − (∂ g++)(∂−a− ) − (∂ g++)(∂−a− ) − (∂ g++)(∂−a− ) µ ν ABE A BE B EA E AB µ µ µ − ∂A ∇ aµBE − ∂B ∇ aµEA − ∂E ∇ aµAB η˜ + − µ ABEµ4...µ7123 δF . (60) 24 µ4...µ7 12 H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30

µ µ µ µ Notice that S 1 2 3 contains ∇ aµρλ and its derivatives. Those terms correspond to the gauge freedom for the three-form gauge potential. Now that we have collected the expressions for the various tensors, we are ready to diagonalize the ﬁeld equations. Recall that the Einstein equation is 1 − ∇σ ∇ h¯ + K + Q − N − L = κ2 [T ] (61) 2 σ µν µν µν µν µν 11 µν S and the Maxwell equation is

Zµ1µ2µ3 + Sµ1µ2µ3 = J µ1µ2µ3 . (62) The right-hand side of these equations is given by specifying the source that we consider (recall 2 that the three-form current J is of order κ11), hence we only need to concentrate on diagonalizing the left-hand sides. As will be seen shortly, it is useful to deﬁne “level” for tensors: lower +/upper − indices contribute +1 to the level; lower −/upper + indices contribute −1 to level; and the upper A/lower A indices contribute zero to the level. We shall see that the ﬁeld equations should be solved in ascending order of their levels. The following is the detailed prescription of the diagonalization procedure. Let us use the shorthand notation (E.E.)µν for the lower (µν) component of the Ein- µ µ µ stein equation, and (M.E.) 1 2 3 for the upper (µ1µ2µ3) component of the Maxwell equation.

• At level −2.

The only ﬁeld equation at this level is (E.E.)−−, which reads, upon using the expressions of σ ¯ the various tensors ∇ ∇σ hµν,Kµν,Qµν,..., etc. that we have given above

1 µν ¯ 2 − g ∂ ∂ h−− + Q−− = κ [T−−] . (63) 2 µ ν 11 S ¯ This equation can be immediately solved for h−− after specifying the source term and the gauge choice term Q−−.

• At level −1.

We have (E.E.)−A, which reads 1 µν ¯ ¯ 2 − g ∂ ∂ h− − (∂ g++)(∂−h−−) + Q− = κ [T− ] , (64) 2 µ ν A A A 11 A S ¯ ¯ which can now be solved for h−A,usingtheh−− found previously. Also at this level is (M.E.)+AB , which reads, µν µ µ µ ¯ ¯ g ∂µ∂νa−ij − ∂− ∇ aµij + ∂i ∇ aµ−j − ∂j ∇ aµ−i + µij k (∂−h−k − ∂kh−−) + = J ij , µν µ µ µ +ib g ∂ ∂ a− − ∂− ∇ a + ∂ ∇ a − − ∂ ∇ a − = J , µ ν ib µib i µ b b µ i µν µ µ µ +bc g ∂µ∂νa−bc − ∂− ∇ aµbc + ∂b ∇ aµ−c − ∂c ∇ aµ−b = J , (65) µ from which we can ﬁnd a−AB , upon specifying the gauge choice ∇ aµρλ for the three-form and ¯ ¯ using the h−A and h−− found previously. H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30 13

• At level 0.

+−A ABE At this level we have (E.E.)+−, (M.E.) , (E.E.)AB , and (M.E.) . (E.E.)+− is of the form

1 µν ¯ − g ∂ ∂ h+− = known terms. (66) 2 µ ν (From now on, we will not bother writing down the detailed equations; “known terms” refers µ to the gauge choice terms Qµν , ∇ aµρλ , source terms, and terms containing previously found ¯ hµν ’s and aµνρ ’s, one can write those down by looking up the expressions given earlier for the ¯ +−A various tensors.) Hence solving it we get h+− and solving (M.E.) gives a−+A. ABE (E.E.)AB and (M.E.) are coupled, so a little more work is needed. The following are the bce details. First notice that the only unknown in (M.E.) is abce, hence solving this equation we bce µν ﬁnd abce.((M.E.) contains the usual term g ∂µ∂νabce and also a term of the form ∂−adf g which comes from the F ∧ F in the Maxwell equation, hence it is not quite a Laplace equation. But, that being said, one should not have any difﬁculty solving it.) ibc µν ij b (M.E.) is of the form g ∂µ∂νaibc = known terms, solving which gives aibc. (M.E.) and (E.E.)kb are coupled in the following manner µν ¯ g ∂µ∂νaij b + µij k ∂−hkb = known terms, 1 µν ¯ 1 − g ∂ ∂ h + µ ∂−a = known terms. (67) 2 µ ν kb 4 klm lmb ¯ Decoupling these two equations is quite easy. Let us take a12b and h3b as the representative case. One sees that these two equations can be recombined to give µν ¯ g ∂ ∂ + iµ∂− (h + ia ) = known terms, µ ν 3b 12b µν ¯ g ∂µ∂ν − iµ∂− (h3b − ia12b) = known terms. (68) ¯ + ¯ − ¯ Solving these equations gives (h3b ia12b) and (h3b ia12b), and in turn h3b and a12b. (M.E.)ij k is coupled to (E.E.) and (E.E.) through the quantity H ≡ 2 3 h¯ − ij bd 3 i=1 ii 1 9 ¯ 3 a=4 haa in the following manner µν g ∂µ∂νa123 + µ∂−H = known terms, 1 µν ¯ 1 − g ∂ ∂ h + µδ ∂−a = known terms, 2 µ ν ij 2 ij 123 1 µν ¯ 1 − g ∂ ∂ h − µδ ∂−a = known terms. (69) 2 µ ν bd 2 bd 123 Combining the last two equations gives

µν −g ∂µ∂νH + 4µ∂−a123 = known terms. (70) Recombining this with ﬁrst equation, we get µν g ∂ ∂ + 2iµ∂− (H + 2ia ) = known terms, µ ν 123 µν g ∂µ∂ν − 2iµ∂− (H − 2ia123) = known terms, (71) ¯ solving which gives H and a123. Using the resulting expression for a123 one can then ﬁnd hij ¯ ABE and hbd . Thus we are done with (E.E.)AB and (M.E.) . 14 H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30

• At level 1.

−AB µν (M.E.) is of the form g ∂µ∂νa+AB = known terms, solving which gives a+AB ,

1 µν ¯ ¯ (E.E.)+ is of the form − g ∂ ∂ h+ = known terms, solving which gives h+ . A 2 µ ν A A • At level 2.

1 µν ¯ ¯ (E.E.)++ is of the form − g ∂ ∂ h++ = known terms, solving which gives h++. 2 µ ν Thus we have diagonalized the whole set of Einstein equations and Maxwell equations.

4.2. Application to a spherical membrane source using the near-membrane expansion

Now let us apply the general formalism of the previous subsection to the case of interest, with the source being a spherical membrane sitting at the origin of the transverse directions, 1 2 + 2 2 + 3 2 = 2 4 = 9 = + = − = i.e., having (X ) (X ) (X ) r0 , X 0,...,X 0, and X t,X 0. The gauge µ choice we shall take is: qα = 0 (hence all the Qαβ ’s vanish); and ∇ aµρλ = 0. The nonzero components of the stress tensor and three-form current for this source are given by

−1 2 − 4 9 µr0 µr0 [T−−] = Tδ(x )δ(r − r )δ x ···δ x , [T+−] =− [T−−] , s 0 3 s 3 s

2 i j 4 µr0 x x ij µr0 [T ] = − δ [T−−] , [T++] = [T−−] ij s 2 s s s (72) 3 r0 3 and

k +ij = 2 − µr0 x [ ] J κ11( 2) ij k T−− s, (73) 3 r0 √ where r ≡ xixi . √ Now let us explain√ what we mean by “near-membrane expansion”. Define w ≡ r − r0, z ≡ xaxa, and ξ ≡ w2 + z2, which parameterize the distance away from the source membrane. We shall assume that w,z,ξ are of the same order of magnitude. The near-membrane expansion is an expansion in ξ/r0. When one sits very close to membrane, one just sees a flat membrane, which is the zeroth order of the expansion. As one moves away from the membrane, one begins to feel the curvature of membrane, which gives the higher order corrections in the expansion. One should also note that the zeroth order of this expansion is just the flat space limit: µ → 0, r0 →∞, with µr0 kept finite. It is instructive to see how the zeroth order works. At this order, in the Einstein equations and ¯ Maxwell equations, the effective source terms (which are of the forms (∂g++)hµν , etc.) arising µνρ from the various tensors Kµν , Nµν , Lµν , Z , etc. are less singular then the delta-functional µνρ sources [Tµν]s and J , and can thus be thrown away. Then the resulting equations are trivially i µνρ decoupled. Also, at this order, we can treat the x ’s in [Tµν]s and J as constant vectors. One finds (using the subscript 0 to denote “zeroth order”)

k ¯ µr0 x ¯ [h−A]0 = 0, [a−ij ]0 = ij k [h−−]0, [a−ib]0 = 0, [a−bc]0 = 0, 3 r0 H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30 15

2 ¯ µr0 ¯ [h+−] =− [h−−] , [a−+ ] = 0, 0 3 0 A 0

2 i j ¯ µr0 x x ij ¯ [h ] = − δ [h−−] , ij 0 2 0 3 r0 [h¯ ] = 0, [h¯ ] = 0, [a ] = 0, ib 0 bc 0 ABE 0 3 k µr0 x ¯ [a+ij ]0 =− ij k [h−−]0, [a+ib]0 = 0, [a+bc]0 = 0, 3 r0

4 ¯ ¯ µr0 ¯ [h+ ] = 0, [h++] = [h−−] , (74) A 0 0 3 0 ¯ where [h−−]0 satisﬁes 2 2 2 2 µr0 2 ∂ ∂ ∂ ¯ − k− + + +···+ [h−−] 3 (∂w)2 (∂x4)2 (∂x9)2 0 − 1 µr 1 =− 0 κ2 T δ(w)δ x4 ···δ x9 (75) πR 3 11 1 (where we have multiplied the right-hand side of the equation by 2πR due to the Fourier transform along the x− direction), and is given by − µr0 2 ¯ exp( 3 k−ξ) µr0 µr0 [h−−] = ∆ 3 + 3 k−ξ + k−ξ , (76) 0 ξ 5 3 3

κ2 T ≡ 11 with ∆ 4 µr0 . 16π R( 3 ) Plugging the above zeroth order solution of the field equations into the light-cone Lagrangian L δ lc given in Eq. (27), for a spherical probe membrane with radius r0, sitting at rest in the 1 2 + 2 2 + 3 2 = 2 1, 2, 3 directions, and moving about in the 4 through 9 directions: (X ) (X ) (X ) r0 , 4 9 + = − = = µΠ− X (t), . . . , X (t), and X t,X 0, one finds (using the facts that T 3r sin θ , and we can = 0 set r0 r0 because we are looking at the zeroth order) 1 ¯ ˙ a ˙ a 2 δL = Π−[h−−] X X . (77) lc 8 0 ¯ It is worth noting that, keeping only the leading order term in k− in [h−−]0,Eq.(77) becomes the v4 Lagrangian for the case of longitudinal momentum transfer between two membranes in the flat space, given in [16]. Now let us go on to consider higher orders in the near-membrane expansion. Since in this paper we do not consider longitudinal momentum transfer, we shall set k− = 0 (which makes many fields equations decouple). Denote

2 ≡ ∂A∂A (when acting on functions of (w, z))

= 20 + δ2, (78) 2 2 with 2 ≡ ∂ + ∂ + 5 ∂ = ∂2 + ∂ ∂ being the zeroth order Laplace operator, and δ2 ≡ 0 ∂w2 ∂z2 z ∂z w a a 2 ∂ being curvature correction to it. r0+w ∂w 16 H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30

• At level −2.

(E.E.)−−

−1 ¯ 1 2 4 9 µr0 2h−− =− κ T δ(w)δ x ···δ x . (79) πR 11 3 Let ¯ ¯ ¯ h−− =[h−−]0 + δh−−, (80) ¯ 3 with [h−−] ≡ ∆ , which satisﬁes 0 ξ 5

−1 ¯ 1 2 4 9 µr0 2 [h−−] =− κ T δ(w)δ x ···δ x . (81) 0 0 πR 11 3 Then (E.E.)−− becomes ¯ ¯ ¯ 20δh−− + δ2[h−−]0 + δ2δh−− = 0. (82)

Now let us look at the order of magnitude of each term in the above equation. Notice that 20 ∼ 1 2 ∼ 1 ∼ 1 ¯ ∼ 1 2 , δ . The second term is thus ∆ 6 , which tells us δh−− ∆ 4 . Solving the ξ r0ξ r0ξ r0ξ equation iteratively we ﬁnd

2 3 4 ¯ ¯ w w w w δh−− =[h−−] − + − + (83) 0 r 2 3 4 0 r0 r0 r0 and thus 5 ¯ 1 r0 ξ h−− = 3∆ + O . (84) 5 r + w 5 ξ 0 r0 ξ 5 We did not compute the O( 5 ) terms, because we are only interested in the part of the solution r0 that is singular as ξ → 0. Solving the other ﬁeld equations is similar, so we just present the results ξ 5 below, omitting the O( 5 ) symbol. r0

• At level −1. ¯ h−A = 0,

a−bc = 0, a− = 0, ib 2 + 2 2 2 2 k 1 1 w 1 w z 1 wz w z a− = x µ∆ 1 − + − + . (85) ij ij k 5 r 2 3 4 ξ 2 0 6 r0 2 r0 r0

• At level 0. 2 2 2 ¯ µr0 3∆ 5 w 7 z r0 h+− =− 1 + + , 5 2 2 + 3 ξ 4 r0 8 r0 r0 w a−+A = 0, H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30 17

aABD = 0, (86) 1 1 5 w 17 w2 1 z2 3 w3 1 wz2 h¯ = xkxbµ2∆ − − + − − + kb 5 1 2 2 3 3 ξ 2 4 r0 12 r 12 r 2 r 4 r 0 0 0 0 3 w4 1 w2z2 + − , (87) 2 r4 2 r4 0 0 xixj µr 2 3 w z2 w3 wz2 w4 w2z2 z4 h¯ = 0 ∆ − − + + − − + ij 2 5 1 2 3 2 3 4 4 4 r 3 ξ r0 r r r r r r 0 0 0 0 0 0 µr 2 3 1 w 7 w2 1 z2 1 w3 3 wz2 1 w4 − δij 0 ∆ + + − − + + 5 1 2 2 3 3 4 3 ξ 2 r0 12 r 24 r 4 r 8 r 4 r 0 0 0 0 0 1 w2z2 1 z4 − + , (88) 24 r4 3 r4 0 0 1 1 w 7 w2 19 z2 7 w3 19 wz2 7 w4 19 w2z2 h¯ = δ (µr )2∆ − − + + − − . bd bd 0 5 r 2 2 3 3 4 4 ξ 2 0 18 r0 72 r0 18 r0 72 r0 18 r0 72 r0 (89) • At level +1.

a+bc = 0, a+ = 0, ib 2 2 2 3 2 4 k µr0 1 w w 5 z 3 w 7 wz 3 w a+ =− x µ∆ + − − + + − ij ij k 5 1 2 2 2 3 3 4 3 ξ r0 r 4 r 2 r 4 r 2 r 0 0 0 0 0 5 w2z2 1 z4 − + , 4 4 (90) 4 r0 2 r0 ¯ h+A = 0. (91)

• At level +2. 4 2 2 3 2 4 ¯ µr0 3 5 w 31 w 1 z 17 w 1 wz 1 w h++ = ∆ + + − + − + 5 1 2 2 3 3 4 3 ξ 2 r0 12 r 24 r 12 r 12 r 3 r 0 0 0 0 0 23 w2z2 17 z4 + + . 4 4 (92) 24 r0 32 r0 (A note in passing: as it turns out, the choice of convention for η˜ in the Maxwell equation (54) does not matter, since it drops out when solving the field equations for our specific source.) = + 1 2 + 2 2 + Again, let the probe membrane have a radius r0 r0 w, with the trajectory√ (X ) (X ) 3 2 = 2 4 9 + = − = ≡ ˙ a ˙ a (X ) r0 , X (t), . . . , X (t), and X t,X 0. We shall take v X X to be of order = 1 µz (recall that the supersymmetric circular orbit has v 6 µz; so here we are considering generically nonsupersymmetric orbits that can be regarded as deformations of the supersymmetric µΠ− circular one). Plugging in the supergravity solution given above into Eq. (27),usingT = , 3r0 sin θ and in the end keeping only the part of δLlc that is singular as ξ → 0, we find the probe’s Llc to 18 H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30 be

Llc = (Llc)pp + δLlc (93) with (Llc)pp being the action in the unperturbed pp-wave background, and µ4(4w2z2 + 7z4) − 72µ2v2(2w2 + 5z2) + 3888v4 δL = Π−∆ . (94) lc 10368ξ 5 L → ∼ 1 Notice that the above δ lc’s singular behavior as ξ 0 is homogeneous: ξ (since v is of order µz). The expression in the numerator: µ4(4w2z2 + 7z4) − 72µ2v2(2w2 + 5z2) + 3888v4 nicely factorizes into (36v2 − z2µ2)[108v2 − (4w2 + 7z2)µ2], which shows that for the special case of = 1 the supersymmetric circular orbit v 6 µz considered in [10], δLlc vanishes as expected. To be more precise, so far we have been talking about Lagrangian density. Since the membrane worldvolume is taken to be a unit sphere, the Lagrangian is given by κ2 T 2 (36v2 − z2µ2)[108v2 − (4w2 + 7z2)µ2] δL = dθ dφδL = 11 lc lc 4π 3Rµ2 1152ξ 5 α (36v2 − z2µ2)[108v2 − (4w2 + 7z2)µ2] = , (95) µ2 1152ξ 5 = µΠ− wheretogettheﬁrstlineweusedT to eliminate Π− in terms of T and r0, and set 3r0 sin θ r ≈ r in the end to remove higher w order curvature correction. To get the second line we used 0 0 r0 2 the expressions for κ11, T , and α in terms of M and R given at the end of Section 2.1. Here we would like to make a brief comparison of the above membrane result with the graviton result given in [2]. First of all, the membrane result contains the variable w (the difference in radius between the probe membrane and the source membrane), which has no counterpart in the graviton case. Secondly, in terms of the x1 through x3 directions, the two membranes are sitting at rest at the i origin; this corresponds to setting x = 0 and vi = 0 in the graviton case. If we set w = 0inEq.(95), i.e., consider two membranes of the same size, then ξ = z, and α (δL ) = 36v2 − z2µ2 108v2 − 7z2µ2 , (96) lc membrane 1152µ2z5 i while for the gravitons, upon setting Np = Ns = N, x = 0, and vi = 0 in Eq. (19) of [2],we have α3N 2 (δL ) = 36v2 − z2µ2 140v2 − 7z2µ2 . (97) lc graviton 5736z7 When comparing the above two expressions, note the difference between the numerators: (108v2 − 7z2µ2) for membrane, and (140v2 − 7z2µ2) for graviton. Also notice their different power law dependence on z: 1 for membrane and 1 for graviton, which cannot be undone z5 z7 by integrating over the membrane (r0, the radius of the membrane, has nothing to do with z,the separation of the membranes in the X4 to X9 directions).

5. Matrix theory computation—the membrane limit

Shin and Yoshida have previously calculated the one-loop effective action for membrane fuzzy spheres extended in the first three directions and having periodic motion in a sub-plane of the H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30 19 remaining six transverse directions. In Ref. [10] they considered the case of supersymmetric µ circular motion for an arbitrary radius and angular frequency 6 (this orbit preserves eight supersymmetries and was first found by [1]). Here we generalize that analysis to orbits which are not required to satisfy the classical equations of motion and are in general nonsupersymmetric. The procedures are: expanding the action to quadratic order in fluctuations, writing the fields in terms of the matrix spherical harmonics introduced in [17],2 diagonalizing the mass matrices of the bosons, fermions, and ghosts, and finally summing up the masses to get the one-loop effective action. In doing so, we shall adopt the notations of [10]. We shall consider the background BI 0 I = (1) B I , (98) 0 B(2) where

i µ i a B = J × ,B= 0 · 1N ×N , (1) 3 (1)N1 N1 (1) 1 1 i µ i i a a B = J × + x (t)1N ×N ,B= x (t)1N ×N (99) (2) 3 (2)N2 N2 2 2 (2) 2 2 with the J i ’s being su(2) generators. The above background has the interpretation of one spherical membrane (labeled by the subscript (1)) sitting at the origin, and the other spherical membrane (labeled by the subscript (2)) moving along the arbitrary orbit given by {xi(t), xa(t)}. The fluctuations of the bosonic fields are given by Z0 Φ0 ZI ΦI = (1) I = (1) A 0 † 0 ,Y I † I . (100) (Φ ) Z(2) (Φ ) Z(2) As it turns out, the part of the bosonic action containing the diagonal fluctuations Z0,ZI does not contain any new terms in addition to those given in [10], so we do not write it out here. (It will be the same for the fermionic and ghost parts of the action; it is the off-diagonal fluctuations that give the one-loop interaction potential between the two membranes.) The action for the off-diagonal fluctuations is µ 2 µ S = dt Tr −Φ˙ 02 + x2Φ02 + J i ◦ Φ02 − 2 xi Re J i ◦ Φ0 Φ0 † OD 3 3

µ 2 µ 2 + Φ˙ i2 − x2Φi2 − Φi + iij k J j ◦ Φk2 + J i ◦ Φi2 3 3 µ + 2 xi Re J i ◦ Φ0 Φ0 † − 2ix˙i Φ0 †Φi − Φi †Φ0 3

µ2 µ 2 + Φ˙ a2 − x2 + Φa2 − J i ◦ Φa2 2 6 3 µ + 2 xi Re J i ◦ Φa Φa † − 2ix˙a Φ0 †Φa − Φa †Φ0 , (101) 3 where x2 ≡ xixi + xaxa, and dots mean time-derivatives.

2 For the computation below, one only needs the transformation of the matrix spherical harmonics under SU(2),however, for detailed construction of the matrix spherical harmonics, see, e.g., Appendix A of [18]. 20 H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30

We specialize to the case where, xi = 0,x8 = b,x9 = vt, with (x8)2 + (x9)2 denoted by z2. The effective potential will be computed by summing over the masses of the fermionic and ghost fluctuations and then subtracting the masses of the bosonic fluctuations. This method, which we will refer to as the sum over mass method, is the same as the one used in [2]. Although the above trajectory has the form of a straight line with constant velocity in the (x8,x9) plane, the a 2 a 2 final expression of Veff in terms of z ≡ (x ) and v ≡ (x˙ ) should suffice for the purpose of comparing with supergravity for arbitrary orbits xa(t). One may ask whether the sum over mass formula is valid when the masses of the fluctuations are time-dependent (one origin for such a time-dependence is the acceleration of the trajectory). In Section 4.2 and Appendix A of [2], it was carefully shown that, in the case of two-graviton interaction in the pp-wave background, the sum over mass formula was sufficient in computing the terms that could occur on the supergravity side. The time-dependence in the masses of the fluctuations will give terms of the form of matrix theory corrections to supergravity (i.e., terms that dominate at extreme short distances and cannot be observed in supergravity), which do not concern us since we are only interested in the comparison with supergravity. Here we expect a similar argument to hold in the case of two-membrane interaction. The rather nontrivial agreement with supergravity presented at the end of this section and also the agreement with the work of Shin and Yoshida [11] in Section 6.2 confirm the validity of the sum over mass method. We expand the fields in terms of matrix spherical harmonics 1 + − 2 (N1N2) 1 j × 0,I = 0,I N1 N2 Φ φjmYjm . (102) = 1 | − | m=−j j 2 N1 N2 For our choice of background the masses of modes in the i = 1, 2, 3 and a = 4, 5, 6, 7, 8 directions are the same as those in [10]. For the gauge field and a = 9 direction we have 1 + − 2 (N1N2) 1 2 ˙ µ S = dt −φ0 2 + z2 + j(j + 1) φ0 2 jm 3 jm 1 j= |N1−N2| 2 µ 2 µ 2 + 9˙ 2 − 2 + + + 0 2 φjm z j(j 1) φjm 6 3 − 0 ∗ 9 − 9 ∗ 0 2iv φjm φjm φjm φjm , (103) where there is an implicit sum over −j m j. It is straightforward to diagonalize the mass matrix for these modes. Combining all contributions from bosonic fluctuations we get a bosonic effective potential3 given by, 1 (N +N )−2 2 12 µ 2 V B =− (2j + 1) z2 + (j + 1)2 eff 3 = 1 | − |− j 2 N1 N2 1 1 (N +N ) 2 1 2 µ 2 − (2j + 1) z2 + j 2 3 = 1 | − |+ j 2 N1 N2 1

3 Note that our convention differs from that of [10] by an overall minus sign. See Section 2. H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30 21 1 (N +N )−1 2 12 µ 2 − (2j + 1) z2 + j(j + 1) 3 = 1 | − | j 2 N1 N2 1 (N +N )−1 2 12 µ 2 1 2 − 5(2j + 1) z2 + j + 3 2 j= 1 |N −N | 2 1 2 1 (N +N )−1 2 12 µ 2 1 1 µ 4 − (2j + 1) z2 + j(j + 1) + + + 16v2 3 8 2 6 j= 1 |N −N | 2 1 2 µ 2 1 1 µ 4 + z2 + j(j + 1) + − + 16v2 . (104) 3 8 2 6 For the fermions we start with the action given in [10]

µ L = Tr iΨ †Ψ˙ − Ψ †γ I Ψ,BI − i Ψ †γ 123Ψ (105) F 4 with

Ψ χ = (1) Ψ † . (106) χ Ψ(2) Again the action for the diagonal ﬂuctuations has no new terms, and for the off-diagonal part we decompose the SO(9) spinor χ using the subgroup SO(3) × SO(6) ∼ SU(2) × SU(4) preserved by pp-wave, 16 → (2, 4) + (2¯, 4¯)

1 χAα χ = √ . (107) 2 χˆ Aα

Substituting this into LF , 1 µ L = Tr i χ† Aαχ˙ − χ† Aαχ − χ† Aα σ i β J i ◦ χ F Aα 4 Aα 3 α Aβ + ˆ † Aα ˙ˆ + 1 ˆ † Aα ˆ + µ ˆ † Aα i β i ◦ˆ i χ χ Aα χ χAα χ σ αJ χAβ 4 3 + xi − χ† Aα σ i β χ + χˆ † Aα σ i β χˆ α Aβ α Aβ + a † Aα a ˆ B + ˆ † α a AB † x χ ρAB χα χ A ρ χBα , (108) with the σ i ’s and ρa’s being the gamma matrices for SO(3) and SO(6), respectively. We now consider our speciﬁc background and expand in modes,

1 (N +N )−3/2 2 1 2 1 3 L = iπ† π˙ + iπˆ † πˆ˙ − j + π † π −ˆπ † πˆ F jm jm jm jm 3 4 jm jm jm jm j= 1 |N −N |−1/2 2 1 2 + a † a ˆ +ˆ† a † x πjmρ πjm πjm ρ πjm 22 H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30

1 (N +N )−1/2 2 1 2 1 1 + iη† η˙ + iηˆ† η˙ˆ − j + η† η −ˆη† ηˆ jm jm jm jm 3 4 jm jm jm jm j= 1 |N −N |+1/2 2 1 2 + a † a ˆ +ˆ† a † x ηjmρ ηjm ηjm ρ ηjm , (109) where again there is an implicit sum over m. This can be diagonalized and contributes to the effective action 1 (N +N )−3/2 2 1 2 µ 2 3 2 V F = 2 (2j + 1) z2 + j + + v eff 3 4 j= 1 |N −N |−1/2 2 1 2

µ 2 3 2 + z2 + j + − v 3 4 1 (N +N )−1/2 2 1 2 µ 2 1 2 + 2 (2j + 1) z2 + j + + v 3 4 j= 1 |N −N |+1/2 2 1 2

µ 2 1 2 + z2 + j + − v . (110) 3 4 There is also the contribution from the ghosts, however, this has no new terms for our choice of background, 1 (N +N )−1 2 12 µ 2 V G = 2 (2j + 1) z2 + j(j + 1) (111) eff 3 = 1 | − | j 2 N1 N2 and the total effective action is the sum of the three pieces = B + F + G Veff Veff Veff Veff. (112) We introduce the variables N and u,

N1 = N + 2u, N2 = N, (113) where u will be related to the difference in radii of the two spheres and from now on we will restore α using dimensional analysis. Deﬁne

4 2 2 1 αµ 2 2 2 2 η± = z ± + 16α v ,ν± = z ± αv, (114) 2 6 and assuming that u 1 (rather than assuming u 0 because the lower limit of the ﬁrst summa- tioninEq.(104) has to be nonnegative) we can write the effective action so that j always starts from 0 and ﬁnishes at N − 1. N−1 2 1 1 αµ 2 V =− (2j + 2u − 1) z2 + (j + u)2 eff α 3 j=0 H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30 23 2 1 αµ 2 + (2j + 2u + 3) z2 + (j + u + 1)2 3 2 1 αµ 2 + (2j + 2u + 1) z2 + (j + u)(j + u + 1) 3 2 1 αµ 2 + 5(2j + 2u + 1) z2 + (j + u + 1/2)2 3 1 2 2 2 αµ 1 + (2j + 2u + 1) η+ + (j + u)(j + u + 1) + 3 8 1 2 2 2 αµ 1 + η− + (j + u)(j + u + 1) + 3 8 1 2 2 2 αµ 2 − 2(2j + 2u) ν+ + (j + u + 1/4) 3 1 2 2 2 αµ 2 + ν− + (j + u + 1/4) 3 1 2 2 2 αµ 2 − 2(2j + 2u + 2) ν+ + (j + u + 3/4) 3 1 2 2 2 αµ 2 + ν− + (j + u + 3/4) 3 2 1 αµ 2 − 2(2j + 2u + 1) z2 + (j + u)(j + u + 1) . (115) 3 Let us write the above summation as

N−1 Veff = V(j) (116) j=0 and use Euler–Maclaurin sum formula to convert it into integrals

N 1 1 V = V(j) dj − V(N) + V(0) + V (N) − V (0) eff 2 12 0 1 − V (N) − V (0) +···. (117) 720 It is useful to ﬁrst see what we are expecting from such an integral. From Eq. (115),wesee that a typical term of V(j) is roughly of the form: 1 j j 2 1 z 2 v V(j) = j z2 + αv + α2µ2j 2 = µN 2 + + . (118) α N N N 2 αµ αµ2 24 H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30

Plugging the above form of V into Eq. (117) and defining new variables γ = j/N and ζ = (( z )2 + v ), not keeping track of the exact coefficients, we have: αµ αµ2 1 ζ 2 ζ 2 V = µN 3 dγ γ γ 2 + + µN 2 1 + eff N N 0 2 1 2 1 2 ζ µ 3 2 ζ + µN ∂γ γ γ + + ∂ γ γ + +··· N N γ N 0 0 ζ 1 ζ 1 ζ = µN 3 F + F + F +··· , (119) 0 N N 1 N N 2 2 N where F are functions of ζ = 1 (( z )2 + v ) originating from the nth derivative of γ (we n N N αµ αµ2 ζ shall see in the next paragraph why we use N as the argument for Fn). Note that each Fn is 1 weighted by a factor Nn . First we look at the term F0. Assuming that in a power expansion of F0(x) there exists a term 4 x3, after expanding in α it would contribute to V a v4 term of the form µN 3( ζ )3 ∼ αv , which eff N µ2z5 has the correct form of membrane interactions and has the correct order of N (see Section 2). 1 If in the power expansion of F0(x) there also exists a term x , then it would contribute to Veff 4 3 ζ ∼ N2α3v4 a v term of the form µN ( N ) z7 . This is the same expression as graviton interactions. The details of the coefficients and whether such terms exist in the power expansion of course has to be seen by actually performing the integration, however as we will see later, the integrals do produce terms of the correct interactions, in the both the membrane and the graviton limit. Next we look at Fn for n>0. The whole argument in the last paragraph goes through, except 1 now every term is weighted by an extra factor of Nn . For example, the membrane like interaction 4 produced by F looks like 1 αv . This factor of 1 means that this term is in fact a matrix n Nn µ2z5 Nn theory correction to supergravity, because it vanishes as N →∞. Therefore, we could see that in converting the summation into a series of integrals, only the first one is needed for comparison with supergravity. All the other Fn with n>0 produces only matrix theory corrections which is not the subject of interest here. Now we go back to Eq. (115). As discussed above, we only need the integral part F0, which we calculate using MATHEMATICA. After calculating the integrals, u is replaced by the supergravity = αµ variable w 3 u, and the answer is expanded first in large N, keeping only the leading order (which is order N 0), and then expanded in small α, keeping only the (α)1 order (which according to Section 2.2 is the appropriate order to be compared to supergravity in the membrane limit). Finally the answer is converted back into Minkowski signature by sending v2 →−v2. We obtain the following one-loop potential in the membrane limit4:

(36v2 − z2w2)(108v2 − (4w2 + 7z2)µ2) V = α . (120) eff 1152(w2 + z2)5/2µ2

4 This is only the effective potential in the membrane limit because when we expand in large N and keeping only the lowest order we essentially send the radius of the spheres r ∼ αµN to inﬁnity, thus in terms of z we have z 1. Note 0 r0 also that the order the limits are taken is important. If the small α limit is taken before the large N limit, implicitly we z would be assuming αµ N which is the graviton limit. H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30 25

We ﬁnd exact agreement when this expression is compared with the supergravity light-cone Lagrangian given in Eq. (95).

6. Interpolation of the effective potential

6.1. The interpolation

The purpose of this section is to find the effective potential that interpolates between the membrane limit and the graviton limit. Due to the complexity of the field equations on the supergravity side, this problem could only be attacked on the matrix theory side. On the matrix theory side, however, there is a subtlety that needs to be taken into account before such a potential could be found. Here we will analyze only the v4 term, the µ2v2 term as well as µ4 term can be studied in exactly the same way. From the supergravity side, what we wish to find is more or less clear. Near the membrane, when z 1, we expect an expansion like: r0

αv4 z z 2 z 3 z 4 V = + + + + +··· eff 2 5 1 (121) µ z r0 r0 r0 r0 and far away from the membrane, when z 1, we expect an expansion: r0

αv4 r2 r r 2 r 3 r 4 V = 0 1 + 0 + 0 + 0 + 0 +··· . (122) eff µ2z5 z2 z z z z

2 3 4 4 r2 We have used N α v = αv 0 to rewrite the graviton result so that it looks similar to the mem- z7 µ2z5 z2 brane effective potential. Therefore, we are basically looking for a function C(x) which appears in the effective potential in the following way:

αv4 z V = C . eff 2 5 (123) µ z r0 As one can see, both the graviton and the membrane actions are of this form. C(x) should have the appropriate limit at x → 0 and x →∞to give the correct potential at the membrane and the graviton limit respectively. Physically it represents curvature corrections due to the ﬁnite size of the spherical membrane. So now we go to the matrix theory side to try to ﬁnd C(x). The subtlety is that the effective potential on the matrix theory side not only includes curvature corrections but also the matrix theory corrections to supergravity which we are not interested in, not to mention that the sum over mass formula is incapable of deducing such matrix theory corrections exactly [2]. For our purpose of comparing with supergravity, therefore, all matrix theory corrections to supergravity should be thrown away. Such corrections appear on the matrix theory side as 1/N corrections, mixed together with the curvature corrections, and it looks something like:

αv4 z 1 z 1 z V = C + C + C +··· . eff 2 5 0 1 2 2 (124) µ z r0 N r0 N r0

In such an expansion, only the C0 term should be kept. The reader is cautioned that naively sending N to inﬁnity will not give us the correct interpolating potential because r0 also depends on N and such limit would only result in the effective potential in the membrane limit. 26 H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30

There are many ways such matrix corrections could appear. For example, in a typical matrix theory computation we may get terms of the form: αv4 V ∼ z2 + α2µ2 . (125) eff µ2z7 Rewriting the above gives:

αv4 α2µ2N 2 1 αv4 r2 1 V ∼ 1 + ∼ 1 + 0 . (126) eff µ2z5 z2 N 2 µ2z5 z2 N 2 The second term can now be identiﬁed as a matrix theory correction and is irrelevant to us. To isolate the curvature corrections (which we want) from the matrix theory corrections (which we do not want), we look at Eq. (124) more carefully. We can see that since 1 = αµ ,all N 6r0 the matrix theory corrections will appear at higher order in α. Therefore to get the interpolating effective potential from the matrix theory side, we can follow the steps below:

(1) change the summation over j in Eq. (115) into an integral over j from 0 to N; 6r0 3w (2) replace N by αµ , and u by αµ; (3) expand in small α and keep only the lowest order (which shall turn out to be order α1, with all lower orders vanishing). This is the interpolating effective potential.

With MATHEMATICA, the interpolating effective potential for two spheres of the same radius (w = 0) in Minkowski signature is found to be: α 36v2 − z2µ2 Veff = µ2z5 1152(4r2 + z2)5/2 ! " 0 # × 2 − 5 + 4 2 + 2 + 2 2 2 + 2 + 4 2 + 2 108v z 16r0 4r0 z 8r0 z 4r0 z z 4r0 z " " # − 2 2 4 2 + 2 + 4 − + 2 + 2 z µ 112r0 4r0 z 7z z 4r0 z " ##$ + 2 2 − + 2 + 2 8r0 z 2z 7 4r0 z . (127)

2 2 2 We see that Veff always carries a factor of 36v − z µ , meaning the effective potential van- = µz ishes whenever v 6 . This is expected because such conﬁgurations correspond to circular orbits which preserve half of the supersymmetries. Expanding this potential in the membrane limit of large r0 we get: α(3888v4 − 360v2z2µ2 + 7z4µ4) α(36v2 − z2µ2)(108v2 − 7z2µ2) V = = . (128) eff 1152µ2z5 1152µ2z5 This result is of course identical to the matrix theory result (with w = 0) in Section 5 where the membrane limit was taken in advance. In the graviton limit of small r0, after replacing r0 by αµN/6, we have: N 2α3(720v4 − 56v2z2µ2 + z4µ4) N 2α3(20v2 − z2µ2)(36v2 − z2µ2) V = = . (129) eff 768z7 768z7 The two limits of the effective action can then be compared with that of the supergravity side. Indeed from the expressions (96) and (97) we see that we have perfect agreement. H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30 27

In above we have only given the expression of the interpolating potential for two membranes with the same radius (w = 0). We have also found the interpolating potential with w included, using the same steps given above. However we choose to omit the rather lengthy expression here for brevity.

6.2. Comparison with Shin and Yoshida

As mentioned in Section 5, Ref. [10], considered the case of supersymmetric circular motion µ with angular frequency 6 and found a ﬂat potential, which agrees with what we have found (see the comment under Eq. (127)). In a subsequent paper, [11], the authors considered the case of a slightly elliptical orbit with separation z, µt µt µt z = (r + )cos2 + (r − )sin2 = r2 + 2 + 2r cos (130) 2 6 2 6 2 2 3 and velocity v, µ µt µt v = (r + )sin2 + (r − )cos2 6 2 6 2 6 µ µt = r2 + 2 − 2r cos , (131) 6 2 2 3 where is the small expansion parameter for the eccentricity of the orbit. They considered the large separation limit, r2 0, and found an effective action, Eq. (1.2) of [11], 2 4 3 4 2 35 1 385(αµN) 1 1 Γeff = dt α µ N − 4 − 27 · 3 r7 211 · 33 N 2 r9 2 2 35 1 11(αµN)2 1 1 = 4 dt α3µ4N 2 − − 7 1 2 9 (132) 384 r2 36 4N r2 4 9 after expanding to O( ) and O(1/r2 ). Note that in the equation above we have restored µ and α, and set N1 = N2 = N, r1 = 0. To compare this with our result we substitute the above expressions of z(t) and v(t) into our interpolating potential equation (127) and expand in the parameters 1/r2 and . As we are only comparing effective actions we average our potential over one period of oscillation. We ﬁnd for our time-averaged potential,

105 1 r2 35 1 11(αµN)2 1 V = α4µ2 r2 − 0 = α34µ4N 2 − eff 0 7 11 9 7 9 (133) 32 r2 r2 384 r2 36 r2 = αµN (where to reach the ﬁnal line we used r0 6 ) which agrees with Eq. (132) after throwing away the 1 matrix theory corrections in the latter. In calculating the matrix theory effective potential N2 in Section 5 we assumed a constant velocity. However, as we can see by this comparison, as long as we ignore matrix theory corrections, it leads to the correct result. Thus we see that we can consistently neglect acceleration terms in the effective potential as discussed earlier. 28 H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30

7. Discussion and future directions

One immediate generalization of the work reported in this paper is to consider more complicated probe configurations (recall that in this paper we have restricted our attention to a probe membrane which is spherical and has no velocity in the x1 through x3 directions). For example, we could consider deforming the probe membrane so that it is no longer a perfect sphere. It means that the coordinates of the membrane XA will now be some general functions of θ and φ, and in particular it no longer has to appear only as a point in the 4 through 9 directions. We can also give the probe membrane a nonzero velocity in the x1 through x3 direction. These generalizations give more interesting dynamics and can be fairly easily carried out. On the supergravity side, this just requires putting the relevant probe configuration into the light-cone Lagrangian; on A the matrix theory side, this requires replacing the background configurations B(2)’s used in this paper with some more general configurations. In this paper we take the source to be a static spherical membrane preserving all of the sixteen linearly realized supersymmetries. As we know, there are other interesting membrane configurations that are permitted by the pp-wave background, for example, the static hyperbolic brane which preserves eight supersymmetries, given by [19]. It would be interesting to investigate gauge/gravity duality in the case where these static objects are sources and interact with some graviton or membrane probe. On the supergravity side, we can apply our general formalism of diagonalizing the field equations as well as the near-membrane expansion to find the metric and three-form perturbations produced by these sources. On the matrix theory side, for the aforemen- tioned hyperbolic membrane, one could use the unitary representations of SO(2, 1) worked out in [20–22] to expand the fluctuating fields and compute the one-loop effective potential. Moreover, it is evidently desirable to push our computation on the supergravity side beyond the near-membrane expansion, finding the solution to the field equations which will then give the δLlc to be compared with the fully interpolating potential (127) found on the matrix theory side. The membrane-interaction we described in this paper has no longitudinal momentum transfer. However, as we can see, our supergravity computation can be quite easily generalized to nonzero longitudinal momentum transfer, by not setting k− to zero when solving the field equations (e.g., see the comments made after Eq. (77)). On the gauge theory side, this requires an instanton computation. We hope to say more on this in the near future. There are also M-theory pp-wave backgrounds with less supersymmetries, and matrix theories in these pp-waves backgrounds have been proposed [23,24]. It would be interesting to investigate the gauge/gravity duality in these less supersymmetric settings. Looking slightly further afield it would be interesting to consider the extension of our results to the tiny graviton matrix theory. The BMN matrix theory used in this paper can be derived from the regularized M2-brane action, specifically the M2-brane in the 11-dimensional pp-wave background, and we may interpret it as a theory of N spherical M2-branes of very small size. The tiny graviton matrix theory of [25] is quite analogous, however instead of spherical M2-branes it describes the dynamics of very small D3-branes on the ten-dimensional pp-wave background. This is then conjectured to be a dual description of DLCQ strings in this background. Evidence for this conjecture was provided in the original paper [25] and subsequently in [26].In[25] the zero energy vacuum configurations were found and it was shown that there are solutions consisting of concentric fuzzy spheres much like those analyzed in this paper (though of course they are now fuzzy three-spheres). It should be possible to modify our calculations to study the analogous dynamics of spherical branes in the tiny graviton matrix theory and compare with what is found from the type IIB supergravity. H.K. Lee et al. / Nuclear Physics B 728 (2005) 1–30 29

Finally we note that while this paper was being written, the work by Lin, Lunin, and Mal- dacena (LLM) [27] appeared, in which all 1/2 BPS excitations of M-theory plane wave were found in a quite elegant manner. We expect that the singular supergravity solution for the spherical membrane presented in this work can be obtained from a certain regular LLM geometry by taking the appropriate limit and reducing to the linearized level, and it would be interesting to see how this can be done in detail.5 Moreover, one should be able to use LLM’s results to compute the interaction between a probe (e.g. a graviton, or a spherical membrane) and an arbitrary 1/2 BPS source, and compare the answer with that of a loop computation in the pp-wave matrix theory. However, how to represent these general 1/2 BPS sources on the matrix theory side is not very clear to us at this point.

Acknowledgements

The authors would like to thank John Schwarz for helpful advice and support during the summer. We would also like to thank I. Swanson for useful discussions and Oleg Lunin for explaining the work [27] to us. This work was supported by DOE grant DE-FG03-92ER40701. X. Wu is also supported in part by DOE grant DE-FG01-00ER45832 and NSF grant PHY/0244811.

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Low-energy photon–photon collisions to two loops revisited

J. Gasser a,M.A.Ivanovb,M.E.Sainioc,d

a Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland b Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna (Moscow region), Russia c Helsinki Institute of Physics, PO Box 64, 00014 University of Helsinki, Finland d Department of Physical Sciences, University of Helsinki, Finland Received 27 June 2005; accepted 1 September 2005 Available online 26 September 2005

Abstract In view of ongoing experimental activities to determine the pion polarizabilities, we have started to recalculate the available two-loop expressions in the framework of chiral perturbation theory, because they have never been checked before. We make use of the chiral Lagrangian at order p6 now available, and of improved techniques to evaluate the two-loop diagrams. Here, we present the result for the neutral pions. The cross section for the reaction γγ → π0π0 agrees with the earlier calculation within a fraction of a percent. We present analytic results for the dipole and quadrupole polarizabilities, and compare the latter with a recent evaluation from data on γγ → π0π0.  2005 Elsevier B.V. All rights reserved.

PACS: 11.30.Rd; 12.38.Aw; 12.39.Fe; 13.60.Fz

Keywords: Chiral perturbation theory; Two-loop diagrams; Pion polarizabilities; Compton-scattering

1. Introduction

We consider the process γγ → π 0π 0 in the framework of chiral perturbation theory (ChPT) [1–3]. The one-loop calculation of the scattering amplitude was performed in Refs. [4,5], and the two-loop amplitude was worked out in [6]. Because the effective Lagrangian at order p6 was not available at that time, the ultraviolet divergences were evaluated in the MS scheme,

E-mail address: [email protected] (J. Gasser).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.09.010 32 J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54 then dropped and replaced with a corresponding polynomial in the external momenta. The three new counterterms which enter at this order in the low-energy expansion were estimated with resonance saturation. Whereas such a procedure is legitimate from a technical point of view, it does not make use of the full information provided by chiral symmetry. The evaluation of the two-loop amplitude involving charged pions was performed later by Burgi [7]. Over the last ten years, considerable progress has been made in this ﬁeld, both in theory and experiment. As for theory, the Lagrangian at order p6 has been constructed [8,9], and its divergence structure has been determined [10]. This provides an important check on the above calculations: adding the counterterm contributions from the p6 Lagrangian to the MS amplitude evaluated in [6] and in [7] must provide a scale independent result. Also in the theory, improved techniques to evaluate the two-loop diagrams that occur in these amplitudes have been developed [11]. The improvement arises mainly in the evaluation of diagrams with four external legs, where the techniques of Ref. [11] allow one to extract the ultraviolet divergences by use of simple recursion relations. We are now able to present the ﬁnal result for the two-loop amplitudes in a rather compact form (in Refs. [6,7], the result was presented partly in numerical form only, because the algebraic expressions were too long to be published). Concerning experiment, quadrupole polarizabilities [12] for the neutral pions have recently been determined from data on γγ → π 0π 0 [13]. Further, the charged pion polarizabilities (α − β)π+ have been determined at the Mainz Microtron MAMI [14], with a result that is at variance with the two-loop calculation presented in [7]. Last but not least, there is an ongoing experiment by the COMPASS Collaboration at CERN to measure the charged pion and kaon polarizabilities [15,16]. In view of these developments, and because the two-loop expressions for the polarizabilities had never been checked, we decided to recalculate these amplitudes, using the improved techniques of Ref. [11] to evaluate the integrals, and invoking the chiral Lagrangian at order p6 [9, 10]. As the calculation in the case of neutral pions involves considerably less diagrams, and because the Fortran code for these amplitudes is still available to us for checks, we have decided to start the program with a re-evaluation of these amplitudes. This is the main purpose of the present work. The evaluation of the corresponding expressions for the charged pions and for the kaons is underway and will be presented elsewhere [17]. The article is organized as follows. Section 2 contains the necessary kinematics of the process γγ → π 0π 0. To make the article self contained, we summarize in Section 3 the necessary ingredients from the effective Lagrangian framework. In Section 4, we display the Feynman diagrams and discuss their evaluation. Section 5 contains a concise representation of the two Lorentz invariant amplitudes that describe the scattering matrix element. In Section 6, we compare the present work with the previous calculation [6], while Section 7 contains explicit expressions for the dipole and quadrupole polarizabilities valid at next-to-next-to-leading order in the chiral expansion, together with a numerical analysis and a comparison with an evaluation from data on γγ → π 0π 0 [13]. The summary and an outlook are given in Section 8. Finally, several technical aspects of the calculation are relegated to Appendices A–E.

2. Kinematics

The matrix element for the reaction

0 0 γ(q1)γ (q2) → π (p1)π (p2) (2.1) J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54 33 is given by 0 0 4 (4) N π (p1)π (p2) out γ(q1)γ (q2) in = i(2π) δ (Pf − Pi)T , (2.2) with T N = e2µνV , 1 2 µν −i(q1x+q2y) 0 0 Vµν = i dxe π (p1)π (p2) out Tjµ(x)jν(y)|0. (2.3)

2 Here jµ is the electromagnetic current, and α = e /4π 1/137. We consider real photons, 2 = · = qi 0, with i qi 0. The decomposition of the correlator Vµν into Lorentz invariant amplitudes reads

Vµν = A(s, t, u)T1µν + B(s,t,u)T2µν + C(s,t,u)T3µν + D(s,t,u)T4µν, 1 T = sg − q q , 1µν 2 µν 1ν 2µ 2 T2µν = 2s∆µ∆ν − ν gµν − 2ν(q1ν∆µ − q2µ∆ν),

T3µν = q1µq2ν,

T4µν = s(q1µ∆ν − q2ν∆µ) − ν(q1µq1ν + q2µq2ν),

∆µ = (p1 − p2)µ, (2.4) where 2 2 2 s = (q1 + q2) ,t= (p1 − q1) ,u= (p2 − q1) ,ν= t − u (2.5) are the standard Mandelstam variables. The tensor Vµν satisﬁes the Ward identities µ = ν = q1 Vµν q2 Vµν 0. (2.6) The amplitudes A and B are analytic functions of the variables s,t and u, symmetric under crossing (t, u) → (u, t). The amplitudes C and D do not contribute to the process considered here, because i · qi = 0. It is useful to introduce in addition the helicity amplitudes 4 − 2 8(Mπ tu) H++ = A + 2 4M − s B, H+− = B. (2.7) π s The helicity components H++ and H+− correspond to photon helicity differences λ = 0, 2, re- | = 3 0 (3) − spectively. With our normalization of states p1 p2 2(2π) p1δ (p1 p2), the differential cross section for unpolarized photons in the centre-of-mass system is

γγ→π0π0 2 dσ α s 2 2 = β(s)H(s,t), H(s,t) =|H++| +|H+−| , (2.8) dΩ 64 = − 2 with β(s) 1 4Mπ /s. The relation between the helicity amplitudes M+± in Ref. [13] and the amplitudes used here is

M++(s, t) = 2παH++(s, t), M+−(s, t) = 16παB(s,t). (2.9) The physical regions for the reactions γγ → π 0π 0 and γπ0 → γπ0 are displayed in Fig. 1, where we also indicate with dashed lines the nearest singularities in the amplitudes A and B. These singularities are generated by two-pion intermediate states in the s,t and u channel. 34 J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54

Fig. 1. Mandelstam plane with three related physical regions. s-channel: γγ → ππ, t-andu-channel: γπ → γπ.We → → = 2 denote the threshold for γγ ππ(γπ γπ) by A (P ). The dashed lines at s,t,u 4Mπ indicate the presence of branch-points in the amplitude, generated by two-pion intermediate states.

3. The effective Lagrangian and its low energy constants

The effective Lagrangian consists of a string of terms. Here, we consider QCD with two ﬂavours, in the isospin symmetry limit mu = md =ˆm. At next-to-next-to-leading order (NNLO), one has

Leff = L2 + L4 + L6. (3.1)

The subscripts refer to the chiral order. The expression for L2 is

F 2 L = D UDµU † + M2 U + U † , 2 4 µ e D U = ∂ U − i(QU − UQ)A ,Q= diag(1, −1), (3.2) µ µ µ 2 where e is the electric charge, and Aµ denotes the electromagnetic field. The quantity F denotes the pion decay constant in the chiral limit, and M2 is the leading term in the quark mass expansion 2 2 = 2 + ˆ ··· of the pion (mass) , Mπ M (1 O(m)). Further, the brackets denote a trace in flavour space. In Eq. (3.2), we have retained only the terms relevant for the present application, i.e., we have dropped additional external fields. We choose the unitary 2 × 2matrixU in the form √ 2 0 + 2 π √π 2π U = σ + iπ/F, σ + = 1 × ,π= − . (3.3) F 2 2 2 2π −π 0 The Lagrangian at NLO has the structure [2]

10 l L = l K = 1 D UDµU † 2 +···, (3.4) 4 i i 4 µ i=1 J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54 35 where li denote low energy couplings (LECs), not ﬁxed by chiral symmetry. At NNLO, one has [9,10] 57 L6 = ciPi. (3.5) i=1

For the explicit expressions of the polynomials Pi , we refer the reader to Refs. [9,10].Thever- 0 0 tices relevant for γγ → π π involve l1,...,l6 from L4 and c29,...,c34 from L6. 4 6 The couplings li and ci absorb the divergences at order p and p , respectively, = d−4 r + li (µc) li (µ, d) γiΛ , (µc)2(d−4) c = cr (µ, d) − γ (2)Λ2 − γ (1) + γ (L)(µ, d) Λ , i F 2 i i i i 1 1 Λ = , ln c =− ln 4π + Γ (1) + 1 . (3.6) 16π 2(d − 4) 2 r r r r The physical couplings are li (µ, 4) and ci (µ, 4), denoted by li ,ci in the following. The coef- (1,2,L) ﬁcients γi are given in [2], and γ are tabulated in [10]. In order to compare the present i ¯ calculation with the result of [6], we will use the scale independent quantities li introduced in [2],

r γi ¯ l = (li + l), (3.7) i 32π 2 = 2 2 where the chiral logarithm is l ln(Mπ /µ ). We will use [18] ¯ ¯ ¯ ¯ l1 =−0.4 ± 0.6, l2 = 4.3 ± 0.1, l3 = 2.9 ± 2.4, l4 = 4.4 ± 0.2, (3.8) and [19] ¯ . ¯ ¯ l∆ = l6 − l5 = 3.0 ± 0.3. (3.9) r The constants ci occur in the combinations ar = 4096π 4 −cr − cr + cr , 1 29 30 34 ar = 256π 4 8cr + 8cr + cr + cr + 2cr , 2 29 30 31 32 33 r =− 4 r + r + r b 128π c31 c32 2c33 . (3.10)

Their values were estimated by resonance exchange in Ref. [6], and in the large Nc framework r in Ref. [20].InRef.[21], c34 was determined from a chiral sum rule. For the present application, we take the values obtained in [6], r + r =− ± a1(Mρ) 8b (Mρ) 14 5, r − r = ± a2(Mρ) 2b (Mρ) 7 3, r b (Mρ) = 3 ± 1,Mρ = 770 MeV. (3.11) We have checked that the couplings from Ref. [20] lead to polarizabilities that are compatible with the results displayed in Table 1.Below,weuseµ = Mρ . As mentioned already in Ref. [6], varying this scale between 500 MeV and 1 GeV leads to a negligible change of, e.g., the cross 0 0 section for the reaction γγ → π π below 400 MeV. Finally, we will use Fπ = 92.4MeV[22] (see [23] for a recent update of this value), and Mπ = 135 MeV. 36 J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54

Table 1 − − The dipole and quadrupole polarizabilities in units of 10 4 fm3 and 10 4 fm5, respectively. The second column contains the two-loop results, including the corresponding uncertainties, and the third one displays the contribution from ω ex- 8 6 + change at order p . Note that this contribution is much larger than the uncertainty at order p in the case of (α2 β2)π0 . [∗] + The symbol refers to the present work. The slight difference in the value of (α1 β1)π0 with the one reported in [6] ¯ is due to the updated values of l2 and of the pion decay constant Fπ used here. See main text for more details ChPT to two loops ω-exchange at order p8 Dispersion relations − − ± − − ± (α1 β1)π0 1.9 0.2 [6] 0.02 1.6 2.2 [26] + ± ± (α1 β1)π0 1.1 0.3 [6] 0.02 0.98 0.03 [26] 1.00 ± 0.05 [27] − ± [∗] ± (α2 β2)π0 37.6 3.3 0.50 39.70 0.02 [13] + ± [∗] − − ± (α2 β2)π0 0.037 0.003 0.25 0.181 0.004 [13]

Fig. 2. The one-loop diagrams.

4. Evaluation of the diagrams

The lowest-order contributions are the one-loop diagrams displayed in Fig. 2. They have been evaluated for the first time in Refs. [4,5], where it was noticed that the sum of these two amplitudes is ultraviolet finite, because there are no contributions from the effective Lagrangian at order p4 at this order. The two-loop diagrams are displayed in the Figs. 3, 5 and 6. The two-loop diagrams in Fig. 3 may be generated according to the scheme indicated in Fig. 4, where the shad- owed blob denotes the d-dimensional elastic ππ-scattering amplitude at one-loop accuracy, with two pions off-shell. As is discussed in Appendix C, the one-loop integrals in the ππ amplitude may be represented in a dispersive manner. This allows one to reduce the two-loop integrals in Fig. 3 to the one-loop ones, where one has to perform at the end an integration over a dispersive parameter. Two further diagrams are displayed in Fig. 5. The first one—called “acnode” in the literature—may again be evaluated by use of a dispersion relation, see Appendix C. The second one is trivial to evaluate, because it is a product of one-loop diagrams. The remaining diagrams at order p6 are shown in Fig. 6. The evaluation of the diagrams was done in the following manner.

(1) We have performed the integration over the loop momenta in the d-dimensional regularization scheme, in particular using the procedure described in Ref. [11] and in Appendix C, and invoking FORM [24]. (2) We have then checked numerically that the amplitude satisfies the Ward identities (2.6) in d dimensions, in the unphysical region, where the amplitudes are real. (3) We have verified that the counterterms from the Lagrangian L6 [10] remove all ultraviolet divergences, which is a very non-trivial check on our calculation. (4) We have checked that the (ultra-violet finite) amplitude so obtained is scale independent. (5) Finally, we have numerically verified that the three lowest partial waves of the helicity non-flip amplitude H++ carry the proper one-loop ππ phase, in conformity with unitarity. J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54 37

Fig. 3. A set of two-loop diagrams generated by L2 and one-loop diagrams generated by L4.

Fig. 4. Construction scheme for the diagrams in Fig. 3. 38 J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54

Fig. 5. Acnode and butterﬂy diagrams.

6 Fig. 6. The remaining diagrams at order p : one-loop graphs generated by L4, and counterterm contributions from L6.

We note that the steps (3) and (4) could not be performed in Refs. [6,7], because the counterterms at order p6 were not yet available [10].

5. The two-loop amplitudes

We give the expression for the amplitudes A and B by using the same notation as in [6], and ¯ ¯ refer the reader to Appendix B for the one-loop integrals J(s)¯ , J(s)¯ , G(s)¯ , G(s)¯ and H(s)¯ that occur. We have

4G¯ (s) A = π s − M2 + U + P + O E4 , sF2 π A A π 2 B = UB + PB + O E . (5.1) ¯ ¯ The quantity Gπ (s) stands for G(s), evaluated with the physical pion mass. The unitary parts UA(B) contain s,t and u-channel cuts, and PA(B) are linear polynomials in s. We ﬁnd

2 l¯ U = G(s)¯ s2 − M4 J(s)¯ + C(s,l¯ ) + ∆ s − M2 J(s)¯ A 4 π i 2 4 π sFπ 24π Fπ (l¯ − 5/6) ¯ ¯ + 2 s − M2 H(s)¯ + sG(s)¯ + M2 G(s)¯ − J(s)¯ d2 2 4 4 π 4 2 π 3 00 144π sFπ + ∆A(s,t,u), J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54 39 1 4 1 5 C(s,l¯ ) = l¯ − s − M2 2 + l¯ − s2 − sM2 + M4 i 2 2 1 2 π 2 4 8 π 16 π 48π 3 3 6 − 4 ¯ + 2 − 2 ¯ − 2 + 4 3Mπ l3 12Mπ s Mπ l4 12sMπ 15Mπ , 1 d2 = 3 cos2 θ − 1 , 00 2 (l¯ − 5/6)H(s)¯ U = 2 + ∆ (s,t,u). B 2 4 B (5.2) 288π Fπ s Here, θ is the scattering angle in the centre-of-mass system,

q1 · p1 =|q1||p1| cos θ. (5.3)

The expressions for ∆A(B) are displayed in Appendices D and E. The polynomial parts are 1 2 PA = a M + a s , (16π 2F 2)2 1 π 2 π 1 4 20 110 a = ar + 4l2 + l 8l¯ + 12l¯ − − l¯ + 12l¯ + , 1 1 18 2 ∆ 3 3 2 ∆ 9 1 4 5 697 a = ar − l2 + l 2l¯ + 12l¯ − − l¯ + 12l¯ + , (5.4) 2 2 18 2 ∆ 3 3 2 ∆ 144 b PB = , (16π 2F 2)2 π 1 2 1 393 b = br − l2 + l 2l¯ + − l¯ + , 36 2 3 3 2 144 M2 l = log π . (5.5) µ2 r r r 6 The constants a1,a2 and b are displayed in terms of the LECs at order p in Eq. (3.10).Using the fact that the bare couplings ci displayed in Eq. (3.6) are scale independent, one indeed ﬁnds that the above expressions for the amplitudes A,B are scale independent as well.

6. Comparison with the previous calculation

We can now compare the amplitudes A,B with the earlier calculation, presented in Section 7 of Ref. [6]. In that reference, the amplitudes were evaluated with a different techniques. Further- more, the Lagrangian L6 was not available in those days, and an important ingredient to check the ﬁnal result was, therefore, missing. We can make the following observations.

(1) The amplitudes A and B consist of a part with explicit analytic expressions, and additional terms ∆A,B, that are given in Appendices D and E of the present work in the form of integrals over Feynman parameters. These latter terms were given only in numerical form in [6]. (2) The explicit analytic expressions agree with the previous calculation, except for the coefficient of the single logarithm in a2 in Eq. (5.4). The factor 2/3in[6] is replaced by −4/3 here. As the present amplitude is scale independent, we conclude that it is the result Eq. (5.4) which is 40 J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54 correct.1 This mistake does not affect the algebraic expressions for the polarizabilities discussed below, for which we fully agree with Ref. [6]. (3) We can compare the quantities ∆A,B in numerical form only. For this purpose, we have made two checks. First, we have evaluated the cross section for the reaction γγ → π 0π 0 below a centre-of-mass energy of 400 MeV, using the same values for the LECs as in [6]. It agrees with the previous one within a fraction of a percent—the difference would not be visible in Fig. 5 of Ref. [6], and we do not, therefore, reproduce that plot here. Second, we have re-evaluated the two-loop contributions to the polarizabilities presented in column 4 of Table 3 in [6].The numbers (0.17, −0.31) in the old calculation become (0.17, −0.30) here. (4) To summarize, we confirm the previous result up to the coefficient in one of the chiral logarithms, and up to minute changes in the numerical values of ∆A,B. Numerically, the results in [6] are not affected in any significant manner by these modifications, whose effect is by far smaller than the uncertainties generated by the (not precisely known) values of the low energy constants.

7. Pion polarizabilities: dipole and quadrupole

The dipole and quadrupole polarizabilities are deﬁned [12,13] through the expansion of the = 2 helicity amplitudes at ﬁxed t Mπ , α = 2 = ± + s ± + O 2 H+∓ s,t Mπ (α1 β1)π0 (α2 β2)π0 s . (7.1) Mπ 12 Because we have at our disposal the helicity amplitudes at two-loop order, we can work out the polarizabilities to the same accuracy. It turns out that all relevant integrals can be performed in closed form. We discuss the results in the remaining part of this section.

7.1. Chiral expansion

Using the same notation as in [6], we ﬁnd for the dipole polarizabilities 2 α Mπ d1± 4 (α ± β ) 0 = c ± + + O M , 1 1 π 2 2 1 2 2 π (7.2) 16π Fπ Mπ 16π Fπ with

c1+ = 0,c1− =−1/3, r 1 ¯ ¯ d + = 8b − 144l(l + 2l ) + 96l + 288l + 113 + ∆+ , 1 648 2 2 r r 1 ¯ ¯ ¯ ¯ ¯ d − = a + 8b + 144l(3l − 1) + 36(8l − 3l − 12l + 12l ) + 43 + ∆− , 1 1 648 ∆ 1 3 4 ∆ 2 2 ∆+ = 13643 − 1395π ,∆− =−3559 + 351π . (7.3) We have split off the numbers 113 and 43, respectively, to illustrate that these expressions completely agree with the ones displayed in Eq. (8.14) of Ref. [6], where, as already mentioned, no

1 Burgi [25] provides in his thesis work the isospin I = 0, 2 amplitudes, and the one for the charged pions. Subtracting the latter from the former reveals that his calculation agrees with the statement just made. J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54 41 explicit expressions for the remainders ∆± were worked out. For the quadrupole polarizabilities, we obtain 2 α Mπ d2± 4 (α ± β ) 0 = c ± + + O M , 2 2 π 2 2 3 2 2 2 π (7.4) 16π Fπ Mπ 16π Fπ with

c2+ = 0,c2− = 156/45, 2 ¯ 5009 13453π 16l2 d + =− + + , (7.5) 2 27 720 45 r r 1 ¯ 2 d − = 12a − 24b + 1280l(1 − 6l ) + 19216 − 1811π 2 2 960 ∆ 4(52l¯ + 5l¯ + 3l¯ − 78l¯ + 105l¯ ) − 1 2 3 4 ∆ . (7.6) 45 7.2. Numerical results

For numerical evaluations of the polarizabilities we use the values of the LECs given in Sec- tion 3. The results are displayed in Table 1. The second column contains the ChPT prediction at two-loop order, including an estimate of the uncertainty at this order. The third column displays the contribution from ω exchange at order p8, see the following subsection. It is an indication of yet higher order contributions. The last column contains the results from dispersive calculations. The following comments are in order.

(1) The slight difference in the value of (α1 + β1) 0 with the one reported in [6] is due to the ¯ π updated values of l2 and of the pion decay constant Fπ used here. (2) Our results for the dipole polarizabilities as well as for the quadrupole polarizabilities − (α2 β2)π0 agree with the results of the recent investigations performed in [26,27] and [13] within the uncertainties quoted. (3) The prediction for the quadrupole polarizabilities (α2 + β2) 0 is positive, in contrast to π ¯ the result reported in Ref. [13]. The ChPT expression contains as the only LEC l2, known rather accurately from ππ scattering [18]. We come back to this point in the following subsection, where we also discuss the uncertainties quoted in the table for the ChPT calculation. (4) We plot the helicity amplitudes in Fig. 7. It illustrates the fact that the helicity flip amplitude H+− is quite flat at this order, in contrast to the non-flip amplitude H++, see the values of the quadrupole polarizabilities in Table 1. (5) For a comparison of the ChPT—predictions of the dipole polarizabilities with calculations performed before 1994, and for additional information on these quantities, we refer the interested reader to Refs. [6,28].

7.3. Estimating the uncertainties

To estimate the uncertainties in the prediction of the polarizabilities, we ﬁrst note that the helicity non-ﬂip amplitude H++ starts out at order p4. We have therefore, for this quantity, a leading and next-to-leading order calculation at our disposal. For the corresponding polarizabilities − − (α1 β1)π0 and (α2 β2)π0 , we thus simply add in quadrature the uncertainties generated by the 42 J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54

Fig. 7. The helicity amplitudes α H++(s, t = M2 ) (solid line), α H+−(s, t = M2 ) (dashed line), plotted as a func- Mπ π Mπ π = 2 tion of s,att Mπ . Compare with the definition of the polarizabilities in Eq. (7.1) The dotted lines are displayed to guide the eye. order p4 and p6 LECs (see Section 3). The resulting numbers are given in column 2 of Table 1. For an indication of yet higher order corrections, see column 3 and comments at the end of this subsection. We now consider the helicity flip amplitude H+−, which starts out at order p6—the two-loop calculation performed here determines therefore only the leading order term in H+−. According to Eq. (2.7), this amplitude is proportional to B(s,t,u), which is an analytic function of the variables s,ν at the Compton threshold and can be, therefore, expanded in a Taylor series, B(s,t,u) = U + Vs+ Wν2 + O s2,ν4,sν2 . (7.7) The relation to the polarizabilities is + = + = (α1 β1)π0 8αMπ U, (α2 β2)π0 96αMπ V. (7.8) The Taylor coefficients themselves have a chiral expansion of the form 1 M2 U U = U + π 1 + O M4 , (16π 2F 2)2 0 16π 2F 2 π π π 1 M2 V V = V + π 1 + O M4 . 2 2 2 2 0 2 2 π (7.9) (16π Fπ ) Mπ 16π Fπ 6 Whereas LECs from order p do contribute to U0, the leading term V0 is a pure loop effect, 2 because V0/Mπ is not analytic in the pion mass and thus cannot receive contributions from polynomial counterterms. To illustrate this point, and to estimate the size of V1, we consider the vector meson exchange amplitudes worked out in [6]. The contribution from ω exchange is dominant and given by C 1 1 − B (s,t,u)= ω + ,C= 0.67 GeV 2. (7.10) ω 2 − 2 − ω 2 Mω t Mω u J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54 43

6 In the language of ChPT, this amplitude starts out at order p . It does contribute to U0—this term is included in the resonance exchange estimates for the O(p6) LECs in (3.11). The corresponding + result of the uncertainty in (α1 β1)π0 , calculated as before, is displayed in the second column of Table 1. Higher order corrections are discussed at the end of this subsection. Finally, we come to the estimate of the uncertainty in (α2 + β2) 0 . The one displayed in π ¯ the second column is exclusively due to the uncertainty in the coupling l2, which is known rather well. In agreement with what is said above, resonance exchange does not contribute to V0. On the other hand, there is no reason why V0 should dominate the contribution from V1. 8 Indeed, if we use (7.10) to also estimate effects from order p , we ﬁnd with Mω = 782 MeV the =− + value V1 2.2. The corresponding contribution to the quadrupole polarizability (α2 β2)π0 is −0.25 × 10−4 fm5—of the order needed to bring the ChPT calculation into agreement with the analysis of Ref. [13]. This result illustrates that the discrepancy between the chiral prediction for the quadrupole po- + 6 larizability (α2 β2)π0 at order p and the dispersion analysis in Ref. [13] is of no signiﬁcance, because the terms neglected may well be much larger than the leading order term, which would only dominate for very small values of the pion mass. On the other hand, to obtain a reliable + estimate of (α2 β2)π0 in the framework of ChPT, one needs to perform a reliable calculation at order p8. This is outside the scope of the present work. As Table 1 summarizes the main result of this work, we quote the contribution from ω exchange in column 3, in order to make it very clear also in the table that there is no contradiction + between the two-loop calculation of (α2 β2)π0 and the dispersive analysis of Ref. [13].In addition, we display in the same column the contribution from ω-exchange to the other polarizabilities at order p8 as well, using the relevant expression for the amplitude A [6], s − 4(t + M2 ) s − 4(u + M2 ) A (s,t,u)= C π + π . (7.11) ω ω 2 − 2 − Mω t Mω u + It is seen that, except for (α2 β2)π0 , these higher order corrections are negligible.

8. Summary and outlook

(1) We have recalculated the two-loop expression for the amplitude γγ → π 0π 0 in the framework of chiral perturbation theory. We have made use of the techniques developed in Ref. [11], and of the effective Lagrangian L6 available now [9,10]. (2) The method has allowed us to evaluate the dipole and quadrupole polarizabilities in closed form. (As far as we are aware, the quadrupole polarizabilities have never been calculated in ChPT before.) The two Lorentz invariant amplitudes A and B are presented as a sum of multiple integrals over Feynman parameters whose numerical evaluation poses no difficulty. This is in contrast to Ref. [6], where part of the amplitudes, denoted by ∆A,B, were published in numerical form only. (3) Our result agrees with the earlier calculation [6] up the coefficient in one of the chiral logarithms in the amplitude A, and up to minute differences in the numerical values of the remainder ∆A,B. The induced changes in the numerics of the cross section and of the dipole polarizabilities are far below the uncertainties generated by the (not precisely known) values of the low energy constants. (4) The values for the dipole and quadrupole polarizabilities are presented in Table 1 and confronted with recent evaluations from data on γγ → π 0π 0. There is reasonable agreement for − the dipole polarizabilities. As for the quadrupole ones, the combination (α2 β2)π0 —related to 44 J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54 the helicity non-flip amplitude—agrees with [13] within the uncertainties quoted. On the other + hand, the sum (α2 β2)π0 —related to the helicity flip amplitude—differs in sign from the one in Ref. [13]. We have shown why this does not contradict the predictions of ChPT: this quantity is a pure two-loop effect, with no contributions from resonance exchange at this order, and one expects from order p8 (three loops) substantial corrections to the leading order result. We have indeed identified ω-exchange as an important contribution at this order. The relevant contributions from ω-exchange to the polarizabilities are collected in column 3 of the Table 1. They may be taken as an indication of the size of terms at order p8. (5) It would be instructive to improve the estimates for the LECs c29,...,c34 in the sense that in these estimates, the constraints from the asymptotics of QCD [21,29] should be respected.

The corresponding calculation of the charged pion polarizabilities is in progress [17].

Acknowledgements

This work was completed while M.A.I. visited the University of Bern. It is a pleasure to thank G. Colangelo, J. Bijnens and B. Moussallam for useful discussions, and S. Bellucci for useful remarks concerning the manuscript. This work was supported by the Swiss National Sci- ence Foundation, by RTN, BBW-Contract No. 01.0357, and EC-Contract HPRN-CT2002-00311 (EURIDICE). M.A.I. also appreciates the partial support by the Russian Fund of Basic Research under Grant No. 04-02-17370. Also, partial support by the Academy of Finland, grant 54038, is acknowledged.

Appendix A. Notation

In order to simplify the expressions, we set the pion mass equal to one in all appendices,

Mπ = 1. (A.1) We use the following notation for d-dimensional one-loop and two-loop integrals, dd l dd l dd l ···= (···), · · · = 1 2 (···). (A.2) (2π)d i (2π)d i (2π)d i In particular, 1 = F [z],n 1, [z − l2]n n + − (n − 2 − w) d F [z]=zw 2 nC(w) ,w= − 2. (A.3) n (n) 2 The measures in the integration over Feynman parameters are deﬁned by 2 3 d x = dx2 dx3,dx = dx1 dx2 dx3. (A.4) In dispersion relations, we use the d-dimensional measure C(w)(3/2) σ w [dσ]= − 1 βdσ, (3/2 + w) 4 1 C(w) = ,β= 1 − 4/σ. (A.5) (4π)2+w J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54 45

Appendix B. Loop-integrals

(1) The loop-integral G(s)¯ is

1 1 2 dx G(s)¯ =− 1 + ln 1 − sx(1 − x) . (B.1) 16π 2 s x 0 G¯ is analytic in the complex s-plane, cut along the positive real axis for Re s 4. At small s, ∞ 1 (n!)2 G(s)¯ = sn . (B.2) 16π 2 (n + 1)(2n + 1)! n=1 The absorptive part is 1 1 + σ Im G(s)¯ = ln ,s>4, 8sπ 1 − σ σ = 1 − 4/s, (B.3) and   + 1 1−σ + 2  1 ln + iπ , 4 s, s 1 σ − 2 ¯ = − 4 2 s 1/2 16π G(s)  1 arctg − , 0 s 4, (B.4)  s 4 s + 1 2 σ −1 1 s ln σ +1 ,s0. In the text we also need ¯ G(s)¯ = G(s)¯ − sG¯ (0). (B.5) (2) The loop-integral J(s)¯ is

1 1 J(s)¯ =− dxln 1 − sx(1 − x) . (B.6) 16π 2 0 J¯ is analytic in the complex s-plane, cut along the positive real axis for Re s 4. At small s, ∞ 1 (n!)2 J(s)¯ = sn . (B.7) 16π 2 n(2n + 1)! n=1 The absorptive part is σ Im J(s)¯ = ,s>4. (B.8) 16π Explicitly,   1−σ  σ ln + + iπ + 2, 4 s,  1 σ 2 ¯ − 1/2 1/2 16π J(s)= 2 − 2 4 s arctg s , 0 s 4, (B.9)  s 4−s  σ −1 + σ ln σ +1 2,s0. 46 J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54

In the text we also need ¯ J(s)¯ = J(s)¯ − sJ¯ (0). (B.10) (3) The loop-function H¯ is deﬁned in terms of G¯ and J¯, H(s)¯ = (s − 10)J(s)¯ + 6G(s).¯ (B.11)

Appendix C. The acnode

The evaluation of the two-loop vertex and box diagrams (5) and (6) in Fig. 3 is described in Ref. [11]. It is based on a d-dimensional dispersive representation of the ﬁsh-type diagram,

1 1 J p2 = = C(w)(−w) dx 1 − p2x(1 − x) w [1 − l2][1 − (l + p)2] 0 ∞ [dσ] = , −1.5 <ω<0, (C.1) σ − p2 4 where the measure [dσ] is given in Appendix A. This representation allows one to reduce the two-loop vertex and box integrals to the one-loop case, with a final integration over the dispersion parameter σ . The ultraviolet divergences can be extracted by invoking recursion relations [11]. While the acnode was treated in a different manner in [11], we evaluate it here analogously to the vertex and box diagrams just mentioned. This results in considerable simplifications in the numerical programs. For this purpose, we invoke a dispersion relation for the function I(m,n; s) defined by

1 I(m,n; s) = dx 1 − sx(1 − x) m x(1 − x) n, −1 4,x−

The integrand is symmetric around x = 1/2, so we may restrict the integration from x− to 1√/2 in the evaluation of the absorptive part of I(m,n; s). The substitution x = (1/2)(1 − (1 − 4/s)(1 − u)) generates a hypergeometric function, and we arrive at the dispersion relation ∞ dσρ(m,n; σ) I(m,n; s) = , σ − s 4 (3/2)δ1/2+m 1 3 3 ρ(m,n; σ)= F , + m + n; + m;−δ , 4n(3/2 + m)(−m) 2 1 2 2 2 σ δ = − 1. (C.4) 4 J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54 47

In particular, we ﬁnd (3/2)βδm ρ(m, ; σ)= , 0 + − (3/2 m)( m) (3/2)βδm 2 ρ(m, ; σ)= + m + , 1 + − 1 4(5/2 m)( m) σ (3/2)βδm 4(1 + m) 12 ρ(m,2; σ)= 2 + 3m + m2 + + , (C.5) 16(7/2 + m)(−m) σ σ 2 with β given by Eq. (A.5). The dispersion relation (C.4) allows us to evaluate the integral that occurs in the evaluation of the acnode diagram (1) in Fig. 5, 4lµlνV V µν = 1 2 L R AN , D1D2D3D4D5 2 VL = (l2 + q2 − l1) − 1, 2 VR = (l1 + q1 − l2) − 1, = − 2 = − + 2 D1 1 l1 ,D2 1 (l1 q1) , = − 2 = − + 2 D3 1 l2 ,D4 1 (l2 q2) , 2 D5 = 1 − (l1 − l2 − p1 + q1) . (C.6)

We combine 1/(D1D2) and 1/(D3D4) by using x1 and x2 as Feynman parameters, respectively. By shifting l1 and l2 and again dropping terms that vanish upon contraction with the polarization vectors, one obtains

1 ν µ l l PN (li,pi,qi) Aµν = d2x 2 · 1 , N [ − 2]2 [ − 2]2[ − − 2] 1 l2 1 l1 1 (l1 r) 0 r = l2 − q, q = x1q1 + x2q2 − p1. (C.7)

Here, PN (li,pi,qi) is a polynomial in the momenta indicated. The integration over l1 is performed by using the dispersion relation (C.4) with s = r2. Then we proceed in a manner which is similar to the case of the box diagram described in Ref. [11]. The ﬁnal expression can be written as a combination of the integrals

∞ 1 1−i σ − A(i, k, m, n) = [dσ] d3x − 1 σ k(1 − x )mF [z ], 4 3 n acn 4 0 = 2 + − − − zacn x3 (1 x3)σ x3(1 x3)a, a = x1x2s + x1(t − 1) + x2(u − 1), (C.8) where Fn[z] is deﬁned in (A.3). The integrals A(i, k, m, n) are convergent at w = 0 in the case i = 1,k= 0,m 1,n 4, i = 1,k 1,m 1,n 3, i = 2,k 0,m 0,n 3. 48 J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54

To single out the divergent part in the remaining integrals, we invoke recursion relations in the following manner. We perform a partial integration in x3, and use (1 − x )m+1 dx (1 − x )m =− 3 . 3 3 m + 1

Then we express (1 − x3)σ through zacn and obtain (m + 3 − n + ω)A(i,k,m,n) = Div(i,k,n)− n A(i, k, m, n + 1) − (1 + a)A(i,k,m + 2,n+ 1) , (C.9) where 2 − − + C (w)(3/2) (n − 2 − ω) Div(i,k,n)= 43 n k ω (3/2 + ω) (n) × B(5/2 − i + ω,n + k − 4 + i − 2ω), (x)(y) B(x,y) = . (C.10) (x + y) The divergences in Div(i,k,n)can be worked out straightforwardly. The integral A(1, 0, 0, 3) must be considered separately. We write

∞ 1 3 A(1, 0, 0, 3) = D(0, 3) + [dσ] d x F3[zacn]−F3[y] , 4 0 = 2 + − y x3 (1 x3)σ. (C.11) The divergent quantity D(0, 3) is worked out in Appendix C.1 of Ref. [11], whereas the integral on the right-hand side is convergent at d = 4. This concludes our discussion of the acnode integral (C.7).

Appendix D. The quantities ∆A and ∆B

Here we display the expressions for the quantities ∆A(B) in Eq. (5.2). 1 689 4π 2 15043 ∆ (s,t,u)= − + s A 4 (4πFπ ) 162 9 64800 1 1 + F acn(s,t,u)+ F ver(s) + F box(s,t,u) , (D.1) (4πF )4 288 A A A π 1 8329 2987 2π 2 1 ∆ (s,t,u)= + − B 4 (4πFπ ) 43200 1350 9 s 1 1 + F acn(s,t,u)+ F ver(s) + F box(s,t,u) , 4 B B B (D.2) (4πFπ ) 288 where

∞ 1 (0) (1) (2) P ; P ; 1 P ; F acn = dσβ d3x I acn + I acn + I acn , I 2 y σ zacn zacn 4 0 J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54 49

∞ 1 ver = dσβ 2 PI;ver FI d x , σ zver 4 0 ∞ 1 2 box = dσβ 3 (n) (n) + (n) (n) = FI d x PI;box+ Dbox+ PI;box− Dbox− ,I A,B, (D.3) σ = 4 0 n 1 and (n) = 1 ± 1 Dbox± n n . zbox;t zbox;u

Here PI are polynomials in s,ν = t − u and in xi . Their explicit expressions are given in Appen- dix E. The quantity zacn is displayed in Eq. (C.8), y is given in (C.11), and = − + 2 = − − zver σ(1 x3) x3 y2,y2 1 sx2(1 x2), z ; = B − A x , box t t t 1 At = x2x3 s(1 − x2)x3 + (1 − t)(1 − x3) ,

Bt = At + zver,

zbox;u = zbox;t |t→u. (D.4) The acnode integrals are easy to evaluate numerically in the physical region for the reaction γγ → ππ, because branch points occur at t = 4,u= 4 only. On the other hand, the vertex and box integrals contain branch points at s = 4. In order to evaluate these integrals at s 4, we invoke dispersion relations in the manner described in [11].

Appendix E. The polynomials PA and PB

Here, we display the polynomials PA(B) that occur in the expressions ∆A(B) in Appendix D. We use the abbreviations

x+ = x1 + x2 − 2x1x2,x− = x1 − x2,

x123 = (1 + x3 − 2x2x3)(1 − x3 + 2x1x2x3). (E.1)

E.1. The polynomials PA

(0) =− − − PA;acn 192x3(1 x3)(sx+ νx−), (1) P ; =−6sνx−(1 − x3) A acn 3 4 3 4 2 3 4 × 1 + 8x + 3x − 4x+ x + 3x + 2x + 2x 1 + 2x − 3x 3 3 3 3 3 − 3 3 − 6s2(1 − x ) x2 1 + 2x + 14x3 + 7x4 + 6x2 x4 3 + 3 3 3 − 3 2 3 2 4 2 − x+ 1 + 2x + 4x 2 + x + 3x 1 + 2x − 3 − 3 − 2 2 3 4 4 + 6ν x (1 − x ) 1 − 2x + 2x + 5x − 12x+x − 3 3 3 3 3 + 12s −2x2 (1 − x )3 1 + 2x + 3x2 + 3 3 3 + + − 2 − 3 + 4 x3 2 33x3 19x3 19x3 15x3 50 J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54 2 3 4 5 + x+ −4 + 2x − 63x + 25x + 51x − 35x 3 3 3 3 3 2 2 3 4 + 2x−x 2 + 9x − 7x + 5x − 3x 3 3 3 3 3 2 3 4 3 4 + 12νx−(1 − x ) 4 − 2x − 5x − 4x + 19x + 2x+ 1 + 8x − 21x 3 3 3 3 3 3 3 − 48x 4 + 4x + 6x2 − 19x3 + 10x4 3 3 3 3 3 + − − 2 + 3 − 4 x+ 2 3x3 19x3 41x3 21x3 , (2) 2 2 3 4 P ; = 2sνx−(1 − x3) −11 − 8x3 − 8x − 24x − 9x A acn 3 3 3 3 4 2 2 3 4 + 12x+ 4 + x + 3x + 2x − 2x 11 + 8x + 8x + 6x − 9x 3 3 3 − 3 3 3 3 − 2s2(1 − x )2 x2 35 − 18x + 34x3 + 21x4 + 18x2 x4 3 + 3 3 3 − 3 3 4 2 3 4 − x+ 11 + 16x + 9x + 2x 11 − 2x + 9x 3 3 − 3 3 2 2 2 3 4 + 2ν x (1 − x ) −13 + 18x − 2x + (15 − 36x+)x − 3 3 3 3 + 4s(1 − x ) x 22 + 11x + 83x2 − 101x3 + 45x4 3 3 3 3 3 3 − 2x2 (1 − x )2 11 1 + x + x2 − 9x3 + 3 3 3 3 2 3 4 5 + x+ 60 − 178x + 43x − 173x + 233x − 105x 3 3 3 3 3 + 2x2 (2 − x )x 11 1 + x + x2 + 9x3 − 3 3 3 3 3 2 2 3 4 + 4νx−(1 − x ) −60 + 74x + 9x − 56x + 57x 3 3 3 3 3 3 4 + x+ 22 + 104x − 126x 3 3 + 16x (1 − x ) 60 + 18x − 108x2 + 93x3 − 30x4 3 3 3 3 3 3 2 3 4 + x+ −22 − 83x + 205x − 187x + 63x , 3 3 3 3 2 4 2 2 2 P ; = 128sx (1 − 2x )x 3 − 2x (2 − 15x )x − 18x − 15x x A ver 2 2 3 2 3 3 3 2 3 − 2 2 − − + 2 6 32s x2 (1 2x2) 30 42x2 7x2 x3 , (1) =− 2 − + + − PA;box+ 96sx2x3 6 9x3 20x2x3 22x1x2x3 − 2 − 2 + 2 − 2 2 + 2 2 − 2 2 2 2x3 40x2x3 6x1x2x3 4x2 x3 52x1x2 x3 8x1 x2 x3 − 3 + 3 + 3 − 2 3 − 2 3 − 2 2 3 15x3 58x2x3 14x1x2x3 12x2 x3 72x1x2 x3 2x1 x2 x3 + 8x x3x3 + 4x2x3x3 + 20x4 − 48x x4 − 36x x x4 + 20x2x4 1 2 3 1 2 3 3 2 3 1 2 3 2 3 + 76x x2x4 + 10x2x2x4 − 24x x3x4 − 24x2x3x4 + 24x2x4x4 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 − 2 2 4 + − − + 2 + 16s x2 x3 12 12x1 12x2 24x1x2 12x1 x2 30x1x3 − − + 2 + 2 + 2 24x2x3 90x1x2x3 27x1 x2x3 30x2 x3 66x1x2 x3 − 2 2 − 3 2 − 2 + 2 + 2 − 2 24x1 x2 x3 10x1 x2 x3 24x3 6x1x3 66x2x3 18x1x2x3 + 45x2x x2 − 48x2x2 + 48x x2x2 − 120x2x2x2 − 28x3x2x2 1 2 3 2 3 1 2 3 1 2 3 1 2 3 − 24x x3x2 + 36x2x3x2 + 56x3x3x2 1 2 3 1 2 3 1 2 3 + 2 3 4 − + − 2 + + 2 + 48ν x2 x3 4 8x1 8x1 2x1x3 7x1 x3 10x2x3 − + 2 − 3 + 2 − 2 + 2 2 30x1x2x3 30x1 x2x3 10x1 x2x3 6x3 14x1x3 3x1 x3 J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54 51

− 12x x2 + 12x x x2 + 12x2x x2 − 12x3x x2 + 24x x2x2 2 3 1 2 3 1 2 3 1 2 3 1 2 3 − 48x2x2x2 + 24x3x2x2 1 2 3 1 2 3 + 2 − + + − + 2 − 2 384x2x3 6 9x3 19x2x3 19x1x2x3 7x3 50x2x3 − 2x x x2 + 52x x2x2 − 15x3 + 44x x3 + 16x x x3 1 2 3 1 2 3 3 2 3 1 2 3 − 2 3 + 4 − 4 − 4 + 2 4 60x1x2 x3 8x3 16x2x3 16x1x2x3 32x1x2 x3 , (2) 2 2 3 2 6 P ; =−24sν (1 − x1) x 1 + x2 + x1x2 − 2x1x x x123 A box+ 2 2 3 + 96sx x4x −7x − 7x x + 14x x2 + 3x + 6x x 2 3 123 2 1 2 1 2 3 2 3 + 6x x x − 12x x2x − 3x2 − 2x x2 − 2x x x2 + 4x x2x2 1 2 3 1 2 3 3 2 3 1 2 3 1 2 3 − 2 2 4 − − − + − 2 + + 48s x2 x3 x123 2 2x1 x2 6x1x2 x1 x2 3x3 3x1x3 − 6x x x − 4x2 − 4x x2 + 2x x2 + 8x x x2 1 2 3 3 1 3 2 3 1 2 3 + 2x2x x2 − 4x x2x2 − 4x2x2x2 + 4x2x3x2 1 2 3 1 2 3 1 2 3 1 2 3 − 3 3 + − + − − 2 6 24s x2 (1 x1 2x1x2) 1 x1 x2 x1 x2 x3 x123 2 2 3 4 2 4 − 48ν (1 − x1) x (1 − x3)x (1 + 4x3)x123 + 2x2(1 − x3) x x123, 2 3 3 (1) =− 2 4 − + + − 2 PA;box− 32sνx2 x3 6 6x1 12x1x2 18x1 x2 − + + 2 − 2 + 2 2 15x1x3 12x2x3 33x1 x2x3 30x1x2 x3 21x1 x2 x3 − 10x3x2x + 12x2 + 3x x2 − 24x x2 − 30x x x2 1 2 3 3 1 3 2 3 1 2 3 + 18x2x x2 + 6x2x2 + 54x x2x2 − 48x2x2x2 + 4x3x2x2 1 2 3 2 3 1 2 3 1 2 3 1 2 3 − 2 3 − + + − + 96νx2 x3 9 11x1 12x3 14x1x3 24x2x3 − + 2 + 2 − 2 − 2 56x1x2x3 28x1 x2x3 15x3 27x1x3 58x2x3 + 2 + 2 2 + 2 2 − 2 2 2 84x1x2x3 8x1 x2x3 60x1x2 x3 80x1 x2 x3 − 16x3 + 28x x3 + 32x x3 − 20x x x3 − 42x2x x3 3 1 3 2 3 1 2 3 1 2 3 2 3 2 2 3 − 64x1x x + 84x x x , 2 3 1 2 3 (2) 2 4 P ; = 48sν(1 − x1)x x x123 −2 + 3x3 + 3x2x3 + 3x1x2x3 A box− 2 3 − 6x x2x − 4x2 − 2x x2 − 2x x x2 + 4x x2x2 1 2 3 3 2 3 1 2 3 1 2 3 + 2 − 2 3 − − 6 24s ν 1 x1 x2 (2 x2 x1x2)x3 x123 + − 2 − 4 − + 3 − 3 4 6 ν(1 x1)x2 (1 x3)x3 (1 2x3)x123 24ν (1 x1) x2 x3 x123.

E.2. The polynomials PB (0) ν P = 96x (1 − x ) x− − x+ , B;acn 3 3 s (1) 6ν 2 3 4 5 P = x− 4 − 2x + 21x − 19x + 3x + 5x B;acn s 3 3 3 3 3 2 3ν 2 3 4 − x−(1 − x ) 1 + 2x + 2x + x s 3 3 3 3 52 J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54

− 24 − 2 − + 2 + − x3 2 x3 2 8x3 5x3 3s(1 x3) s 2 3 3 4 2 4 × x (1 − x ) (1 + x ) − x+ 1 + 8x + 3x + 6x x + 3 3 3 3 − 3 2 2 3 + 6 x+(1 − x ) −4 − 2x − 3x + 13x 3 3 3 3 + − − + 2 + 3 − 4 x3 2 33x3 19x3 19x3 15x3 , (2) 2ν 2 3 4 5 P =− x−(1 − x ) −34 + 152x − 43x + 21x − 27x + 15x B;acn s 3 3 3 3 3 3 2 ν 2 2 3 4 − x−(1 − x ) 35 − 18x − 2x + 3x s 3 3 3 3 + 8 − − + 2 − 3 + 4 x3(1 x3) 34 67x3 95x3 56x3 15x3 s − s(1 − x )2 x2 (1 − x )2 13 + 8x + 3x2 3 + 3 3 3 3 4 2 4 + x+ 11 + 16x + 9x − 18x x 3 3 − 3 2 3 4 4 + νx−(1 − x ) 11 + 16x + 9x + 12x+ 4 − 3x − x 3 3 3 3 3 + 2(1 − x ) −x 22 + 11x + 83x2 − 101x3 + 45x4 3 3 3 3 3 3 2 3 4 5 + x+ 34 − 108x + 65x + 73x − 103x + 39x , 3 3 3 3 3 = 2 − 5 2 − − − + PB;ver 16sx2 (1 2x2)x3 33x2 x3 6(4 5x3) 2x2(4 9x3) − 2 − − − 4 1536x2 (1 2x2)(1 x3)(1 2x3)x3 , 24ν2 P (1) =− x3(1 − x )x4 4 + 8x − 10x − 2x x − 3x2x B;box+ s 2 3 3 1 3 1 3 1 3 − 1152 − 2 2 − + 2 x2(1 x3) x3 1 2x3 2x3 s − 2 4 + − − − + 24sx2 x3 4 12x1 4x2 12x3 14x1x3 10x2x3 − 14x x x + 17x2x x − 14x2x2x + 8x2 − 2x x2 − 6x x2 1 2 3 1 2 3 1 2 3 3 1 3 2 3 + 14x x x2 − 3x2x x2 + 4x x2x2 − 12x2x2x2 + 12x2x3x2 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 − 48x (1 − x )x2 −6 + 19x + 16x x x − 27x2 − 2x x2 2 3 3 3 1 2 3 3 2 3 − 2 − 2 2 + 3 − 3 + 2 3 18x1x2x3 28x1x2 x3 20x3 32x1x2x3 64x1x2 x3 ,

2 (2) =−24ν − 2 3 − 3 4 + + 1 − 4 4 PB;box+ (1 x1) x2 (1 x3) x3 (1 4x3) x2(1 x3) x3 s s − 2 − 2 4 − − − + − 2 24sx2 (1 x3) x3 2 2x1 x2 6x1x2 x1 x2 + 3x + 3x x − 6x x x − 4x2 − 4x x2 + 2x x2 3 1 3 1 2 3 3 1 3 2 3 + 8x x x2 + 2x2x x2 − 4x x2x2 − 4x2x2x2 + 4x2x3x2 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 − 12s2x3(1 + x − 2 x x ) 1 + x − x − x2x (1 − x )2x6 2 1 1 2 1 2 1 2 3 3 + 12ν2(1 − x )2x3 −1 − x − x x + 2x x2 (1 − x )2x6 1 2 2 1 2 1 2 3 3 + 48x (1 − x )2x4 −7x − 7x x + 14x x2 + 3x 2 3 3 2 1 2 1 2 3 + + − 2 − 2 − 2 − 2 + 2 2 6x2x3 6x1x2x3 12x1x2 x3 3x3 2x2x3 2x1x2x3 4x1x2 x3 , J. Gasser et al. / Nuclear Physics B 728 (2005) 31–54 53

(1) = 48ν 2 − 2 3 + − − PB;box− x2 (1 x3) x3 (5 11x1 28x3 4x1x3) s + 2 4 + + − − − 48νx2 x3 2 2x1 8x1x2 6x3 3x1x3 2x2x3 − 4x x x + 3x2x x − 12x x2x − 3x2x2x + 4x2 1 2 3 1 2 3 1 2 3 1 2 3 3 + 2 + 2 − 2 + 2 2 x1x3 2x2x3 10x1x2x3 18x1x2 x3 , 3 (2) = 144ν − 2 − 3 4 − + 12ν − 3 4 − 2 6 PB;box− (1 x1)x2 (1 x3) x3 (1 2x3) (1 x1) x2 (1 x3) x3 s s − 12sν 1 − x2 x3(−2 + x + x x )(1 − x )2x6 1 2 2 1 2 3 3 + 24ν(1 − x )x2(1 − x )2x4 −2 + 3x + 3x x + 3x x x 1 2 3 3 3 2 3 1 2 3 − 2 − 2 − 2 − 2 + 2 2 6x1x2 x3 4x3 2x2x3 2x1x2x3 4x1x2 x3 .

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A model for leptogenesis at the TeV scale

Asmaa Abada a, Habib Aissaoui a,b,MartaLosadac

a Laboratoire de Physique Théorique, Université de Paris XI, Bâtiment 210, 91405 Orsay cedex, France b Laboratoire de Physique Mathématique et Physique Subatomique, Mentouri University, Constantine 25000, Algeria c Centro de Investigaciones, Universidad Antonio Nariño, Cll. 58A No. 37-94, Santa Fe de Bogotá, Colombia Received 5 October 2004; accepted 3 August 2005 Available online 24 August 2005

Abstract We consider the mechanism of thermal leptogenesis at the TeV scale in the context of an extension of the Standard Model with 4 generations and the inclusion of four right-handed Majorana neutrinos. We obtain a value for the baryon asymmetry of the Universe in accordance with observations by solving the full set of coupled Boltzmann equations for this model.  2005 Elsevier B.V. All rights reserved.

PACS: 12.60.Jv; 14.60.Pq; 11.30.Fs

Keywords: Leptogenesis; Neutrino physics; Baryon asymmetry of the universe

1. Introduction

The attractive mechanism of thermal leptogenesis has been thoroughly investigated in recent years. Most of the focus has been on the extended model of the Standard Model (SM) with 3 right-handed (RH) neutrinos [1]. This simple model which can explain the baryon asymmetry of the Universe (BAU) and provide appropriate values for neutrino masses and mixing angles is based on the out-of-equilibrium decay of a RH neutrino, in a CP violating way, such that an asymmetry in the leptonic decay products is produced. The leptonic asymmetry is then converted into a baryonic asymmetry due to (B + L)-violating sphaleron interactions which are in equilibrium above the electroweak breaking scale.

E-mail addresses: [email protected] (A. Abada), [email protected] (H. Aissaoui), [email protected] (M. Losada).

From the experimental point of view the recent best fit values of the ratio of the baryon to photon density η = nB /nγ , = +0.4 × −10 η 6.5−0.3 10 , (1) is given by WMAP [2]. The typical scale at which the standard scenario described above occurs corresponds to 1010–1015 GeV, and the induced neutrino masses arise via the see-saw mechanism [3–5] with a common interaction being at the origin of the asymmetry and mass generation. Recently, detailed calculations have been performed solving the corresponding Boltzmann equations of the system taking into account the different interactions which contribute both to the production and potential washout of the leptonic asymmetry. The implications of these analyses have provided strong constraints on the values of the light left-handed neutrino masses ∼ 8 i mi 0.2eV[6–8] and a lower bound on the mass of the lightest RH neutrino M 10 GeV [6,9,10]. The more recent work of Ref. [11] has included the effect of scattering with gauge bosons which had been neglected before and in Ref. [12] a comprehensive calculation including effects from renormalization group (RG) running, finite temperature background effects, other possible decays producing a leptonic asymmetry (decay of a scalar particle), a correction to the appropriate subtraction that must be performed for scattering via N-exchange was presented. A thorough recalculation of this last issue was done in Ref. [13], coinciding with Ref. [12] that the washout contribution from this type of diagram had been overestimated. The less standard scenario of leptogenesis occurring at the TeV scale has been investigated, for example, in Refs. [14–21]. In particular, in Ref. [14] a TeV scale model which is a simple extension of the Fukugita–Yanagida (FY) model including one more generation to the SM and four gauge singlets was discussed and estimates of the produced CP asymmetry and final baryon asymmetry were obtained. Both supersymmetric and non-supersymmetric versions of the model were analyzed. In this paper we focus on this model of Ref. [14] and solve the corresponding Boltzmann equations (BE) in order to reliably obtain the final values of the asymmetry and determine the region of parameter space which is available for the model. We dedicate Section 2 to a careful discussion of the way we set up our calculations of the Boltzmann equations making emphasis on a few subtle points. In Section 3, we recall the main features of the model and establish our notation. Section 4 presents the Boltzmann equations valid for this model. Section 5 is devoted to our results and conclusions.

2. Preliminary considerations

In Ref. [22] it was pointed out that the relevant object that should be studied in thermal leptogenesis is the density matrix ρ [23]. In the FY model ρ is a (3 × 3) matrix with all entries different from zero for temperatures greater than 1012 GeV. However, as the temperature decreases and each of the different interactions defined by a specific charged Yukawa coupling enters equilibrium the corresponding off-diagonal elements of the density matrix go to zero. In our model at the TeV scale all of the charged Yukawa couplings are in equilibrium and thus ρ will be a diagonal matrix. The matrix ρ is normalized such that Tr ρ = Y ≡ (Y − Y ¯ ), (2) Lα α α α α where the sum over α indicates a sum over flavour. A. Abada et al. / Nuclear Physics B 728 (2005) 55–66 57

Another important issue is the relationship between the produced lepton asymmetry and the induced B − L asymmetry which is conserved by sphaleron interactions. Again a careful consideration of the temperatures and which interactions are in equilibrium changes the equation relating these asymmetries, see Ref. [22]. In general, the procedure that is employed is through the establishment of equations for the chemical potentials reflecting which interactions are in equilibrium. Thus the chemical equilibrium equations allow us to write the asymmetries in number densities in terms of some specific asymmetry, say YLα . For the Fukugita–Yanagida model then generically there would be a (3 × 3) matrix Aαβ of the form [22] = YB/3−Lα Aαβ YLβ , (3) and the numerical values for the elements of A are determined from the equations for the chemical potentials. If all interactions related to charged Yukawa couplings are in equilibrium, as well as sphaleron interactions, the equation for the case of N generations simplifies to Y 4 8 1 B − Y = − + Y − 3Y , (4) N Lα 3N 3 4N + 2 Lβ Lα β or, adding over all generations, 22N + 13 Y − =− Y . (5) B L 6N + 3 Lα α Some comments are in order. In the standard scenario of the FY model with three RH neutrinos, where the production of the asymmetry occurs at some high scale for which not all charged Yukawa couplings are in equilibrium, one should use Eq. (3) to relate the asymmetries. The point is that usually what is solved in the system of coupled Boltzmann equations, under certain as- = sumptions, is an equation for YL α YLα , see below for a discussion on this point, and then the asymmetry in B − L is calculated using Eq. (5). From our previous discussion we see that it is not quite correct to do this unless some approximations are done concerning the different YLα . For our model at the TeV scale,1 all charged Yukawa couplings are in equilibrium and it would be correct to use Eqs. (4) and (5). Another feature of our model is that the Yukawa coupling of the heavy LH neutrino is of the order of 1 to ensure that the masses of the fourth generation leptons are heavy enough, and also to give an enhancement to the value of the CP asymmetry produced in the model. However, then the scattering interaction which is associated to the Yukawa coupling of the heavy LH neutrino will also be in equilibrium in our model. This changes the chemical equilibrium equations such that the relationship between B − L (B) and L is modified to be: 13 8 (13 − 3N) 22N + 13 Y − =− + Y =− Y . (6) B L 3 3 (12N − 2) Lα 6N − 1 Lα α=e,µ,τ α=e,µ,τ Thus, as was estimated in Ref. [14] there is no net asymmetry produced in the heavy left- handed lepton. Consequently, we can put YLσ to zero and make the usual assumptions about the other YLα when constructing the Boltzmann equation for YL provided we use Eq. (6) to relate the produced leptonic asymmetry to the final net B − L asymmetry. Below, and from now on, YL denotes the sum over the asymmetries of only the three light leptonic flavours.

1 In this model, we denote by σ the ﬂavour of the fourth leptonic generation. 58 A. Abada et al. / Nuclear Physics B 728 (2005) 55–66

In Ref. [12] the effect of renormalization group (RG) running for masses and couplings was included. In particular, one of the important effects is that the diagrams with L = 1 proportional to gauge couplings can become sizeable compared to those proportional to the top Yukawa coupling when considering values for these couplings at the high scale at which the lepton asymmetry is being produced. In our model at the TeV scale, we have checked that this is a very small effect and we will not include RG effects in our analysis. We also have not included the L = 1 proportional to gauge couplings as we are not in the resonant leptogenesis case of Ref. [11]. Also in Ref. [12] thermal background effects have been included into the calculation, it was pointed out that numerically these effects are particularly relevant when correcting the propagator of the Higgs scalar field. We include only this thermal correction into our calculation. Another important point that should be mentioned, which is crucial for our model, is the fact that the actual decay temperature at which the lepton asymmetry is produced is just above the electroweak scale; thus one must be careful and check that for our choice of parameters the sphaleron interactions are still in equilibrium. Although, in the SM there is no first order phase transition, just a crossover, and this is not modified for our model with four generations, sphaleron interactions are switched off for temperatures below ∼100 GeV [24]. A careful calculation of the chemical equilibrium equations in the region close to the electroweak phase transition was performed in [25].

3. The model

The relevant part of the Lagrangian of the model we consider is given by ¯ MNi ¯ c ν ¯ L = LSM + ψR i/∂ψR − ψ ψR + h.c. − λ LαψR φ + h.c. , (7) i i 2 Ri i iα i where ψRi are two-component spinors describing the right-handed neutrinos and we deﬁne a Majorana 4-component spinor, N = ψ +ψc . Our index i runs from 1 to 4, and α = e,µ,τ,σ. i Ri Ri The σ component of Lα corresponds to a left-handed lepton doublet which must satisfy the ν LEP constraints from the Z-width on a fourth left-handed neutrino [26].Theλiα are Yukawa couplings and the ﬁeld φ is the SM Higgs boson doublet whose vacuum expectation value is denoted by v. We work in the basis in which the mass matrix for the right-handed neutrinos M is diagonal and real,

M = diag(M1,M2,M3,M4), (8)

ν and deﬁne mD = λ v. To leading order,2 the induced see-saw neutrino mass matrix for the left-handed neutrinos is given by

ν† −1 ν 2 mν = λ M λ v , (9) which is diagonalized by the well-known PMNS matrix.

2 Given the fact that in our model, at least one of the RH neutrino masses take values ∼TeV, one has to go to next order in the general see-saw formula. However, these corrections are small and not relevant for our purpose. A. Abada et al. / Nuclear Physics B 728 (2005) 55–66 59

We consider the out-of-equilibrium decay of the lightest of the gauge singlets Nj , which we take to be N1. The decay rate at tree-level is given by

† 4 ∗ ˜ (4G) (λ λ)jj (λαj λαj ) mj M1Mj ΓN = Mj = Mj = , (10) j 8π 8π 8πv2 α=1 where λ ≡ λν and we have defined3 4 (λ∗ λ ) ˜ (4G) = αj αj 2 mj v . (11) M1 α=1 Γ To ensure an out-of-equilibrium decay of N it is necessary that N1 1, where H is 1 H(T=M1) the Hubble expansion rate at T = M1. This condition will impose an upper bound on the usual effective mass parameter m˜ 1 which is defined in, for example, [27]. The CP asymmetry calculated from the interference of the tree diagrams with the one-loop diagrams (self-energy and vertex corrections) is [28] M2 M2 1 1 ν† 2 j j i = Im λ λν f + g , (12) (8π) [ † ] ij M2 M2 λνλν ii j i i where √ √ 1 + x x f(x)= x 1 − (1 + x)ln ,g(x)= . (13) x 1 − x In Ref. [14] it was shown that the upper bound of the CP asymmetry produced in the decay of the lightest right-handed neutrino N1 is 3 M m | | 1 4 , (14) 1 8π v2 where m4 denotes the largest eigenvalue of the left-handed neutrino mass matrix mν .Dueto experimental constraints we require that m4 > 45 GeV, which implies that the bound on 1 is irrelevant. In general, there will be regions of parameter space in which the produced CP asymmetry can be very large, thus the final allowed region of parameter space could include areas in which the washout processes can be very large as well. In Ref. [14] possible textures for the λν matrix were given to illustrate how to obtain masses consistent with low energy data and the appropriate amount of CP asymmetry from the decay of N1.

4. Boltzmann equations

We will now write the Boltzmann equations for the RH neutrino abundance and the lepton densities. With all the above arguments in consideration, the main processes we consider in the thermal bath of the early universe are decays, inverse decays of the RH neutrinos, and the lepton

3 To simplify our notation, we write the sums over α = 1–4 denoting the sums over all flavours e,µ,τ,σ,andsums ∗ (λ λ ) (3G) 3 αj αj 2 over α = 1 to 3 to denote sums over the three lightest flavours. For instance, m˜ = = v is the usual j α 1 M1 effective mass parameter m˜ 1 defined in the literature for j = 1. 60 A. Abada et al. / Nuclear Physics B 728 (2005) 55–66 number violation L = 1 and L = 2, Higgs and RH neutrinos exchange scattering processes, respectively. In our analysis we stick to the case where the asymmetry is due only to the decay of the 4 lightest RH neutrino N1. The first BE, which corresponds to the evolution of the abundance of = the lightest RH neutrino YN1 involving the decays, inverse decays and L 1 processes is given by dY z Y N1 =− N1 − + eq 1 (γD1 γS1 ), (15) dz sH(M1) YN = where z M1/T, s is the entropy density and γDj ,γSj are the interaction rates for the decay and L = 1 scattering contributions, respectively. The BE for the lepton asymmetry is given by dY z Y Y L =− N1 − + L 1γD1 eq 1 γW eq , (16) dz sH(M1) YN YL where 1 is the CP violation parameter given by Eq. (12) and γW , is a function of γDj and = γSj and L 2 interaction rate processes, called the washout factor which is responsible for eq the damping of the produced asymmetry. In Eqs. (15) and (16), Yi is the equilibrium number density of a particle i, which has a mass mi , given by 45 g m 2 m z Y eq(z) = i i z2K i , i 4 2 (17) 4π g∗ M1 M1 = = where gi is the internal degree of freedom of the particle (gNi 2, g 4) and g∗ 108. Explicit expressions of the interaction rates γDj , γSj and γW are given below.

The reaction density γDj is related to the tree level total decay rate of the RH neutrino Nj of Eq. (10) by

= eq K1(z) γDj nN ΓNj , (18) j K2(z) where Kn(z) are the K-type Bessel functions. We deﬁne the reaction density of any process a + b → c + d by ∞ M4 1 √ √ γ (i) = 1 dxσˆ (i)(x) xK xz , (19) 64π 4 z 1 + 2 2 (Ma Mb) /M1 where σˆ (j)(x) are the reduced cross sections for the different processes which contribute to the Boltzmann equations. For the L = 1 Higgs boson exchange processes, we have 4 2 ˜ (4G) 2 ∗ x − aj mj M1 x − aj σˆ (1) = 3α λ λ = 3α , (20) tj u αj αj x u v2 x α=1

4 We have checked that the asymmetry produced by the decay of the second lightest RH neutrino is very tiny in our model compared to the one produced by the decay of N1. A. Abada et al. / Nuclear Physics B 728 (2005) 55–66 61 4 x − a x − a + a a − a x − a + a ˆ (2) = ∗ j 2 j 2 h + j 2 h j h σtj 3αu λαj λαj ln x x − aj + ah x − aj ah α=1 (4G) m˜ M1 x − a x − 2a + 2a a − 2a x − a + a = α j j j h + j h j h , 3 u 2 ln (21) v x x − aj + ah x − aj ah where Tr(λ†λ ) m2 M 2 µ 2 α = u u t ,a= j ,a= , u 2 j h (22) 4π 4πv M1 M1 the infrared cutoff µ is chosen to be equal to 800 GeV due to phenomenological considerations. So we have = (1) + (2) γSj 2γtj 4γtj . (23) The expressions of reduced cross sections of L = 2 RH neutrino exchange processes, which washout a part of the asymmetry are given by 4 3 4 4 3 4 ˆ (1) = ∗ ∗ (1) + ∗ ∗ (1) σN λαj λαj λβj λβj Ajj Re λαnλαj λβnλβj Bnj , (24) α=1 β=1 j=1 α=1 β=1 n

Fig. 1. s and t channel L = 2 Majorana exchange diagrams.

There are several important comments to make at this point. First of all, notice that in Eqs. (10), (20) and (21),wehavesummedoverallstatesα. However, in Eqs. (24) and (25) terms which correspond to β = 4 (that is the σ flavour, see Fig. 1), do not contribute to Eq. (16) = because they are multiplied by YLσ 0. Indeed, as was mentioned before, the processes involving at the same time only this flavour in the initial and final state are in thermal equilibrium, and so there is no asymmetry in this flavour. Second, as in the case of FY model, one can parametrize ˜ (4G) = γDj and γSj by M1 and m1 (see Eqs. (10), (20), (21)). Third, the L 2 processes can be divided and treated in two separate regimes. The contribution in the temperature range from z = 1 ˜ (4G) ˜ (3G) ˜ to z 10 is proportional to m1 m1 instead of m1. For the range of temperature (z 1), ¯ 2 ≡ † the dominant contribution at leading order would have been proportional to m tr(mνmν), had thesummationontheindexβ gone from 1 to 4. In addition, we emphasize that the so-called RIS (real intermediate states) in the L = 2 interactions have to be carefully subtracted, to avoid double counting in the Boltzmann equations. In our calculation, we have followed Ref. [13] − 1 where the authors have shown how the appropriate ( 8 γDj in Eq. (31)) subtraction must be done.

5. Results and conclusions

5.1. Constraints on the fourth generation

Our model is based on the addition of a fourth doublet of leptons and quarks (in order to be anomaly-free) to the particle content of the Standard Model, and of four RH neutrino singlets. The masses and couplings of the singlets are constrained by the left-handed neutrino masses (from solar, atmospheric and WMAP data) and the right amount of CP asymmetry 1. Con- cerning the fourth generation of leptons and quarks, the direct constraint is that they must be heavier than MZ/2. There are stronger constraints from electroweak precision tests: for example, the lower bound on the charged lepton σ from LEP II is approximately 80 GeV [29].As is well known, the SM central value of the Higgs mass is lower than the direct lower bound set by LEP II [30] and the existence of a fourth generation is one possible way to increase the Higgs mass [31–33].TheZ-lineshape versus fourth generation masses have been extensively studied in, for example, Ref. [34], where we can see that the fourth left-handed neutrino is excluded at 95% CL if its mass is lower than 46.7 ± 0.2 GeV. In our analysis the left-handed neutrino masses are always constrained in order to ﬁt the data (solar + atmospheric + WMAP) as well as this latter bound for the 4th LH neutrino. There are other constraints on the fourth generation, the generalized CKM matrix elements, etc., which can be found in [26] and in Ref. [35], where present and future experimental searches at Tevatron and LHC are widely discussed. A. Abada et al. / Nuclear Physics B 728 (2005) 55–66 63

5.2. Numerical solutions of BE

In our numerical analysis we use one of the possible textures for the Yukawa matrix (λν )inthe four generation case which can produce the needed amount of CP asymmetry and fit the neutrino data [14]:   α  11 0 λ = C   , (32) ν 11 0 001/C where C, and α are parameters ( and α are complex numbers). This texture will induce to first order the following mass matrix for the three lightest LH neutrinos 2 mν ∝ 11. (33) 11 This is a typical texture for a hierarchical spectrum that has to fit the oscillation data (atmospheric, solar, CHOOZ) and the constraints on the absolute mass. To show the feasibility of our scenario we chose different values of the parameters C, , α and values of the RH neutrino masses, Mi , i = 1, 4, and we constrain the inputs given the data for the LH neutrino masses and the out-of-equilibrium condition. It is precisely a hierarchy in the Yukawa couplings which allows for an out-of-equilibrium decay without a suppression of the induced CP asymmetry. In Fig. 2, we illustrate for a given set of input parameters the different thermally averaged reaction rates contributing to BE as a function of z = M1/T: γX ΓX = ,X= D,S, L = 2. (34) neq N1 It is clear from this plot that for this set of parameters, and this is true for a wide range of parameter space, all rates at z = 1 fulfill the out-of-equilibrium condition (i.e., ΓX

Fig. 2. Various thermally averaged reaction rates ΓX contributing to BE normalized to the expansion rate of the Universe H(z= 1).

Fig. 3. Abundance YN1 and the baryon asymmetry YB−L for the four generation model. involving the heavy fourth generation leptonic ﬁelds and consistently written the BE which contribute to the ﬁnal baryon asymmetry together with the appropriate conversion factor from the lepton asymmetry to the baryonic one. Our results show that in this simple extension of the Standard Model it is possible to produce the right amount of the BAU in a generic way. A. Abada et al. / Nuclear Physics B 728 (2005) 55–66 65

Acknowledgements

We would like to thank S. Davidson for extremely useful discussions. M.L. thanks the Michi- gan Center for Theoretical Physics and the Laboratoire de Physique Théorique, Université de Paris XI, Orsay, for hospitality during the completion of this work. H.A. would like to thank M. Plumacher for useful communications.

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Comments on the D-instanton calculus in (p, p + 1) minimal string theory

Masafumi Fukuma, Hirotaka Irie, Shigenori Seki

Department of Physics, Kyoto University, Kyoto 606-8502, Japan Received 13 June 2005; received in revised form 18 August 2005; accepted 7 September 2005 Available online 20 September 2005

Abstract The FZZT and ZZ branes in (p, p + 1) minimal string theory are studied in terms of continuum loop equations. We show that systems in the presence of ZZ branes (D-instantons) can be easily investigated within the framework of the continuum string ﬁeld theory developed by Yahikozawa and one of the present authors [M. Fukuma, S. Yahikozawa, Phys. Lett. B 396 (1997) 97, hep-th/9609210]. We explicitly calculate the partition function of a single ZZ brane for arbitrary p. We also show that the annulus amplitudes of ZZ branes are correctly reproduced.  2005 Elsevier B.V. All rights reserved.

1. Introduction

Since conformally invariant boundary states were constructed in the worldsheet description [2–4], renewed interest in noncritical string theory has arisen [5–17]. These boundary states lead to two types of branes. One is of FZZT branes given by Neumann-like boundary states. They are extended in the weak coupling region of Liouville coordinate and naturally correspond to unmarked macroscopic-loop operators [18] in matrix models. The other is of ZZ branes given by Dirichlet-like boundary states. They are localized in the strong coupling region of Liouville coordinate and are shown in [5–7] to correspond to “eigenvalue instantons” in matrix models [19]. In particular, the decay amplitude of ZZ branes is identiﬁed with that of eigenvalues rolling from

E-mail addresses: [email protected] (M. Fukuma), [email protected] (H. Irie), [email protected] (S. Seki).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.09.007 68 M. Fukuma et al. / Nuclear Physics B 728 (2005) 67–82 a local maximum of matrix-model potential, and the ZZ branes are identified with D-instantons in noncritical string theory [20–22].1 In this paper, we show that these FZZT and ZZ branes can be naturally understood within the framework of a continuum string field theory for macroscopic loops [1,23]. The Fock space of this string field theory is realized by p pairs of free chiral fermions ca(ζ ) and c¯a(ζ ) (a = 0, 1,...,p− 1), living on the complex plane whose coordinate is given by the boundary cosmological constant ζ .Itisshownin[1] that their diagonal bilinears c¯a(ζ )ca(ζ ) (bosonized as ∂ϕa(ζ )) can be identified with marked macroscopic loops, ∞ −ζl ∂ϕa(ζ ) =:¯ca(ζ )ca(ζ ):= dle Ψ(l) (1.1) 0 with Ψ(l)being the operator creating the boundary of length l. This implies that the unmarked macroscopic loops (FZZT branes) are described by ∞ dl − ϕ (ζ ) ∼ e ζlΨ(l). (1.2) a l 0

Furthermore, it is shown in [1] that their off-diagonal bilinears c¯a(ζ )cb(ζ ) (a = b) (bosonized as − eϕa(ζ ) ϕb(ζ )) can be identiﬁed with the operator creating a soliton at the “spacetime coordinate” ζ [22]. In order for this operator to be consistent with the continuum loop equations (or the W1+∞ constraints) [24–28], the position of the soliton must be integrated as dζ dζ − D ≡ c¯ (ζ )c (ζ ) = eϕa(ζ ) ϕb(ζ ). (1.3) ab 2πi a b 2πi This integral can be regarded as deﬁning an effective theory for the position of the soliton. In the weak coupling limit g → 0(g: the string coupling constant) the expectation value of c¯a(ζ )cb(ζ ) −1 0 behaves as exp(g Γab(ζ ) + O(g )), where the “effective action” Γab is expressed as the difference of the disk amplitudes:

(0) (0) Γab =ϕa −ϕb . (1.4)

Thus, in the weak coupling limit, the soliton will get localized at a saddle point of Γab and behave as a D-instanton (a ZZ brane). The relation (1.4) evaluated at the saddle point gives a well-known relation between disk amplitudes of FZZT and of ZZ branes [6]. One shall be able to extend this relation to arbitrarily higher-order amplitudes. The main aim of the present paper is to elaborate the calculation around the saddle point performed in [1], and to establish the above relationship with a catalog of possible quantum numbers. We show that the partition functions of D-instantons for generic (p, p + 1) minimal string theories can be calculated explicitly, and also demonstrate that many of the known results obtained in Liouville ﬁeld theory [6,11,15] and/or in matrix models [13,14,19] can be reproduced easily. This paper is organized as follows. In Section 2 we give a brief review on the continuum string ﬁeld theory for macroscopic loops [1,23]. In Section 3 we evaluate one-instanton partition function for (p, p + 1) minimal string theory, and show that it correctly reproduces the value

1 In what follows, the terms “ZZ branes” and “D-instantons” will be used interchangeably. M. Fukuma et al. / Nuclear Physics B 728 (2005) 67–82 69 obtained in [19] (see also [13,14]) for the pure gravity case (p = 2). In Section 4 we make a detailed comparison of our analysis with that performed in Liouville ﬁeld theory. In Section 5 we calculate the annulus amplitudes of ZZ branes, and show that it reproduces the values obtained in [6,15]. Section 6 is devoted to conclusion and discussions. In the present paper, we exclusively consider the unitary case (p, q) = (p, p + 1). General (p, q) minimal strings can also be treated in a similar manner, and detailed analysis will be reported in the forthcoming paper.

2. Review of the noncritical string ﬁeld theory

From the viewpoint of noncritical strings, (p, p + 1) minimal string theory describes two- dimensional gravity coupled to matters of minimal unitary conformal ﬁeld theory (cmatter = 1 − 6/p(p + 1)). There the basic physical operators are the macroscopic loop operator Ψ(l)which creates a boundary of length l. It is often convenient to introduce their Laplace transforms ∞ −ζl ∂ϕ0(ζ ) ≡ dle Ψ(l), (2.1) 0 and ζ is called the boundary cosmological constant. The left-hand side is written with a derivative for later convenience. Note that ζ can take a different value on each boundary. The connected correlation functions of macroscopic loop operators ··· ∂ϕ0(ζ1) ∂ϕ0(ζn) c (2.2) are expanded with respect to the string coupling constant g(>0) as ··· = −2+2h+n ··· (h) ∂ϕ0(ζ1) ∂ϕ0(ζn) c g ∂ϕ0(ζ1) ∂ϕ0(ζn) c , (2.3) h0

(0) (0) where h is the number of handles. Note that ∂ϕ0(ζ ) and ∂ϕ0(ζ1)∂ϕ0(ζ2)c correspond to the disk and annulus (cylinder) amplitudes, respectively. The operator formalism of the continuum string ﬁeld theory [1] is constructed with the following steps:

Step 1. We introduce p pairs of free chiral fermions ca(ζ ) and c¯a(ζ ) (a = 0, 1,...,p− 1) which have the OPE δ c (ζ )c¯ (ζ ) ∼ ab ∼¯c (ζ )c (ζ ). (2.4) a b ζ − ζ a b This can be bosonized with p free chiral bosons ϕa(ζ ) (deﬁned with the OPE: ϕa(ζ )ϕb(ζ ) ∼ +δab ln(ζ − ζ ))as

ϕa(ζ ) −ϕa(ζ ) c¯a(ζ ) = Ka:e :,ca(ζ ) = Ka:e :, (2.5) where Ka are cocycles which ensure the correct anticommutation relations between fermions with different indices a = b.2 In this paper the normal ordering ::are always taken so as to respect the SL(2, C)-invariant vacuum |0. The chiral ﬁeld ϕa(ζ ) can in turn be expressed as

∂ϕa(ζ ) =:¯ca(ζ )ca(ζ ):. (2.6)

− 2 ≡ a 1 − pb In [1] Ka are chosen to be Ka b=0( 1) with pa being the fermion number of the ath species. 70 M. Fukuma et al. / Nuclear Physics B 728 (2005) 67–82

Step 2. We impose the Zp-twisted boundary conditions on the fermions as 2πi 2πi ca e ζ = c[a+1](ζ ), c¯a e ζ =¯c[a+1](ζ ) [a]≡a(mod p) , (2.7) and correspondingly, 2πi ∂ϕa e ζ = ∂ϕ[a+1](ζ ). (2.8)

This can be realized by inserting the Zp-twist ﬁeld σ(ζ) at ζ = 0 and at ζ =∞, with which the chiral ﬁeld ∂ϕa(ζ ) is expanded as 1 − − − σ |···∂ϕ (ζ ) ···|σ =σ |··· ω naα ζ n/p 1 ···|σ ω ≡ e2πi/p , (2.9) a p n n∈Z [αn,αm]=nδn+m,0. (2.10) Here σ |≡0|σ(∞) and |σ ≡σ(0)|0.

{ k} = ∈ Z Step 3. We introduce the generators of the W1+∞ algebra [29], Wn (k 1, 2,...; n ), that are given by the mode expansion of the currents p−1 k ≡ k −n−k = :¯ k−1 : = W (ζ ) Wn ζ ca(ζ )∂ζ ca(ζ ) (k 1, 2,...). (2.11) n∈Z a=0

Finally we introduce the state |Φ that satisﬁes the vacuum condition of the W1+∞ constraints: k| = − + Wn Φ 0 (k 1,n k 1). (2.12) This constraint is shown to be equivalent to the continuum loop equations [24–28,30]. In addition 3 to the W1+∞ constraints, we further require that |Φ be a decomposable state.

Step 4. One can prove that the connected correlation functions of macroscopic loop operators are given as cumulants (or connected parts) of the following correlation functions [1]4: − B |: ··· :| g ∂φ0(ζ1) ∂φ0(ζn) Φ ∂ϕ (ζ ) ···∂ϕ (ζ ) ≡ . (2.13) 0 1 0 n − B | g Φ Here the state ∞ B 1 − ≡σ | exp − Bnαn (2.14) g g n=1 characterizes the theory, and the (p, q) minimal string is realized by taking Bp+q = 0 and Bn = 0 (∀n p + q + 1).

3 | | = H | A state Φ is said to be decomposable if it can be written as Φ e σ ,whereH is a bilinear form of the fermions, = ¯ = | { ∞ }| H dζ dζ a,b ca(ζ )hab(ζ, ζ )cb(ζ ). This is equivalent to the statement that τ(x) σ exp n=1 xnαn Φ is a τ function of the pth reduced KP hierarchy [31]. It is proved in [27,28] that this set of conditions (W1+∞ constraints and decomposability) is equivalent to the Douglas equation, [P,Q]=1 [32]. 4 Since the normal ordering differs from that for the twisted vacuum |σ , the two-point function acquires a ﬁnite renormalization. This turns out to be the so-called nonuniversal term in the annulus amplitude [1]. The representation of loop correlators with free twisted bosons can also be found in [33], where open-closed string coupling is investigated. M. Fukuma et al. / Nuclear Physics B 728 (2005) 67–82 71

From the viewpoint of two-dimensional gravity, α1 creates the lowest-dimensional operator on random surfaces, so that B1 (≡ µ) should correspond to the bulk cosmological constant in the unitary minimal strings. In fact, if we choose B1 = µ, B2p+1 =−4p/(p + 1)(2p + 1) and Bn = 0(n = 0or2p + 1), then we obtain the following disk and annulus amplitudes for marked macroscopic loops [1]: − 2(p 1)/p + + ∂ϕ (ζ ) (0) = ζ + ζ 2 − µ (p 1)/p + ζ − ζ 2 − µ (p 1)/p , (2.15) 0 p + 1 (0) = + 2 − 1/p + − 2 − 1/p ∂ϕ0(ζ1)∂ϕ0(ζ2) c ∂ζ1 ∂ζ2 ln ζ1 ζ1 µ ζ1 ζ1 µ − + 2 − 1/p − − 2 − 1/p − − ζ2 ζ2 µ ζ2 ζ2 µ ln(ζ1 ζ2) , (2.16) which agree with the matrix model results [18]. The amplitudes of the corresponding FZZT branes are obtained by integrating the above amplitudes. In particular, the annulus amplitudes of the FZZT branes are given by (0) = + 2 − 1/p + − 2 − 1/p ϕ0(ζ1)ϕ0(ζ2) c ln ζ1 ζ1 µ ζ1 ζ1 µ − + 2 − 1/p − − 2 − 1/p − − ζ2 ζ2 µ ζ2 ζ2 µ ln(ζ1 ζ2), (2.17) with nonuniversal additive constants. Moreover, by combining the W1+∞ constraint with the KP equation, one can easily show ≡ − ∂ 2 − | that the two-point function of cosmological term u(µ, g) ( g ∂µ) ln B/g Φ satisﬁes the Painlevé-type equations [34,35]5

2g2 (p = 2) 4u2 + ∂2 u = µ, (2.18) 3 µ 3g2 g4 (p = 3) 4u3 + (∂ u)2 + 3g2u∂2 u + ∂4 u =−µ, (2.19) 2 µ µ 6 µ . . As for the solutions with soliton backgrounds, the crucial observation made in [1] is that the commutators between the W1+∞ generators and c¯a(ζ )cb(ζ ) (a = b) give total derivatives: k ¯ = ∗ Wn , ca(ζ )cb(ζ ) ∂ζ ( ), (2.20) and thus the operator dζ D ≡ c¯ (ζ )c (ζ ) (2.21) ab 2πi a b commutes with the W1+∞ generators: k = Wn ,Dab 0. (2.22)

5 If one takes account of the doubling in matrix models with even potentials, the two-point function will be replaced by f ≡ 2u. 72 M. Fukuma et al. / Nuclear Physics B 728 (2005) 67–82

Here the contour integral in (2.21) needs to surround the point of infinity (ζ =∞) p times in order to resolve the Zp monodromy. Eq. (2.22) implies that if |Φ is a solution of the W1+∞ con- ··· | straints (2.12), then so is Da1b1 Dar br Φ . We will see that the latter can actually be identified with an r-instanton solution.6 By using the weak field expansions, the expectation value of Dab can be expressed as dζ − D = eϕa(ζ ) ϕb(ζ ) ab 2πi dζ − = exp eϕa(ζ ) ϕb(ζ ) − 1 2πi c dζ 1 = exp ϕ (ζ ) − ϕ (ζ ) + ϕ (ζ ) − ϕ (ζ ) 2 +··· . (2.23) 2πi a b 2 a b c Since connected n-point functions have the following expansion in g: ∞ ··· = −2+2h+n ··· (h) ∂ϕa1 (ζ1) ∂ϕan (ζn) c g ∂ϕa1 (ζ1) ∂ϕan (ζn) c , (2.24) h=0 leading contributions to the exponent of (2.23) in the weak coupling limit come from spherical topology (h = 0): dζ + + D = e(1/g)Γab(ζ ) (1/2)Kab(ζ ) O(g) (2.25) ab 2πi with ≡ (0) − (0) ≡ − 2 (0) Γab(ζ ) ϕa(ζ ) ϕb(ζ ) ,Kab(ζ ) ϕa(ζ ) ϕb(ζ ) c . (2.26) Thus, in the weak coupling limit the integration is dominated by the value around a saddle point in the complex ζ plane.

3. Saddle point analysis

In this section, we make a detailed calculation of the integral (2.25) around a saddle point, up 1 to eO(g ). The functions Γab(ζ ) and Kab(ζ ) can be calculated, basically by integrating the disk and 2πia annulus amplitudes ((2.15) and (2.16)), followed by analytic continuation ζ1 → e ζ1, ζ2 → 2πib e ζ2 and by taking the limit ζ1,ζ2 → ζ . For example, Γab is calculated as (0) (0) 2πia (0) 2πib (0) Γab(ζ ) = ϕa(ζ ) − ϕb(ζ ) = ϕ0 e ζ − ϕ0 e ζ

e2πiaζ (0) = dζ ∂ϕ0(ζ ) , (3.1) e2πibζ

6 Note that if the decomposability condition is further imposed, the only possible form for the collection of instanton | ≡ | solutions should be Φ,θ a=b exp(θabDab) Φ with fugacity θab [23]. M. Fukuma et al. / Nuclear Physics B 728 (2005) 67–82 73 and we obtain

p (2p+1)/2p Γab(ζ ) = µ 21/p(p + 1) 1 + − − − + × ωa − ωb s(2p 1)/p + ω a − ω b s (2p 1)/p 2p + 1 − − − − ωa − ωb s1/p + ω a − ω b s 1/p , (3.2) where ω ≡ e2πi/p, and s = s(ζ) is deﬁned as 2 2 ζ ζ − ζ ζ s = √ + √ − 1,s1 = √ − √ − 1. (3.3) µ µ µ µ

On the other hand, the calculation of Kab needs a special care because ϕa(ζ1)ϕb(ζ2) does not obey simple monodromy. This is due to the fact that the two-point function ϕa(ζ1)ϕb(ζ2) is deﬁned with the normal ordering ::that respects the SL(2, C) invariant vacuum: −B/g|:ϕ (ζ )ϕ (ζ ):|Φ ϕ (ζ )ϕ (ζ ) = a 1 b 2 . (3.4) a 1 b 2 −B/g|Φ

− δab ln(ζ1 − ζ2). (3.5) We thus obtain7 (0) = a + 2 − 1/p + −a − 2 − 1/p ϕa(ζ1)ϕb(ζ2) c ln ω ζ1 ζ1 µ ω ζ1 ζ1 µ − b + 2 − 1/p − −b − 2 − 1/p ω ζ2 ζ2 µ ω ζ2 ζ2 µ

− δab ln(ζ1 − ζ2), (3.6)

7 Although these amplitudes may have nonuniversal additive corrections, they will be totally canceled in the calculation of Γab and Kab. In this sense, the value of the partition function of a D-instanton must be universal as in the matrix model cases [13]. 74 M. Fukuma et al. / Nuclear Physics B 728 (2005) 67–82 from which Kab(ζ ) are calculated to be a 1/p −a −1/p b 1/p −b −1/p −1 Kab(ζ ) = ln ω s − ω s + ln ω s − ω s − 2ln s − s − − − − ln ωa − ωb s1/p + ω a − ω b s 1/p − − − 2 − ln ωb − ωa s1/p + ω b − ω a s 1/p + 2ln √ . (3.7) p µ In order to make a further calculation in a well-defined manner, it is convenient to introduce new variables z and τ [11], for which the functions Γab and Kab are single-valued: 2 ζ ζ 1 − s = √ + √ − 1 ≡ eipτ ,z≡ cos τ = s1/p + s 1/p . (3.8) µ µ 2 ζ is then expressed as ζ √ = cos pτ = T (z), (3.9) µ p where Tn(z) (n = 0, 1, 2,...) are first Tchebycheff polynomials of degree n defined by Tn(cos τ)≡ cos nτ . The monodromy of the operator φa(z) ≡ ϕa(ζ ) under twisted vacuum |σ is expressed as

φa(z)|σ =φ0(za)|σ (3.10) with za ≡ cos τa ≡ cos(τ + 2πa/p). We return to the calculation of the partition√ function in the presence of one soliton, Dab . Since the measure is written as dζ = p µUp−1(z) dz, we need to calculate the following integral: √ p µ −(1/g)Γ (z)+(1/2)K (z)+O(g) D = dzU − (z)e ab ab . (3.11) ab 2πi p 1

Here Un(z) (n = 0, 1, 2,...) are second Tchebycheff polynomials of degree n deﬁned by Un(cos τ)≡ sin(n + 1)τ/ sin τ . Tn(z) and Un(z) are related as 1 T (z) = nUn(z), U (z) = zUn(z) − (n + 1)Tn+ (z) . (3.12) n n 1 − z2 1

The function Γab and their derivatives are easily obtained by using the formulas dz sin(τ + 2πa/p) d2z sin(2πa/p) a = , a = , (3.13) dz sin τ dz2 sin3 τ and are found to be (0) (0) Γab(z) = φa(z) − φb(z) (p−1)/p 2 p (2p+1)/2p 1 = µ T + (z ) − T + (z ) − (z − z ) , (3.14) p + 1 2p + 1 2p 1 a 2p 1 b a b (2p+1)/2p 2 p (2p+1)/2p b a −p−1 2(p+1) −a−b Γ (z) =− µ ω − ω U − (z)y y − ω ab p + 1 p 1 y ≡ eiτ , (3.15) M. Fukuma et al. / Nuclear Physics B 728 (2005) 67–82 75

z Γ (z) = Γ (z) ab 1 − z2 ab (p−1)/p pω + 1 − µ(2p 1)/2p p + 1 1 − z2 × (2p + 1) T2p+1(za) − T2p+1(zb) − (za − zb) . (3.16)

As for Kab, one obtains the following formula from (3.6): − − (0) φa(z1) φb(z1) φc(z2) φd (z2) c (z1a − z2c)(z1b − z2d ) √ = ln − (δac + δbd − δad − δbc) ln µ Tp(z1) − Tp(z2) , (z1a − z2d )(z1b − z2c) (3.17) and thus Kab is expressed with z as = − − (0) Kab(z) lim φa(z1) φb(z1) φa(z2) φb(z2) c z1,z2→z (z − z )(z − z ) = lim ln 1a 2a 1b 2b → 2 z1,z2 z (z1a − z2b)(z1b − z2a)(Tp(z1) − Tp(z2)) µ √ 2 =−ln −(za − zb) Up−1(za)Up−1(zb) − 2lnp µ. (3.18) = = = The saddle points z z∗ cos τ∗ are determined by solving the equation Γab(z∗) 0, and are found to be8 a + b a + b + l τ∗ = − + π, l ∈ Z. (3.19) p p + 1

They acquire the following changes under the transformation z∗ → z∗a = cos(τ∗ + 2πia/p): b − a a + b + l m n z∗ = − π = − π, a cos + cos + (3.20) p p 1 p p 1 b − a a + b + l m n z∗ = cos + π = cos + π. (3.21) b p p + 1 p p + 1 Here we have introduced two integers m and n as m ≡ b − a, n ≡ a + b + l. (3.22)

It is easy to see that T2p+1(z) takes the following values at those shifted points: m n T + (z∗ ) = + π, 2p 1 a cos + (3.23) p p 1 m n T + (z∗ ) = cos − π. (3.24) 2p 1 b p p + 1 From this, one easily obtains 8p (2p+1)/2p n m Γab(z∗) =− µ sin π sin π , (3.25) 21/p(2p + 1) p + 1 p

8 = There are other possible saddle points determined by√Up−1(z∗) 0. However, they only give irrelevant contributions to the integral because of the vanishing measure dζ = p µUp−1(z) dz at such saddle points. 76 M. Fukuma et al. / Nuclear Physics B 728 (2005) 67–82 and = 8p (2p+1)/2p n m Γab(z∗) µ sin π sin π . (3.26) 21/p sin2 τ∗ p + 1 p By using the relation Up−1(z∗a) = (sin pτ∗/ sin τ∗a)Up−1(z∗), Kab(z∗) can also be calculated easily, and is found to be 2nπ 2mπ cos( + ) − cos( ) = √ p 1 p Kab(z∗) 2ln nπ mπ . (3.27) 2p 2µ sin τ∗Up−1(z∗) sin( p+1 ) sin( p ) In order for the integration to give such nonperturbative effects that vanish in the limit g →+0, we need to choose a contour along which Re Γab(z) takes only negative values. In particular, (m, n) should be chosen such that Γab(z∗) is negative. This in turn implies that Γab(z∗) is positive, and thus the corresponding steepest descent path intersects the saddle point in the pure-imaginary direction in the complex z plane. We thus take z = z∗ + it around the saddle point, so that the Gaussian integral becomes √ ∞ p µ (1/2)K (z∗) (1/g)Γ (z∗) −(1/2g)Γ (z∗)t2 D = U − (z∗)e ab e ab dt e ab ab 2π p 1 −∞ µg Up−1(z∗) = p e(1/2)Kab(z∗)e(1/g)Γab(z∗). (3.28) 2π Γab(z∗) Substituting all the values obtained above, we ﬁnally get 1/2p √ cos( 2nπ ) − cos( 2mπ ) 2 −(2p+1)/4p p+1 p 1 D = √ gµ exp − Γ (z∗) (3.29) ab 3 nπ 3 mπ g ba 8 2πp sin ( p+1 ) sin ( p ) with =− Γba(z∗) Γab(z∗) 8p + n m =+ µ(2p 1)/2p sin π sin π (> 0). (3.30) 21/p(2p + 1) p + 1 p Note that the expression (3.29) is invariant under the change of (m, n) into (p − m − 1,p− n). Thus we can always restrict the values of (m, n) to the region 0 0. (3.31) For example, in the pure gravity case (p = 2), the only possible choice is (m, n) = (1, 1) or (a, b; l) = (0, 1; 0), for which we obtain √ 1/4 1/2 2 g − 4 6 D = µ 5/8 exp − µ5/4 . (3.32) 01 8π 1/233/4 5g √ 9 By rescaling the string coupling constant as g = gs/ 2, it becomes √ 1/2 g − 8 3 D = s µ 5/8 − µ5/4 . 01 1/2 3/4 exp (3.33) 8π 3 5gs

9 Such relations among various parameters can be best read off by looking at the string equations. M. Fukuma et al. / Nuclear Physics B 728 (2005) 67–82 77

This coincides, up to a factor of i, with the partition function of a D-instanton evaluated by resort- ing to one-matrix model [19] (see also [13,14]). We shall make a comment on this discrepancy in Section 6.

4. Comparison with Liouville ﬁeld theory

In our analysis made in the preceding sections, the operators which are physically meaningful are the macroscopic loop operator ϕ0(ζ ) and the soliton operators Dab = dζ c¯a(ζ )cb(ζ ) (a = b). They should have their own correspondents in Liouville field theory. The former evidently corresponds to FZZT branes. For example, with the parametrization (3.8) and (3.9),the annulus amplitude of FZZT branes (Eq. (2.17)) is expressed (up to nonuniversal additive constants) as − (0) = z1 z2 ϕ0(ζ1)ϕ0(ζ2) c ln , (4.1) Tp(z1) − Tp(z2) and agrees with the calculation based on Liouville field theory [6,15]. On the other hand, the relation between the soliton operators and ZZ branes is indirect. In fact, our solitons can take arbitrary positions for finite values of g, but in the weak coupling limit they get localized at saddle points and become ZZ branes. In this section we shall establish this relationship between the localized solitons and the ZZ branes with explicit correspondence between their quantum numbers. A detailed analysis of FZZT and ZZ branes in (p, q) minimal string theory is performed in [11]. According to this, the BRST equivalence classes of ZZ branes are labeled by two quantum numbers (m, n), and their boundary states can be written as differences of two FZZT boundary states: | = + − − m, n ZZ ζ zmn FZZT ζ zmn FZZT. (4.2) √ = ± Here ζ(z) µTp(z) denotes the boundary cosmological constant of an FZZT brane. zmn are the singular points of the Riemann surface Mp,q which z uniformizes, and are given by

± π(mq ± np) z = cos , (4.3) mn pq m = 1,...,p− 1,n= 1,...,q− 1,mq− np > 0. (4.4) From this, one obtains the relation (m,n) = + − − ZZZ ZFZZT ζ zmn ZFZZT ζ zmn , (4.5) (m,n) where ZZZ is the disk amplitude of a ZZ brane with quantum number (m, n), and ZFZZT(ζ ) is that of an FZZT brane with boundary cosmological constant ζ . On the other hand, our analysis shows that in the weak coupling limit g →+0, the partition function in the presence of a soliton, Dab, is dominated by a saddle point z∗ of the function (0) (0) (1/g)Γ (z∗) Γab(z) =φa(z) −φb(z) , and is expressed as Dab∼e ab . This implies that Γab(z∗)(<0) should be regarded as the disk amplitude of a D-instanton [20]. Furthermore, = ; = − a+b + a+b+l it was explicitly evaluated at the saddle point z z∗(a, b l) cos( p p+1 )π in the previous section as 78 M. Fukuma et al. / Nuclear Physics B 728 (2005) 67–82 (0) (0) Γab(z∗) = φa(z∗) − φb(z∗) (0) (0) = φ0 z∗a(a, b; l) − φ0 z∗b(a, b; l) (0) (0) = ϕ ζ z∗ (a, b; l) − ϕ ζ z∗ (a, b; l) 0 a 0 b = ZFZZT ζ z∗a(a, b; l) − ZFZZT ζ z∗b(a, b; l) . (4.6)

Here z∗a and z∗b are calculated in (3.20) and (3.21) as m n m n z∗ (a, b; l) = cos − π, z∗ (a, b; l) = cos + π, (4.7) a p p + 1 b p p + 1 with (a, b; l) being related to (m, n) as in (3.22). We thus see that the shifted points z∗a(a, b; l) ; ± and z∗b(a, b l) correspond to the singular points zmn as ; = − ; = + z∗a(a, b l) zmn,z∗b(a, b l) zmn. (4.8) (m,n) Then (4.5), (4.6) and (4.8) lead to the following equality between Γab(z∗) and ZZZ : = − − + =− (m=b−a,n=a+b+l) Γab(z∗) φ0 zmn φ0 zmn ZZZ . (4.9) Therefore, each saddle point corresponds to a ZZ brane.10 The relative minus sign between (m,n) Γab(z∗) (taken to be negative for the convergence in the weak coupling limit) and ZZZ (con- ventionally normalized to be positive) appearing in (4.9) is naturally derived in our analysis and matched with the argument given in [15]. For the rest of this section, we illustrate the above correspondence along the line of the geometric setting introduced in [11]. We only consider the pure gravity case (p = 2),butthe generalization to other cases must be straightforward. = k For p 2, the amplitude of a ZZ brane is given by Γ10. Furthermore, using the Wn constraint with k = 1, one can easily show that ∂ϕ1(ζ )=−∂ϕ0(ζ ). Thus the saddle points of Γ10 can be determined simply by solving the following equation: √ 8 µ 2 √ 0 = ∂ Γ (ζ ) = 2 ∂ϕ (ζ ) = f (ζ ), f (ζ ) ≡ ζ − (ζ + µ). (4.10) ζ 01 0 3 3 3 2 √ =− Here f3(ζ ) is√ a degree-three polynomial of ζ , and has a single root at ζ µ and a double 2 root at ζ = µ/2. The algebraic curve deﬁned by y = f3(ζ ) is thus a torus with a pinched cycle, as depicted in Fig. 1.

Fig. 1. Pinched Riemann surface.

10 We should stress that it is not the saddle point z∗ (the “position” of the D-instanton) but the shifted points z∗a and z∗b which actually correspond to the singular points in the Riemann surface Mp,p+1, although the set of the saddle points coincides with that of the singular points. M. Fukuma et al. / Nuclear Physics B 728 (2005) 67–82 79

√ 2 Using the relation ζ/ µ = T2(z) = 2z − 1, these roots are expressed in terms of z as √ √ √ µ 3 ζ∗ =− µ ⇐⇒ z∗ = 0,ζ∗ = ⇐⇒ z∗ =± . (4.11) 2 2

The ﬁrst saddle point z∗ = 0 corresponds to the zero of Up−1(z) = U1(z) = 2z which was dis- carded in our analysis because such saddle points give rise to vanishing measure. We thus see that the pinched cycle corresponds to the D-instanton and thus to the ZZ brane. √ If we take (a, b; l) = (0, 1; 0) as before, this selects the saddle point at z∗(0, 1; 0) = 3/2. Then its shifted points are calculated as √ √ 3 − 3 + z∗ = = z ,z∗ =− = z . (4.12) 0 2 11 1 2 11

5. Annulus amplitudes of D-instantons

The annulus amplitudes of distinct D-instantons (ZZ branes) can also be calculated easily. These amplitudes correspond to the states

DabDcd|Φ, (5.1) which appear, for example, when two distinct solitons are present in the background: + eDab Dcd |Φ. The two-point function of two solitons, DabDcd, can be written as ϕ (ζ )−ϕ (ζ ) ϕ (ζ )−ϕ (ζ ) −B/g|:e a b ::e c d :|Φ D D = dζ dζ ab cd − | B/g Φ + − − − − + − = dζ dζ e(δac δbd δad δbc) ln(ζ ζ ) eϕa(ζ ) ϕb(ζ ) ϕc(ζ ) ϕd (ζ ) + − − − − + − = (δac δbd δad δbc) ln(ζ ζ ) ϕa(ζ ) ϕb(ζ ) ϕc(ζ ) ϕd (ζ ) − dζ dζ e exp e 1 c + = dζ dζ e(1/g)Γab(ζ ) (1/g)Γcd (ζ )e(1/2)Kab(ζ )e(1/2)Kcd (ζ )

+ − − − − − (0) × e(δac δbd δad δbc) ln(ζ ζ )e (ϕa(ζ ) ϕb(ζ ))(ϕc(ζ ) ϕd (ζ )) c eO(g). (5.2) Since Dab and Dcd may have their own saddle points ζ∗ and ζ∗ in the weak coupling limit, the two-point function will take the following form: DabDcd=DabDcd exp (δac + δbd − δad − δbc) ln(ζ∗ − ζ∗) (0) + ∗ − ∗ − ϕa(ζ ) ϕb(ζ ) ϕc(ζ∗) ϕd (ζ∗) c . (5.3) We thus identify the annulus amplitude of D-instantons as (0) | ∗ = ∗ − ∗ − Kab cd(z ,z∗) ϕa(ζ ) ϕb(ζ ) ϕc(ζ∗) ϕd (ζ∗) c + (δac + δbd − δad − δbc) ln(ζ∗ − ζ∗) (0) = φ (z∗) − φ (z∗) φ (z ) − φ (z ) a b c ∗ d ∗ c √ + (δac + δcd − δad − δbc) ln µ Tp(z∗) − Tp(z∗) . (5.4) 80 M. Fukuma et al. / Nuclear Physics B 728 (2005) 67–82

The right-hand side can be simpliﬁed by using (3.17), and we obtain − − + + (z∗ − z )(z∗ − z ) (z − z )(z − z ) = a ∗c b ∗d = mn m n mn m n Kab|cd(z∗,z∗) ln ln − + + − ∗ − ∗ − − − (z a z∗d )(z b z∗c) (zmn zmn )(zmn zmn ) = (m,n|m ,n ) Zannulus (5.5) where we have used the identiﬁcation (see (4.8)) = − = + z∗a zmn,z∗b zmn, (5.6) = − = + z∗c zmn ,z∗d zmn . (5.7) This expression correctly reproduces the annulus amplitudes of ZZ branes obtained in [6,15].

6. Conclusion and discussions

In this paper, we have studied D-instantons of unitary (p, p + 1) minimal strings in terms of the continuum string field theory [1]. In this formulation, there are two types of operators that have definite physical meanings; one is the (unmarked) macroscopic loop operator ϕ0(ζ ), and the other is of the soliton operators Dab. We have calculated the expectation value of the soliton operator Dab in the weak coupling limit g → 0. Dab is then expanded as Dab= dζ exp[(1/g)Γab + (1/2)Kab + O(g)], and is dominated by saddle points. We have carefully identified the valid saddle points, and found that Dab has a well-defined finite value. Since there is no ambiguity in obtaining it, this expectation value must be universal. Each saddle point denoted by z∗(a, b; l) corresponds to the location of a localized soliton. On the other hand, ZZ branes in Liouville field theory are characterized by a pair of FZZT ± M cosmological constants zmn [6], which correspond to singular points on a Riemann surface p,q [11]. We have shown that the free energy of the localized soliton, Γab(z∗), can be identified with (m,n) minus the partition function of the ZZ brane ZZZ with the following relation: b − a = m, a + b + l = n, ; = − ; = + z∗a(a, b l) zmn,z∗b(a, b l) zmn. (6.1) This identification is actually valid for any (p, q) minimal string theories. We pointed out that the set {(a, b; l)} are redundant and can be restricted to a smaller set with 1 b − a(= m) p − 1, 1 a + b + l(= n) q − 1 and mq − np > 0. With the identification (6.1) we have shown that the annulus amplitudes of localized soli- (m,n|m,n) tons, Kab|cd(z∗,z∗), coincide with those of ZZ branes, Zannulus [6,15]. There is no ambiguity in deriving this equivalence, we thus can conclude that these annulus amplitudes also must be universal and can be derived from the saddle point analysis. For the pure gravity case (p = 2) one can compare the resulting value Dab with the one- instanton partition function obtained with the use of matrix models [13,14,19]. We found that they coincide up to a single factor i. In fact, the overall normalization of Dab is not fixed only from the continuum loop equations. However, if Dab creates a soliton with a definite charge, then Dba creates an antisoliton with the opposite charge. Moreover, with the present normalization of Dab, the operator DabDba almost gives identity for the 0-soliton sector, so that the present normalization seems to be natural. In this sense, this discrepancy between our result and the matrix model result deserves explanation. Among possible ones are (i) that it may be natural to M. Fukuma et al. / Nuclear Physics B 728 (2005) 67–82 81 introduce i in defining the fugacity, or (ii) that the D-instanton calculation in matrix models may choose a path different from the steepest-descent one for Γab. In this paper, we mainly consider the unitary (p, p + 1) minimal strings. The analysis can be easily extended to general (p, q) minimal strings, as will be reported in our future communica- tion.

Acknowledgements

The authors would like to thank Hikaru Kawai, Ivan Kostov and Yoshinori Matsuo for useful discussions. This work was supported in part by the Grant-in-Aid for the 21st Century COE “Center for Diversity and Universality in Physics” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. M.F. and S.S. are also supported by the Grant- in-Aid for Scientiﬁc Research No. 15540269 and No. 16740159, respectively, from MEXT.

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Moduli ﬁxing in realistic string vacua

Alon E. Faraggi

Theoretical Physics Department, University of Oxford, Oxford OX1 3NP, UK Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK Received 18 April 2005; received in revised form 22 June 2005; accepted 16 August 2005 Available online 8 September 2005

Abstract I demonstrate the existence of quasi-realistic heterotic-string models in which all the untwisted Kähler and complex structure moduli, as well as all of the twisted sectors moduli, are projected out by the generalized GSO projections. I discuss the conditions and characteristics of the models that produce this result. The existence of such models offers a novel perspective on the realization of extra dimensions in string theory. In this view, while the geometrical picture provides a useful mean to classify string vacua, in the phenomenologically viable cases there is no physical realization of extra dimensions. The models under consideration correspond to Z2 × Z2 orbifolds of six-dimensional tori, plus additional identiﬁcations by internal shifts and twists. The special property of the Z2 × Z2 orbifold is that it may act on the compactiﬁed dimensions as real, rather than complex, dimensions. This property enables an asymmetric projection on all six internal coordinates, which enables the projection of all the untwisted Kähler and complex structure moduli.  2005 Elsevier B.V. All rights reserved.

1. Introduction

String theory continues to serve as the most compelling framework for the unification of all the fundamental matter and interactions. Amazingly, preservation of some classical string symmetries necessitates the introduction of a fixed number of degrees of freedom, beyond the observable ones. One interpretation of these additional degrees of freedom is as extra space–time dimensions, which clearly is in contradiction with the observed facts and diminishes from the appeal of string theory. Thus, the need arises to hide the additional dimensions by compactifying the superstring on a six-dimensional compact manifold. This in turn leads to the problem of fixing

E-mail address: [email protected] (A.E. Faraggi).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.08.028 84 A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108 the parameters of the extra dimensions in such a way that it does not conflict with contemporary experimental limits. Additionally, the extra dimensions give rise to light scalar fields whose VEV parameterizes the shape and size of the extra dimensions. This is typically dubbed as the moduli stabilization problem, and preoccupies much of the activity in string theory [1] ever since the initial realization of its potential relevance for particle phenomenology [2,3]. A particular class of string compactifications that exhibit promising phenomenological properties are the so-called free fermionic heterotic-string models [4–13]. These models are related to Z2 × Z2 orbifold compactifications [14], and many of their appealing phenomenological properties are rooted in the underlying Z2 × Z2 orbifold structure [14]. Recently, a classification of this class of models was initiated and it was revealed that in a large class of models the reduction to three generations necessitates the utilization of a fully asymmetric shift [15]. This indicates that the geometrical objects underlying these models do not correspond to the standard complex geometries that have been the most prevalent in the literature. Furthermore, the utilization of the asymmetric shift has important bearing on the issue of moduli fixing in string theory [16], and in this class of string compactifications and in particular. The operation of an asymmetric shift can only occur at special points in the moduli space, and results in fixation of the moduli at the special points. It is therefore of interest to investigate this aspect in greater detail. In the context of the free fermionic models the marginal operators that correspond to the moduli deformations correspond to the inclusion of Thirring interactions among the world-sheet fermions [17–20]. The allowed world-sheet Thirring interactions must be invariant under the GSO projections that are induced by the boundary condition basis vectors that define the free fermionic string models. Thus, depending on the assignment of boundary conditions in specific models, some of the world-sheet Thirring interactions are forbidden, and their corresponding moduli are projected from the physical spectrum. In this manner the GSO projection induced by the boundary condition basis vectors acts as a moduli fixing mechanism. In this paper I discuss the existence of quasi-realistic three generation free fermionic models, in which all the untwisted Kähler and complex structure moduli are projected out from the low energy effective field theory. This is a striking result with crucial implications for the premise of extra dimensions in physical string vacua. This result indicates that the interpretation of the additional degrees of freedom needed to obtain a conformally invariant string theory as additional continuous spatial dimensions beyond the observable ones, may in fact not be realized in the phenomenologically relevant cases. Furthermore, the fact that the projection is achieved by an asymmetric orbifold action indicates that the treatment of the moduli problem in the low energy effective supergravity theory is inadequate. The reason being that the effective supergravity theories relate to the additional dimensions as classical geometries, and hence necessarily as left–right symmetric. On the other hand the basic property of string theory is that it allows for the left–right world-sheet asymmetry, which in the context of the phenomenological free fermionic models is instrumental in fixing the moduli.

2. General structure of free fermionic models

In the free fermionic formulation [17] of the heterotic string in four dimensions all the world- sheet degrees of freedom required to cancel the conformal anomaly are represented in terms of free fermions propagating on the string world-sheet. In the light-cone gauge the world-sheet ﬁeld content consists of two transverse left- and right-moving space–time coordinate bosons, µ ¯ µ µ X1,2 and X1,2, and their left-moving fermionic superpartners ψ1,2, and additional 62 purely internal Majorana–Weyl fermions, of which 18 are left-moving, χI , and 44 are right-moving, A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108 85

φa. In the supersymmetric sector the world-sheet supersymmetry is realized nonlinearly and the world-sheet supercurrent is given by

µ I J K TF = ψ ∂Xµ + fIJKχ χ χ , (2.1) I where fIJK are the structure constants of a semi-simple Lie group of dimension 18. The χ (I = 1,...,18) world-sheet fermions transform in the adjoint representation of the Lie group. In the realistic free fermionic models the Lie group is SU(2)6.TheχI (I = 1,...,18) transform in the adjoint representation of SU(2)6, and are denoted by χI , yI , ωI (I = 1,...,6). Under parallel transport around a noncontractible loop on the toroidal world-sheet the fermionic ﬁelds pick up a phase

f →−eiπα(f)f. (2.2) The minus sign is conventional and α(f ) ∈ (−1, +1]. A model in this construction [17] is defined by a set of boundary conditions basis vectors and by a choice of generalized GSO projection coefficients, which satisfy the one-loop modular invariance constraints. The boundary conditions basis vectors bk span a finite additive group Ξ = nibi, (2.3) k = − where ni 0,...,Nzi 1. The physical massless states in the Hilbert space of a given sector α ∈ Ξ are then obtained by acting on the vacuum state of that sector with the world-sheet bosonic and fermionic mode operators, with frequencies νf , νf ∗ and by subsequently applying the generalized GSO projections

iπ(bi Fα) ∗ α e − δαc |s=0 (2.4) bi with (biFα) ≡ − bi(f )Fα(f ) , (2.5) real+complex real+complex left right where Fα(f ) is a fermion number operator counting each mode of f once (and if f is complex, f ∗ minus once). For periodic complex fermions (i.e., for α(f ) = 1) the vacuum is a spinor in order to represent the Clifford algebra of the corresponding zero modes. For each periodic complex fermion f , there are two degenerate vacua |+, |−, annihilated by the zero modes f0 ∗ = − =− µ and f0 and with fermion number F(f) 0, 1, respectively. In Eq. (2.4), δα 1ifψ is µ periodic in the sector α, and δα =+1ifψ is antiperiodic in the sector α. Each complex world- sheet fermion f givesrisetoaU(1)ff∗ current in the Cartan subalgebra of the four-dimensional gauge group, with the charge given by 1 Q(f ) = α(f ) + F(f). (2.6) 2 Alternatively, a real left-moving fermion may be combined with a real right-moving fermion to form an Ising model operator [21,22], in which case the U(1) generators are projected out, and the rank of the four-dimensional gauge group is reduced. This distinction between complex and real world-sheet fermion has important bearing on the assignment of asymmetric versus 86 A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108 symmetric boundary conditions, and hence on the issue of moduli ﬁxing in the Z2 ×Z2 fermionic models. The δα in Eq. (2.4) is a space–time spin statistics factor, equal to +1 for space–time bosons and −1 for space–time fermions, and is determined by the boundary conditions of the space–time µ fermions ψ1,2 in the sector α, i.e.,

iπα(ψµ) δα = e . The ∗ α c bi are free phases in the one-loop string partition function. For the Neveu–Schwarz untwisted sector we have the general result that ∗ NS δ c = δ . NS b b Hence, the free phases do not play a significant role in the determination of the untwisted moduli. The existence of the untwisted moduli in a model depends solely on the boundary conditions. The free phases of the models will therefore be suppressed in the following. The boundary condition basis defining a typical realistic free fermionic heterotic string model is constructed in two stages. The first stage consists of the NAHE set, which is a set of five boundary condition basis vectors, {1,S,b1,b2,b3} [4,8,13,14]. The gauge group in- 3 duced by the NAHE set is SO(10) × SO(6) × E8 with N = 1 supersymmetry. The space– time vector bosons that generate the gauge group arise from the Neveu–Schwarz sector and from the sector ξ2 ≡ 1 + b1 + b2 + b3. The Neveu–Schwarz sector produces the generators of 3 SO(10) × SO(6) × SO(16).Theξ2-sector produces the spinorial 128 of SO(16) and completes the hidden gauge group to E8. The NAHE set divides the internal world-sheet fermions in the ¯1,...,8 ¯ 1,...,5 following way: φ generate the hidden E8 gauge group, ψ generate the SO(10) gauge group, and {¯y3,...,6, η¯1}, {¯y1, y¯2, ω¯ 5, ω¯ 6, η¯2}, {¯ω1,...,4, η¯3} generate the three horizontal SO(6) symmetries. The left-moving {y,ω} states are divided into {y3,...,6}, {y1,y2,ω5,ω6}, {ω1,...,4} and χ12, χ34, χ56 generate the left-moving N = 2 world-sheet supersymmetry. At the level of the NAHE set the sectors b1, b2 and b3 produce 48 multiplets, 16 from each, in the 16 representation of SO(10). The states from the sectors bj are singlets of the hidden E8 gauge group and transform under the horizontal SO(6)j (j = 1, 2, 3) symmetries. This structure is common to a large set of realistic free fermionic models. The second stage of the construction consists of adding to the NAHE set three (or four) additional basis vectors. These additional vectors reduce the number of generations to three, one from each of the sectors b1, b2 and b3, and simultaneously break the four-dimensional gauge group. The assignment of boundary conditions to {ψ¯ 1,...,5} breaks SO(10) to one of its subgroups [11]. 3 Similarly, the hidden E8 symmetry is broken to one of its subgroups, and the flavor SO(6) symmetries are broken to U(1)n, with 3 n 9. For details and phenomenological studies of these three generation string models interested readers are referred to the original literature and review articles [23]. The correspondence of the free fermionic models with the orbifold construction [24] is facilitated by extending the NAHE set, {1,S,b1,b2,b3}, by at least one additional boundary condition A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108 87 basis vector [14] = | ξ1 (0,...,0 1,..., 1 , 0,...,0). (2.7) ψ¯ 1,...,5,η¯1,2,3 3 With a suitable choice of the GSO projection coefficients the model possesses an SO(4) × E6 × 2 U(1) × E8 gauge group and N = 1 space–time supersymmetry. The matter fields include 24 generations in the 27 representation of E6, eight from each of the sectors b1 ⊕b1 +ξ1, b2 ⊕b2 +ξ1 and b3 ⊕ b3 + ξ1. Three additional 27 and 27 pairs are obtained from the Neveu–Schwarz (NS) ⊕ ξ1 sector, that correspond to the untwisted sector of the orbifold models. To construct the model in the orbifold formulation one starts with the compactification on a torus with nontrivial background fields [25]. The subset of basis vectors

{1,S,ξ1,ξ2} (2.8) generates a toroidally-compactified model with N = 4 space–time supersymmetry and SO(12) × E8 ×E8 gauge group. The same model is obtained in the geometric (bosonic) language by tuning the background fields to the values corresponding to the SO(12) lattice. The metric of the six- dimensional compactified manifold is then the Cartan matrix of SO(12), while the antisymmetric tensor is given by   Gij ,i>j, = 0,i= j, Bij  (2.9) −Gij ,i

Table 1 ψµ χ12 χ34 χ56 ψ¯ 1,...,5 η¯1 η¯2 η¯3 φ¯1,...,8 α 1 1 0 0 11100 1 0 0 11000000 β 1 0 1 0 11100 0 1 0 11000000 1 1 1 1 1 1 1 1 1 1 1 1 γ 1001 2 2 2 2 2 2 2 2 2 2 2 2 1000

y3y¯3 y4y¯4 y5y¯5 y6y¯6 y1y¯1 y2y¯2 ω5ω¯ 5 ω6ω¯ 6 ω1ω¯ 1 ω2ω¯ 2 ω3ω¯ 3 ω4ω¯ 4 α 1001001 0 0 0 0 1 β 0001011 0 1 0 0 0 γ 1100100 0 0 1 0 0

This choice also projects out the massless vector bosons in the 128 of SO(16) in the hidden- sector E8 gauge group, thereby breaking the E6 × E8 symmetry to SO(10) × U(1) × SO(16). However, the Z2 × Z2 twist acts identically in the two models, and their physical characteristics differ only due to the discrete torsion Eq. (2.11). This analysis confirms that the Z2 × Z2 orbifold on the SO(12) lattice is at the core of the realistic free fermionic models. The set of real fermions {y,ω|¯y,ω¯ } correspond to the six-dimensional compactified coordinates in a bosonic formulation. The assignment of boundary conditions to this set of internal fermions therefore correspond to the action on the internal six-dimensional compactified manifold. Consequently, the assignment of boundary conditions to this set of real fermions determines many of the phenomenological properties of the low energy spectrum and effective field theory. Examples include the determination of the number of light generations [8,13]; the stringy doublet–triplet splitting mechanism [26]; and the top–bottom-quark Yukawa coupling selection mechanism [27]. In the last two cases, it is the left–right asymmetric boundary conditions that enables the doublet–triplet splitting, as well as the Yukawa coupling selection mechanism [26, 27]. In this paper I discuss how the asymmetric assignment of boundary conditions to the set of internal world-sheet fermions {y,ω|¯y,ω¯ }, also fixes the untwisted moduli. It is therefore found that the same condition, namely the left–right orbifold asymmetry, is the one that plays the crucial role, both in the determination of the physical properties mentioned above, as well as in the fixation of the moduli parameters. The symmetric versus asymmetric orbifold action is determined in the free fermionic models by the pairing of the left–right real internal fermions from the set {y,ω|¯y,ω¯ }, into real pairs, that pair a left-moving fermion with a right-moving fermion, versus complex pairs, that combine left– left, or right–right, moving fermions. The real pairs preserve the left–right symmetry, whereas the complex pairs allow for the assignment of asymmetric boundary conditions, that correspond to asymmetric action on the compactified bosonic coordinates. The reduction of the number of generations to three is illustrated in Tables 1 and 2. together with a consistent choice of one-loop GSO phases [9]. Both models produce three generations and the SO(10) GUT gauge group is broken to SU(3) × SU(2) × U(1)2. The model of Table 1 produces three pairs of SU(3) color triplets from the untwisted sector, whereas that of Table 2 produces the corresponding three pairs of electroweak Higgs doublets [9,26]. The set of boundary conditions in Table 1 is symmetric between the left and right movers, whereas that of Table 2 is asymmetric. Hence, these two models demonstrate that the reduction to three generations in itself does not require asymmetric boundary conditions. This seems to be in contradiction to the conclusion reached in [15]. However, I note that in Table 1 the gauge group is broken in two stages, whereas in [15] the SO(10) symmetry remained unbroken. In particular, the breaking pat- A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108 89

Table 2 ψµ χ12 χ34 χ56 ψ¯ 1,...,5 η¯1 η¯2 η¯3 φ¯1,...,8 α 0 0 0 0 11100 0 0 0 11110000 β 0 0 0 0 11100 0 0 0 11110000 1 1 1 1 1 1 1 1 1 1 1 1 γ 00 0 0 2 2 2 2 2 2 2 2 2 011 2 2 2 0

y3y6 y4y¯4 y5y¯5 y¯3y¯6 y1ω5 y2y¯2 ω6ω¯ 6 y¯1ω¯ 5 ω2ω4 ω1ω¯ 1 ω3ω¯ 3 ω¯ 2ω¯ 4 α 1000001 10 0 1 1 β 0011100 00 1 0 1 γ 0101010 11 0 0 0 tern of the hidden gauge group SO(16) → SO(4) × SO(12) in Table 1 was a case not considered in the classification of [15]. This breaking pattern is, however, allowed if the SO(10) GUT symmetry is broken to SO(6) × SO(4),asinTable 1. The conclusion is therefore that, whereas the classification of Ref. [15] found that the reduction to three generations necessitates the use of an asymmetric shift in the scanned space of models, more complicated symmetry breaking patterns allow for the reduction also with symmetric actions. Hence, it is concluded that the reduction to three generations in itself does not necessitate the use of an asymmetric shift, in agreement with the findings of Ref. [28]. However, the distinction between the symmetric versus asymmetric boundary conditions is manifested in the models of Table 1 versus Table 2 at the more detailed level of the spectrum and the phenomenological consequences. Hence, the world-sheet left–right symmetric model of Table 1 produces SU(3) color triplets and does not give rise to untwisted– twisted–twisted Standard Model fermion mass terms, whereas the asymmetric model of Table 2 does. As I discuss in the following this distinction has a crucial implication also for the issue of moduli fixing in these models. To translate the fermionic boundary conditions to twists and shifts in the bosonic formulation we bosonize the real fermionic degrees of freedom, {y,ω|¯y,ω¯ }. Defining, 1 1 − ξ = (y + iω ) = eiXi ,η= i (y − iω ) = ie iXi (2.12) i 2 i i i 2 i i {¯ ¯ } I ¯ = I + I ¯ with similar definitions for the right movers y,ω and X (z, z) XL(z) XR(z). With these definitions the world-sheet supercurrents in the bosonic and fermionic formulations are equivalent, int = = = TF χiyiωi χiξiηi i χi∂Xi. i i i The momenta P I of the compactified scalars in the bosonic formulation are identical with the U(1) charges Q(f ) of the unbroken Cartan generators of the four-dimensional gauge group, Eq. (2.6). The internal coordinates can be complexified by forming the combinations (X = XL + XR) ± = 2k−1 ± 2k ± = 2k−1 ± 2k = Zk X iX ,ψk χ iχ (k 1, 2, 3), (2.13) ± where the Zk are the complex coordinates of the six compactified dimensions, now viewed as ± three complex planes, and ψk are the corresponding superpartners. 90 A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108

3. Moduli in free fermionic models

The relevant moduli for our discussion here are the untwisted Kähler and complex structure moduli of the six-dimensional compactified manifold. Additionally, the string vacua contain the dilaton moduli whose VEV governs the strength of the four-dimensional interactions. The VEV of the dilaton moduli is a continuous parameter from the point of view of the perturbative heterotic string, and its stabilization requires some nonperturbative dynamics, or some input from the underlying quantum M-theory, which is not presently available. The problem of dilaton stabilization is therefore not addressed in this paper, as the discussion here is confined to perturbative heterotic string vacua. Since the moduli fields correspond to scalar fields in the massless string spectrum, the moduli space is determined by the set of boundary condition basis vectors that define the string vacuum and encodes its properties. The first step therefore is to identify the fields in the fermionic models that correspond to the untwisted moduli. The subsequent steps entail examining which moduli fields survive successive GSO projections and consequently the residual moduli space. The four-dimensional fermionic heterotic string models are described in terms of two- dimensional conformal and superconformal field theories of central charges CR = 22 and CL = 9, respectively. In the fermionic formulation these are represented in terms of world-sheet fermions. A convenient starting point to formulate such a fermionic vacuum is a model in which all the fermions are free. The free fermionic formalism facilitates the solution of the conformal and modular invariance constraints in terms of simple rules [17]. Such a free fermionic model corresponds to a string vacuum at a fixed point in the moduli space. Deformations from this fixed point are then incorporated by including world-sheet Thirring interactions among the world- sheet fermions, that are compatible with the conformal and modular invariance constraints. The coefficients of the allowed world-sheet Thirring interactions correspond to the untwisted moduli fields. For symmetric orbifold models, the exactly marginal operators associated with the untwisted moduli fields take the general form ∂XI ∂X¯ J , where XI , I = 1,...,6, are the coordinates of the six-torus T 6. Therefore, the untwisted moduli fields in such models admit the geometrical interpretation of background fields [25], which appear as couplings of the exactly marginal operators in the nonlinear sigma model action, which is the generating functional for string scattering amplitudes [29,30]. The untwisted moduli scalar fields are the background fields that are compatible with the orbifold point group symmetry. It is noted that in the Frenkel–Kac–Segal construction [31] of the Kac–Moody current algebra from chiral bosons, the operator i∂XI is a U(1) generator of the Cartan subalgebra. Therefore, in the fermionic formalism the exactly marginal operators are given by Abelian Thirring oper- i ¯j ¯ i ¯j ¯ ators of the form JL(z)JR(z), where JL(z), JR(z) are some left- and right-moving U(1) chiral currents describe by world-sheet fermions. It has been shown that [18] Abelian Thirring interactions preserve conformal invariance, and are therefore marginal operators. One can therefore use the Abelian Thirring interactions to identify the untwisted moduli in the free fermionic models. The untwisted moduli correspond to the Abelian Thirring interactions that are compatible with the GSO projections induced by the boundary condition basis vectors, which define the string models. The set of Abelian Thirring operators, and therefore of the untwisted moduli fields, is consequently restricted by progressive boundary condition basis vector sets. The minimal basis set is given by the basis that contains only the two vectors {1,S}. 1 = ψ1,2,χ1,...,6,y1,...,6,ω1,...,6y¯1,...,6, ω¯ 1,...,6, ψ¯ 1,...,6, η¯1,2,3, φ¯1,...,8 , (3.1) A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108 91 S = ψ1,2,χ1,...,6 . (3.2) This set generates a model with N = 4 supersymmetry and SO(44) right-moving gauge group and correspond the Narain model at an enhanced symmetry point. Accordingly, we can identify the six χI with the fermionic superpartners of the six compactified bosonic coordinates. Therefore, each pair {yI ,ωI } is identified with the fermionized version of the corresponding left- I I ∼ I I moving bosonic coordinate X , i.e., i∂XL y ω . The two-dimensional action for the Abelian Thirring interactions is = 2 i ¯j ¯ S d zhij (X)JL(z)JR(z), (3.3)

i = 6 ¯j = where JL (i 1,...,6) are the chiral currents of the left-moving U(1) and JR (j 1,...,22), 22 are the chiral currents of the right-moving U(1) . The couplings hij (X), as functions of the space–times coordinates Xµ, are four-dimensional scalar fields that are identified with the scalar components of the untwisted moduli fields. In the simplest model with the two basis vectors of Eq. (3.2) the 6×22 fields hij (X) in Eq. (3.3) are in one-to-one correspondence with the 21 and 15 components of the background metric GIJ and antisymmetric tensor BIJ (I, J = 1,...,6),plus the 6 × 16 Wilson lines AIa.Thehij (X) fields therefore parameterize the SO(6, 22)/SO(6) × SO(22) coset-space of the toroidally compactified manifold. The hij untwisted moduli fields arise from the Neveu–Schwarz sector, I ⊗ ¯ +J ¯ −J χ L Φ Φ R, (3.4) given in terms of the 22 complex right-handed world-sheet fermions Φ¯ +J and their complex conjugates Φ¯ −J . The corresponding marginal operators are given as i ¯j ¯ ≡: i i :: ¯ +j ¯ ¯ −j ¯ : JL(z)JR(z) y (z)ω (z)(z) Φ (z)Φ (z) . (3.5) It is noted that the transformation properties of χi , which appear in the moduli (3.4),arethesame as those of yiωi , which appear in the Abelian Thirring interactions (3.5). The next stages in the free fermionic model building consist of adding consecutive boundary condition basis vectors. In the first instance we can add basis vectors that preserve the N = 4 space–time supersymmetry, and with no periodic left-moving fermions. Those that produce massless states may contain either four, or eight, right-moving periodic fermions. At least one basis vector with eight periodic right-moving fermions is needed to produce space–time spinors under the GUT gauge group. The free fermionic models correspond to Z2 × Z2 orbifolds and can then be classified into models that produce spinorial representations from one, two, or three of the twisted planes of the Z2 ×Z2 orbifold [15]. It turns out that the entire space of models may be classified using a specific set of basis vectors, and the varying cases are produced by the various choices of the GSO projection coefficients. This covering basis contains two basis vectors with eight nonoverlapping right-moving periodic fermions, and all left-moving world-sheet fermions are antiperiodic. Hence, with no loss of generality, these are the vector ξ1 in Eq. (2.7) and the vector ξ2 = | ξ2 (0,...,0 0,...,0, 1,..., 1), (3.6) φ¯1,...,8 where the notation introduced in Section 2 has been used. We note that all the untwisted moduli and marginal operators in Eqs. (3.4) and (3.5) are invariant under the projections induced by (2.7) and (3.6). The four-dimensional right-moving gauge group is now SO(12) × E8 × E8. 92 A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108

The next step in the construction is the inclusion of the basis vectors b1 and b2 that correspond to the action of the Z2 × Z2 orbifold twists. The assignment of boundary condition in b1 and b2 may vary [15], and one specific choice is given by = | b1 ( 1,..., 1 , 0,...,0 1,..., 1, 0,...,0), (3.7) ψµ,χ12,y3,...,6,y¯3,...,6 ψ¯ 1,...,5,η¯1 = | b2 ( 1,..., 1 , 0,...,0 1,..., 1, 0,...,0). (3.8) ψµ,χ34,y1,2,ω5,6,y¯1,2ω¯ 5,6 ψ¯ 1,...,5,η¯2 In the notation of Section 2 we can write the untwisted moduli fields, Eq. (3.4) in the form χi ⊗ y¯j ω¯ j , (3.9) L R i ⊗ ¯ +J ¯ −J χ L Φ Φ R, (3.10) and the (3.5) i ¯j ¯ ≡: i i :: ¯j ¯ j : = JL(z)JR(z) y (z)ω (z) y (z)ω (z) (j 1,...,6), (3.11) i ¯J ¯ ≡: i i :: ¯ +J ¯ ¯ −J ¯ : = JL(z)JR (z) y (z)ω (z) Φ (z)Φ (z) (J 7,...,22). (3.12) i The effect of the additional basis vectors b1 and b2 is to make some of the chiral currents, JL ¯j or JR, antiperiodic. As a result some of the Abelian Thirring interaction terms in Eq. (3.3) are not invariant when the world-sheet fermions are parallel transported around the noncontractible loops of the world-sheet torus. The corresponding moduli fields are projected from the massless spectrum by the GSO projections. Under the basis vector b1 as defined above the chiral currents transform as 1,2 → 1,2 3,4,5,6 →− 3,4,5,6 JL JL ,JL JL , (3.13) ¯1,2 → ¯1,2 ¯3,4,5,6 →−¯3,4,5,6 JR JR , JR JR (3.14) ¯j = and JR (j 7,...,22) are always periodic. Similarly, under b2 we have 3,4 → 3,4 1,2,5,6 →− 1,2,5,6 JL JL ,JL JL , (3.15) ¯1,2 → ¯3,4 ¯1,2,5,6 →−¯1,2,5,6 JR JR , JR JR . (3.16) As a consequence the only allowed Thirring interaction terms are

1,2 ¯1,2 3,4 ¯3,4 5,6 ¯5,6 JL JR ,JL J ,JL JR . (3.17) Correspondingly there are three sets of untwisted moduli scalars (i, j = 1, 2), = i ⊗ ¯j ¯ j = = hij χ L y ω R (i, j 3, 4), (3.18) (i, j = 5, 6), which parameterize the moduli space SO(2, 2) 3 M = . (3.19) SO(2) × SO(2) A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108 93

We can deﬁne six complex moduli from the six real ones. For example, on the ﬁrst complex plane we can write (1) 1 1 1 2 1 1 H = √ (h11 + ih21) = √ χ + iχ ⊗ y¯ ω¯ , (3.20) 1 2 2 L R (1) 1 1 1 2 2 2 H = √ (h12 + ih22) = √ χ + iχ ⊗ y¯ ω¯ . (3.21) 2 2 2 L R

These can be combined into three Kähler (Ti ) structure and three complex structure (Ui ) moduli. For example, on the first complex plane we define 1 (1) (1) 1 1 1 1 2 2 T1 = √ H − iH = √ χ + iχ2 |0L ⊗ y¯ ω¯ − iy¯ ω¯ |0R, (3.22) 2 1 2 2 1 (1) (1) 1 1 1 1 2 2 U1 = √ H + iH = √ χ + iχ2 |0L ⊗ y¯ ω¯ + iy¯ ω¯ |0R (3.23) 2 1 2 2 and similarly for T2,3 and U2,3. These span the coset moduli space SU(1, 1) SU(1, 1) 3 M = ⊗ , (3.24) U(1) U(1) which is the untwisted moduli space of the symmetric Z2 × Z2 orbifold. We can write the allowed Thirring interaction terms, for the untwisted moduli of (3.18), in terms of the complex ± coordinates Zk . For example, for the first set we have 1 − + + − − − + + h J i J¯j = T ∂Z ∂Z¯ + T¯ ∂Z ∂Z¯ + U ∂Z ∂Z¯ + U¯ ∂Z ∂Z¯ , (3.25) ij L R 2 1 1 1 1 1 1 1 1 1 1 1 i,j=1,2 where T and U are the complex fields defined in Eqs. (3.22) and (3.23). The Thirring interactions for the other two complex planes can be written similarly. The boundary condition basis vectors b1 and b2 translate into twists of the complex planes [14]. Thus, b1 leaves the first complex plane invariant, and twists the second and third plane, i.e., Z1 → Z1 and Z2,3 →−Z2,3, whereas b2 leaves the second plane invariant and twists the first and third plane. It is apparent that the Thirring terms in Eq. (3.25) are invariant under the action of the Z2 × Z2 twists, whereas all the mixed terms are projected out by the GSO projections. From Eq. (3.25) it is seen that the T field is associated the Kähler structure moduli, whereas the U field is associated with a complex structure moduli. This agrees with the fact that the untwisted sector of the Z2 × Z2 orbifold produces three complex and three Kähler structure moduli.

4. Moduli in the three generation free fermionic models

Next I turn to examine the untwisted moduli in the quasi-realistic three generation free fermionic models. As discussed in Section 2 the reduction to three generation is achieved by the inclusion of three or four additional boundary condition basis vectors, beyond the NAHE-set. This are typically denoted in the literature as bi (i = 4, 5,...) for basis vectors that do not break the SO(10) GUT symmetry, and by α,β,γ,..., for basis vectors that do. From the point of view of the untwisted moduli beyond the NAHE-set models, the relevant boundary conditions are those of the world-sheet fermions {y,ω|¯y,ω¯ }1,...,6. Left–right symmetric boundary conditions cannot eliminate any additional moduli beyond those that exist in the NAHE-set model. This is a general consequence of the requirement that 94 A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108

Table 3 y1ω1 y¯1ω¯ 1 y2ω2 y¯2ω¯ 2 y3ω3 y¯3ω¯ 3 y4ω4 y¯4ω¯ 4 y5ω5 y¯5ω¯ 5 y6ω6 y¯6ω¯ 6 α 00 00 00 00 10 10 01 01 01 01 10 10 β 01 01 10 10 00 00 00 00 01 01 10 10 γ 10 10 01 01 10 10 10 10 00 00 00 00 the supercurrent (2.1) is well defined under the parallel transport of the world-sheet fermions, as well as the requirement of N = 1 space–time supersymmetry. It is instructive to recall that the moduli of the NAHE-based models correspond to the nonvanishing world-sheet Thirring interactions, Eq. (3.17), and are those of the Z2 × Z2 orbifold, Eq. (3.25). An example of a left–right three generation free fermionic model is given in Table 1. Rewriting the boundary conditions for the internal world-sheet fermions in the notation of Table 3 it is evident that all the Thirring interaction terms in Eq. (3.17) are invariant under the transformations in Table 3. As discussed in [20] and Section 3 there is a one-to-one correspondence between the Thirring terms, Eq. (3.17) and the moduli fields, Eq. (3.18). Therefore, it is sufficient to examine either the moduli or the Thirring terms, and this relation guarantees the existence, or exclusion, of the corresponding Thirring terms/moduli fields. Left–right symmetric boundary conditions preserve the untwisted moduli space of the Z2 ×Z2 orbifold. To see that this is indeed a general result, and a consequence of world-sheet, and space– time, supersymmetry, let us recall that the world-sheet supercurrent Eq. (2.1) restricts that each µ triplet χiyiωi must transform as ψ ∂Xµ. The boundary conditions of each triplets are therefore restricted to be b ψµ = 1 → b(χ ,y ,ω ) = (1, 0, 0); (0, 1, 0); (0, 0, 1); (1, 1, 1), i i i µ b ψ = 0 → b(χi,yi,ωi) = (0, 1, 1); (1, 0, 1); (1, 1, 0); (0, 0, 0). (4.1) The requirement of N = 1 supersymmetry further restricts that

b(χ2k−1) = b(χ2k), k = 1, 2, 3. (4.2)

Symmetric boundary conditions means that the boundary condition of the left-moving b{yi,ωi} are identical to their corresponding right-moving fields b{¯yi, ω¯ i}. In terms of the internal conformal field theory this means that symmetric boundary conditions correspond to the case in which all the internal left-moving world-sheet fermions from the set {y,ω}1,...,6 are paired with 12 right-moving world-sheet real fermions {¯y,ω¯ }1,...,6, to form 12 Ising model operators [21, 22]. Enumerating all the possible boundary conditions for the internal fermions, subject to these restrictions we have   0000  µ = = = → { } { } = 0011 b ψ 1&b(χ2k−1 χ2k) 1 b y,ω 2k−1, y,ω 2k   1100 1111   1010  µ = = = → { } { } = 1001 b ψ 1&b(χ2k−1 χ2k) 0 b y,ω 2k−1, y,ω 2k   0110 0101 A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108 95   1010  µ = = = → { } { } = 1001 b ψ 0&b(χ2k−1 χ2k) 1 b y,ω 2k−1, y,ω 2k   0110 0101   0000  µ = = = → { } { } = 0011 b ψ 0&b(χ2k−1 χ2k) 0 b y,ω 2k−1, y,ω 2k  (4.3)  1100 1111 i.e., there are in total four different possibilities for the transformation of the pairs {yi,ωi} b {yi,ωi} = (0, 0); (1, 0); (0, 1); (1, 1). (4.4) Using the relations (2.12) these translate to transformations of the corresponding bosonic coordinates, i.e., Table 4. In terms of the complexified coordinates Zk,Eq.(2.13) these translate into

Zk →±Zk + δ12π + δ2i2π, (4.5) where δ1,2 = 0; 1 signify possible shifts in the real and complex directions. Examining the complexiﬁed Thirring terms, Eq. (3.25), it is noted that these always involve pairs of complex coordinates that transform with the same sign, and hence are always invariant under symmetric boundary conditions. Relaxing the condition of N = 1 space–time supersymmetry means that we may have

b(χ2k−1) = b(χ2k) (4.6) for some k. For example, for k = 1 we may have y ω y ω 1 1 2 2 . (4.7) 1110

In this case the moduli ﬁelds h11 and h22 are retained, whereas h12 and h21 are projected out. Hence, if we break N = 1 space–time supersymmetry by the assignment of boundary conditions we may project six additional untwisted moduli, and six will remain. Next, I turn to examine the presence of untwisted moduli in models with asymmetric boundary conditions. The assignment of asymmetric boundary conditions entails that a left-moving real fermion from the set {y,ω} is paired with another left-moving real fermion from this set, and with which it has identical boundary conditions in all basis vectors. For every such pair of left- moving fermions, there is a corresponding pair of right-moving fermions. These combinations therefore give rise to a global U(1)L symmetry, and a corresponding gauged U(1)R symmetry. These complex pairings therefore allow the assignment of asymmetric boundary conditions, with important phenomenological consequences [11,13,26,27]. The simplest case is that of Table 5 which involves a single such complexiﬁed fermion.

Table 4

y,ω XL XR X = XL + XR 00 XL + πXR + πX+ 2π 10 −XL −XR −X 01 −XL + π −XR + π −X + 2π 11 XL XR X 96 A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108

Table 5 ψµ χ12 χ34 χ56 ψ¯ 1,...,5 η¯1 η¯2 η¯3 φ¯1,...,8 α 1 1 0 0 11100 1 0 1 11110000 β 1 0 1 0 11100 0 1 1 11110000 1 1 1 1 1 1 1 1 1 1 1 1 γ 1001 2 2 2 2 2 2 2 2 2 011 2 2 2 0

y3y¯3 y4y¯4 y5y¯5 y6y¯6 y1y¯1 y2y¯2 ω5ω¯ 5 ω6ω¯ 6 ω2ω3 ω1ω¯ 2 ω4ω¯ 4 ω¯ 2ω¯ 3 α 1001001 0 0 0 1 1 β 0001011 0 0 1 0 1 γ 1100110 0 0 0 0 1

Table 6

Z1 Z2 Z3 + L − R + + − L + R − + − + αX1 i(X2 X2 ) 2π i2π X3 X3 iX4 i2π Z3 2π − + − L + R + L − R − + + − + β X1 i( X2 X2 ) 2πX3 X3 iX4 2π i2π Z3 2π − + − L + R − L + R − + + γ X1 i( X2 X2 ) X3 X3 iX4 Z3 2π i2π

With some choice of GSO projection coefﬁcients. The model of Table 5 was published in [11], and the entire spectrum is given there. In this model it is easily seen that the moduli ﬁelds

h11,h21,h34,h44,h55,h56,h65,h66, (4.8) and their corresponding Thirring terms, are retained in the spectrum, whereas the moduli ﬁelds,

h12,h22,h33,h43, (4.9) are projected out, and the corresponding Thirring terms are not invariant under the transformations. Using the bosonic coordinates Eq. (2.13) we can translate the transformations of the internal fermions to the corresponding action on bosonic variables. In terms of = + = L + R + L + R = Zi X2i−1 iX2i X2i−1 X2i−1 i X2i X2i (i 1, 2, 3), (4.10) we have Table 6. From Table 6 we see that the complex structure of two of the complex planes is broken, whereas the third is retained. Furthermore, the transformations with respect to two of the internal bosonic coordinates are asymmetric between the left- and the right-moving part, and are therefore not geometrical. This entails that the corresponding moduli must be projected out, and the coordinates are frozen at fixed radii. Indeed, using the identity (3.25), and the definitions of the complex moduli fields in Eqs. (3.20)–(3.23) we relate the projection of the moduli fields hij to constraints on the complex and Kähler moduli of the Z2 × Z2 orbifold, Ui and Ti (i = 1, 2, 3). Thus, in the case of the model of Table 5, the projection of the h11, h22, h33, h43 moduli fields translate to the conditions

T1 = U1,T2 =−U2 whereas T3 = U3 are unconstrained. (4.11) The next example is a model with two complexiﬁed left- and right-moving internal fermions from the set {y,ω|¯y,ω¯ }. Table 7 provides an example of such a model. With some choice of generalized GSO coefﬁcients. The model of Table 7 produces three generations from the twisted sector b1, b2 and b3, and a standard-like observable gauge group. A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108 97

Table 7 ψµ χ12 χ34 χ56 ψ¯ 1,...,5 η¯1 η¯2 η¯3 φ¯1,...,8 α 0 0 0 0 11100 0 0 1 11110000 β 0 0 0 0 11100 0 0 1 11110000 1 1 1 1 1 1 1 1 1 1 1 1 γ 00 0 0 2 2 2 2 2 2 2 2 2 2 000 2 2 1

y3y6 y4y¯4 y5y¯5 y¯3y¯6 y1ω¯ 5 y2y¯2 ω6ω¯ 6 y¯1ω¯ 5 ω1ω¯ 1 ω2ω¯ 2 ω3ω¯ 3 ω4ω¯ 4 α 1000001 10 0 1 0 β 0011100 01 0 0 0 γ 0101000 10 0 0 1

Table 8

Z1 Z2 L − R + + + L − R + + + α(X1 X1 ) iX2 (1 2i)π (X3 X3 ) iX4 (1 2i)π L − R + + + L − R + + + β(X1 X1 ) iX2 (1 2i)π (X3 X3 ) iX4 (1 2i)π L − R + + + L − R + + γ(X1 X1 ) iX2 (1 2i)π (X3 X3 ) iX4 π

Table 9

Z3 L − R + L − R + + α(X5 X5 ) i(X6 X6 ) (2 i)π L − R + L − R + β(X5 X5 ) i(X6 X6 ) iπ L − R + L − R + + γ(X5 X5 ) i(X6 X6 ) (2 i)π

It produces electroweak super-Higgs doublets from the ﬁrst two untwisted planes, and a color triplet from the third. In this model it is easily seen that the moduli ﬁelds,

h12,h22,h34,h44, (4.12) and their corresponding Thirring terms, are retained in the spectrum, whereas the moduli ﬁelds,

h11,h22,h33,h43,h55,h56,h65,h66, (4.13) are projected out, and the corresponding Thirring terms are not invariant under the transformations. In terms of the complex bosonic coordinates (4.10) we have on the ﬁrst two complex planes (Table 8) and on the third (Table 9) the projection of the moduli ﬁelds in Eq. (4.13) translate in this case to the conditions

T1 =−U1,T2 =−U2 whereas T3 = U3 = 0. (4.14) It is seen here that the third complex plane is completely ﬁxed, whereas on the ﬁrst and second planes X2 and X4 retain their geometrical character, and X1 and X3 do not. From the two examples above, and Eq. (3.25) we can draw the general constraints on the complex and Kähler moduli, which can be written for k = 1, 2, 3as

Uk = h2k−12k−1 + h2k 2k + i(h2k 2k−1 − h2k−12k), (4.15)

Tk = h2k−12k−1 − h2k 2k + i(h2k 2k−1 + h2k−12k). (4.16) 98 A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108

Table 10 ψµ χ12 χ34 χ56 ψ¯ 1,...,5 η¯1 η¯2 η¯3 φ¯1,...,8 b4 1 1 0 0 11111 1 0 0 00000000 α 1 0 0 1 11100 1 0 1 11110000 1 1 1 1 1 1 1 1 1 1 1 1 β 1010 2 2 2 2 2 2 2 2 2 011 2 2 2 1

y3y6 y4y¯4 y5y¯5 y¯3y¯6 y1ω6 y2y¯2 ω5ω¯ 5 y¯1ω¯ 6 ω1ω3 ω2ω¯ 2 ω4ω¯ 4 ω¯ 1ω¯ 3 α 1001001 00 0 1 0 β 0001010 11 0 1 0 γ 0011100 10 1 0 0

From the identity Eq. (3.25), the vanishing of certain real Thirring terms on the left-hand side, translate to the conditions

Tk + Uk = 0 ⇔ h2k−12k−1 & h2k 2k−1 are projected out, (4.17)

Tk − Uk = 0 ⇔ h2k 2k & h2k−12k are projected out. (4.18)

Hence, if both Tk + Uk = 0 and Tk − Uk = 0 hold, then both the Kähler and complex structure moduli of the kth plane are projected out, and the radii and angles of the corresponding internal torus are frozen. In this case there is no extended geometry in the effective low energy field theory. This situation is similar to the manner in which gauge symmetries are broken in string theory by the GSO projections. Namely, the gauge symmetry is not realized in the effective low energy field theory, but exists as an organizing principle at the string theory level. That is parts of the string spectrum obeys the symmetry, but the entire string spectrum does not. The above two examples demonstrates the interesting possibility of correlating between the number of complexified fermions and the surviving untwisted moduli, which would suggest that for every complex fermion, four additional untwisted moduli are projected out. However, in the following I show that this is not necessarily the case, and the situation is more intricate. The retention or projection of untwisted moduli in the case with three complex fermions depends on the choice of pairings of the left-moving real fermions from the set {y,ω}1,...,6.The distinction between the different choices of pairings, and some phenomenological consequences, was briefly discussed in Ref. [13]. To demonstrate the effect on the untwisted moduli I consider the two models in Tables 2 [9] and 10 [6]. The choice of generalized GSO projection coefficients of the model of Table 10,aswellas the complete mass spectrum and its charges under the four-dimensional gauge group are given in Ref. [6]. One of the important constraints in the construction of the free fermionic models is the requirement that the supercurrent, Eq. (2.1), is well defined. In the models that utilize only periodic and anti-periodic boundary conditions for the left-moving sector, the eighteen left-moving fermions are divided into six triplets in the adjoint representation of the automorphism group SU(2)6. These triplets are denoted by {χi,yi,ωi}, i = 1,...,6, and their boundary conditions are constrained as given in Eq. (4.1). In the type of models that are considered here a pair of real fermions which are combined to form a complex fermion or an Ising model operator must have the identical boundary conditions in all sectors. In practice it is sufficient to require that a pair of such real fermions have the same boundary conditions in all the boundary basis vectors which span a A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108 99

Table 11

Z1 Z2 L − R + L − R + + L − R + L − R + + α(X1 X1 ) i(X2 X2 ) (1 2i)π (X3 X3 ) i(X4 X4 ) (1 2i)π L − R + L − R + + L − R + L − R + + β(X1 X1 ) i(X2 X2 ) (1 2i)π (X3 X3 ) i(X4 X4 ) (1 2i)π L − R + L − R + L − R + L − R + γ(X1 X1 ) i(X2 X2 ) π(X3 X3 ) i(X4 X4 ) π

Table 12

Z3 L − R + L − R + + α(X5 X5 ) i(X6 X6 ) (2 i)π L − R + L − R + β(X5 X5 ) i(X6 X6 ) iπ L − R + L − R + + γ(X5 X5 ) i(X6 X6 ) (2 i)π given model. In the model of Table 2 the pairings are y3y6,y4y¯4,y5y¯5, y¯3y¯6 , y1ω6,y2y¯2,ω5ω¯ 5, y¯1ω¯ 6 , ω2ω4,ω1ω¯ 1,ω3ω¯ 3, ω¯ 2ω¯ 4 , (4.19) where the notation emphasizes the original division of the world-sheet fermions by the NAHE- set [8,13]. Note that with this pairing the complexified left-moving pairs mix between the six SU(2) triplets of the left-moving automorphism group. That is the boundary condition of y3y6 fixes the boundary condition of y1ω5. In this model we find that under the α projection 1,...,6 →− 1,...,6 ¯1,...,6 → ¯1,...,6 JL JL , JR JR . (4.20) As a result all of the Thirring terms, and hence all the untwisted moduli are projected out in this model. In terms of the complex bosonic coordinates (4.10) the fermionic boundary conditions on the first two complex planes translate to Table 11 and on the third (Table 12). Hence, the boundary conditions in this case correspond to asymmetric action on all six real internal coordinates. In terms of the Kähler and complex structure fields we have that the projection of the 12 real hij translate to

Tk + Uk = 0 and Tk − Uk = 0fork = 1, 2, 3. (4.21) Therefore,

Tk = Uk = 0 and, and all the untwisted geometrical moduli are projected out in this model. On the other hand the pairings in the model of Table 10 are y3y6,y4y¯4,y5y¯5, y¯3y¯6 , y1ω6,y2y¯2,ω5ω¯ 5, y¯1ω¯ 6 , ω1ω3,ω2ω¯ 2,ω4ω¯ 4, ω¯ 1ω¯ 3 . (4.22) Note that with this pairing the complexified left-moving pairs in Eqs. (4.22) mix between the first, third and sixth SU(2) triplets of the left-moving automorphism group. In this model it is easily seen that the moduli fields,

h11,h12,h21,h22,h34,h44,h55,h65 (4.23) 100 A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108

Table 13

Z1 Z2 αZ1 + (2 + i2)π −Z2 + i2π β −Z1 + π −Z2 + (2 + i)π − + L − R + + + γ Z1 i2π(X3 X3 ) iX4 (1 i2)π

Table 14

Z3 α −Z3 + 2π βZ3 + (2 + i)π − + − L + R + γ X5 i( X6 X6 ) iπ

Table 15 ψµ χ12 χ34 χ56 ψ¯ 1,...,5 η¯1 η¯2 η¯3 φ¯1,...,8 α 0 0 0 0 11100 0 0 0 11110000 β 0 0 0 0 11100 0 0 0 10000001 1 1 1 1 1 1 1 1 1 1 1 1 γ 0000 2 2 2 2 2 2 2 2 2 011 2 2 2 1

y3y6 y4y¯4 y5y¯5 y¯3y¯6 y1ω6 y2y¯2 ω5ω¯ 5 y¯1ω¯ 6 ω1ω3 ω2ω¯ 2 ω4ω¯ 4 ω¯ 1ω¯ 3 α 1000100 01 0 0 0 β 0101000 10 0 1 1 γ 0011001 10 0 0 1 and their corresponding Thirring terms, are retained in the spectrum, whereas the moduli ﬁelds,

h33,h43,h56,h66 (4.24) are projected out, and the corresponding Thirring terms are not invariant under the transformations. In terms of the bosonized variables the boundary conditions translate to Table 13 and on the third (Table 14). Hence, in this case the asymmetric fermionic boundary conditions translate to asymmetric action only on two of the six real bosonic coordinates, whereas the other four retain their geometrical interpretation. In terms of the Kähler and complex structure ﬁelds the projection of the 4 real hij ﬁelds in Eq. (4.24) translate to

T1; U1 are unconstrained,T2 =−U2,T3 = U3. (4.25) Furthermore, with the choice of pairings in Eq. (4.22) we can construct a model in which all the untwisted moduli are retained in the spectrum. An example of such a model is given in Table 15. with a choice of generalized GSO projection coefﬁcients. The model in Table 15 is a variant of the model of Ref. [6]. This model utilizes asymmetric boundary conditions. There are three complexiﬁed internal fermions, which are the same as those of the model of Table 10, and the pairing of the left-moving world-sheet fermions is identical. In terms of its phenomenological characteristics the model is a three generation model, each arising from the sectors b1, b2 and b3. The four-dimensional gauge group is the same as that of Ref. [6]. The model utilizes the doublet– triplet splitting mechanism of Ref. [26], which arises from the asymmetric boundary condition assignments in the basis vectors that break the observable SO(10) → SO(6) × SO(4). Similarly, A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108 101

Table 16 y1ω1 y¯1ω¯ 1 y2ω2 y¯2ω¯ 2 y3ω3 y¯3ω¯ 3 y4ω4 y¯4ω¯ 4 y5ω5 y¯5ω¯ 5 y6ω6 y¯6ω¯ 6 α 11 00 00 00 11 00 00 00 00 00 11 00 β 00 11 00 00 00 11 11 11 00 00 00 11 γ 00 11 00 00 00 11 00 00 11 11 00 11

Table 17

Z1 Z2 Z3 αZ1 + (1 + i2)π Z2 + (1 + i2)π Z3 + (2 + i)π βZ1 + (1 + i2)π Z2 + 2πZ3 + (2 + i)π γZ1 + (1 + i2)π Z2 + (2 + i2)π Z3 + iπ the model yields trilevel Yukawa couplings to the three +2/3 charged quarks, but not to the −1/3 charged quarks, which is a result of the asymmetric boundary condition assignment in the basis vector that breaks SO(10) → SU(5) × U(1). In the model of Table 15 all the untwisted moduli are left in the massless spectrum. It is instructive to rewrite the boundary conditions of the internal fermions in this model in the notation of Table 3 (see Table 16). In this notation it is apparent that despite the utilization of asymmetric boundary conditions, the specific pairing of world-sheet fermions allows the retention of all the untwisted moduli. In terms of the bosonized variables the boundary conditions translate to Table 17. From Table 17 it is evident that all three complex planes retain the complex geometry, and hence the three Kähler and three complex moduli remain in the spectrum. In this model the reduction to three generations is attained solely by the shift identifications in the real and complex directions, which are asymmetric in terms of the fermionic boundary conditions, but symmetric in terms of the bosonic variables. The investigation above of the moduli in the three generation free fermionic models is, of course, not exhaustive, but rather illustrative. The moduli of other models in this class may be similarly investigated. I give here a cursory view of several additional models. As illustrated above the determinantal factor in regard to the untwisted moduli is the pairing of the left- and right-moving fermions from the set {y,ω|¯y,ω¯ }1,...,6. In the case of symmetric boundary conditions, with 12 left-moving real fermions combined with 12 real right-moving fermions to form 12 Ising model operators, the moduli fields always remain in the spectrum and the six compactified coordinates maintain their geometrical character. When the real fermions are combined to form complex fermions the situation is more varied, as illustrated above. The determination of the moduli, however, does not depend on the choice of the observable four-dimensional gauge group. To exemplify that I consider the model of Ref. [7], which utilizes the same complexification of the real fermions as that of the model of Table 5. It is then found that the retained and projected moduli in this model are those in Eqs. (4.8) and (4.9), respectively. Of course, there may be a correlation between the assignment of boundary conditions to the real fermions {y,ω}L,R and those that determine the four-dimensional gauge group. Such dependence may arise because of the modular invariance constraints [17]. But typically we may relegate this correlation to the hidden sector, and therefore the observable sector is not affected. In the case of three complex right-moving fermions, and their corresponding left-moving complexified fermions, each pair being associated with a distinct complex planes, we noted in Eqs. (4.19) and (4.22), two cases of pairings, with differing consequences for the untwisted moduli fields. The case of (4.22) was further investigated in the model of Table 15. The moduli 102 A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108

Table 18 ψµ χ12 χ34 χ56 ψ¯ 1,...,5 η¯1 η¯2 η¯3 φ¯1,...,8 α 0 0 0 0 11100 0 0 0 11110000 β 0 0 0 0 11100 0 0 0 11110000 1 1 1 1 1 1 1 1 1 1 1 1 γ 0000 2 2 2 2 2 2 2 2 2 011 2 2 2 0

y3y6 y4y¯4 y5y¯5 y¯3y¯6 y1ω5 y2y¯2 ω6ω¯ 6 y¯1ω¯ 5 ω2ω4 ω1ω¯ 1 ω3ω¯ 3 ω¯ 2ω¯ 4 α 1110111 01 1 1 0 β 0101010 11 0 0 0 γ 0011100 00 1 0 1 with the pairing of Eq. (4.19) may be similarly investigated in models that utilize this pairing. An example of such a model is the model in Table 18, with a suitable choice of GSO projection coefficients. This model is published in [10], and has similar features to the model of Table 2, with the difference being that Table 18 yields bottom-quark and tau-lepton Yukawa couplings at the quartic level of the superpotential [10], whereas Table 2 yields such couplings only at the quintic order [9]. In the model of Table 18 one finds again that all the untwisted moduli fields are projected out. Another example of a model in this class is the model on page 14 of [13], which utilizes the pairings of Eq. (4.19). This model is a variation of the model of Table 18, with the boundary conditions in the vector γ chosen such that the sector b2 produces trilevel bottom-type Yukawa coupling rather than top-type. Again in this model all the untwisted moduli are projected out. In fact, we may conjecture that in general in free fermionic models that utilize the NAHE-set of boundary condition basis vectors, and the pairing of Eq. (4.19), the moduli are always projected out. The reason is that in order to reduce the number of generations to three, one from each of the twisted sectors, b1, b2 and b3, we have to break the degeneracy between the complexified left-moving and right-moving fermions in each sector. This entails the assignment of asymmetric boundary conditions with respect to these complexified fermions, in at least one of the basis vectors, α, β or γ . In the case of the pairing of Eq. (4.19) this leads to the projection of all the untwisted moduli, as noted above, because of the fact that this pairing mixes all the six left-moving triplets of the SU(2)6 automorphism algebra.

5. Twisted moduli

I now turn to show that is the class of three generation string models under consideration here moduli which arise from the twisted sectors are also projected from the massless spectrum. To see how this comes about we start with the set of basis vectors {1,S,ξ1,ξ2,b1,b2} [14].This 2 3 set of basis vectors generates a model with E6 × U(1) × E8 × SO(4) , with 24 matter states in the 27 representation of the observable E6 gauge group, arising from the twisted sectors. These states are decomposed in the following way under E6 → SO(10). The spinorial 16 representations of SO(10) arise from the sectors b1, b2 and b3, whereas the vectorial 10 representations arise from the sectors bj + ξ1. Here, the basis vector ξ1 produces the space–time vector gauge bosons that enhance the N = 4 observable SO(16) gauge group to E8. In addition to the vectorial 10 representation the sectors bj + ξ1, also produce a pair of SO(10) singlets, one of which is embedded in the 27 of E6, whereas the second is identiﬁed with a twisted moduli. Therefore, the models with an E6 observable gauge group contain additional 24 twisted moduli, which matches the number of chiral matter states in the model, as it should. A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108 103

In the realistic free fermionic models the observable gauge group is broken from E6 to SO(10) × U(1). This can be achieved in two equivalent ways. One possibility is to replace the vector ξ1 with the vector 2γ . In this case the gauge group of the N = 4 vacuum, generated by the subset of basis vectors {1,S,2γ,ξ2},isSO(12) × SO(16) × SO(16). The space–time vector bosons of the four-dimensional gauge group in this case are obtained from the NS sector, the sector ξ2, and the sector ξ2 + 2γ . An equivalent way to produce the same N = 4 string vacuum is to break the E8 × E8 gauge group by the choice of GSO phase Eq. (2.11) in the one-loop string partition function. One choice of this phase preserves the states from the sectors ξ1 and ξ2, and therefore enhances the NS SO(16) × SO(16) gauge symmetry to E8 × E8. The second choice projects out those states. In the second case the sectors bj + 2γ ,orbj + ξ1 produce states in the 16 vectorial representation of the hidden SO(16) gauge group, whereas the vectorial states in the (5 + 5¯) and (1 + 1¯), of the observable SO(10) group, generated by the world-sheet fermions ψ¯ 1,...,5 are projected out from the spectrum. Hence, in this case all the moduli that arise from twisted sectors are projected out. The only states that arise from the twisted sectors in three generation models are observable or hidden matter states.

6. Discussion and conclusions

In this paper I investigated the untwisted moduli fields in the three generation free fermionic heterotic-string models. This class of string vacua produced some of the most realistic string models constructed to date. Not only does it produce three generations of chiral fermions under the Standard Model gauge group with potentially phenomenologically viable couplings to the electroweak Higgs doublet fields, but it also affords the attractive embedding of the Standard Model spectrum in SO(10) representations. This property ensures the canonical GUT normalization of the weak hypercharge, which in turn facilitates the agreement of the heterotic-string coupling unification with the experimental data. Thus, these models produce a gross structure, which is compelling from a phenomenological point of view. It ought to be emphasized that one should not regard any of the existing models as providing a completely realistic phenomenology, but merely as providing a probe into what may be some of the ingredients of the eventually true vacuum. From this perspective, a vital property of the free fermionic models is their connection to Z2 × Z2 orbifold compactifications. The free fermionic formalism facilitated the construction of the three generation models and the analysis of some of their phenomenological characteristics. However, this method is formulated, a priori, at a fixed point in the moduli space and the immediate notion of the underlying geometry of the six-dimensional compactified manifold is lost. In particular, the identification of the untwisted moduli fields, and their role in the low energy effective theory, is encumbered. These are reincorporated into the models by identifying the moduli fields as the coefficients of the Abelian Thirring interactions [20]. The moduli fields in the free fermionic models are therefore in correspondence with the Abelian Thirring terms that are invariant under the GSO projections, induced by the basis vectors that define the models. In this paper the issue of untwisted moduli in the realistic three generation free fermionic models was investigated. The existence of models in which all the untwisted Kähler and complex structure moduli are projected out by the generalized GSO projections was demonstrated. The conditions for the projection of all the moduli were identified, and compared to other similar models in which the untwisted moduli are retained. The basic condition that enable the projection of all the untwisted moduli is when the fermionic boundary conditions are such that they correspond to left–right asymmetric boundary conditions with respect to all the six real coordi- 104 A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108 nates of the six-dimensional internal manifold. Additionally, it was shown that in this class of three generation models the E6 symmetryisbrokentoSO(10) × U(1) by a GSO projection. As a consequence all the moduli that arise from the twisted sectors are also projected out in these models. The existence of quasi-realistic models in which all the untwisted Kähler and complex structure moduli are projected out is fascinating. It offers a novel perspective on the existence of extra dimensions in string theory, and on the problem of moduli stabilization. The untwisted moduli are those that govern the underlying geometry, and hence the physics of the extra dimensions. Reparameterization invariance in string theory introduces the need for additional degrees of freedom, beyond the four space–time dimensions, to maintain the classical symmetry in the quantized theory. These additional degrees of freedom may be interpreted as extra dimensions, which are compactified and hidden from contemporary experimental observations. Thus, the consistency of string theory gives rise to the notion of extra dimensions, which in every other respect is problematic. In particular, it raises the issue of what is the mechanism that selects and fixes the parameters of the compactified space. However, if there exist string models in which all the untwisted moduli are projected out by the GSO projections, it means that in these vacua the parameters of the extra dimensions are frozen. In fact, in these string vacua the extra degrees of freedom needed for consistency cannot be interpreted as extra dimensions, as it is not possible to deform from their fixed values, and there is no notion of a continuous classical geometry. The problem of moduli stabilization in string theory attracted considerable attention in the literature [1]. Most of the studies have been directed toward stabilization of the extra dimensions in the effective low energy field theory that emerges from the underlying string theory. The primary obstacle to this is the fact that there are no potential terms for the moduli fields to all orders in perturbation theory. One then, in general, has to rely on the appearance of nonperturbative potential terms, or the utilization of internal fluxes [1]. However, the mechanism that fixes the moduli in the free fermionic models is an intrinsically string theoretic mechanism. The reason is that this mechanism utilizes asymmetric boundary conditions. The possibility to separate the internal dimensions into left- and right-movers, and to assign different transformation properties to them, is intrinsically string theoretic and is nonsensical in the effective low energy point quantum field theories, considered to date. String theory provides a consistent framework for perturbative quantum gravity. In this context it provides the quantum, albeit perturbative, probe of the underlying geometry. The effective low energy point quantum field theory, on the other hand, treats the underlying geometry as a classical geometry. The existence of quasi-realistic string vacua in which all the untwisted moduli are fixed at the string level may tell us that, although string theory requires the additional degrees of freedom for its consistency, these degrees of freedom are not necessarily realized as extra continuous classical dimensions, in the phenomenologically viable cases. These vacua live intrinsically in four space–time dimensions, and there is no notion of extra geometrical dimensions in the low energy effective point quantum field theory. Thus, while the geometrical notion of the extra degrees of freedom provides a useful mean to classify the string vacua, these do not have a physical realization. This situation is similar to the way in which the GUT symmetries are broken in the quasi-realistic free fermionic heterotic-string models by the generalized GSO projections. Namely, also in this case, while the Standard Model states fall into representations of the underlying SO(10) symmetry, the GUT group is broken directly at the string level, and is not realized as a gauge symmetry in the effective low energy point quantum field theory. The existence of quasi-realistic string vacua in which all the untwisted moduli are projected out by the GSO projections is fascinating and intriguing. As discussed in this paper, such string A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108 105 vacua exist among the so-called realistic free fermionic models. We may then ask what are the properties of these models that enabled this outcome, and whether it is unique to this class of models. It should be emphasized that although the free fermionic formalism is formulated at a fixed point in the moduli space, this does not yet entail the absence of an underlying geometry, and the geometrical degrees of freedom are reincorporated in the form of the Abelian world-sheet Thirring interactions. The string vacua in which all the untwisted moduli are fixed represent a special subclass of the three generation models, and the projection of the moduli is highly correlated with the reduction of the number of generations. That is, it is only in the special case of the three generation models that one may expect to find vacua in which all the untwisted moduli are projected out. The defining property of the three generation free fermionic heterotic string models is their relation to Z2 ×Z2 orbifold compactifications [14]. We may therefore ask whether the Z2 projections, and the Z2 × Z2 orbifold possess some special properties that enables the projection of all the untwisted geometrical moduli, and distinguishes it from other compactifications. Naturally, an answer to this question requires further investigation. However, we may note that the special property of the Z2 × Z2 orbifold is that it may act on the internal dimensions as real, rather than complex, dimensions. As discussed in Section 4 it is this property that enables the projection of all the untwisted Kähler and complex structure moduli in the free fermionic models that utilize the pairings of Eq. (4.19). Whether or not similar results may be obtained in other classes of string compactifications is an interesting question that requires further research. It should be emphasized that the results of this paper do not imply a complete solution to the moduli problem in string theory. A moduli field, in general, is a field that does not have a potential to all orders in perturbation theory, and therefore its vacuum expectation value is unconstrained, and it does not get a mass term. The string models contain other sources of moduli fields, aside from the perturbative untwisted geometrical moduli, and the twisted sectors moduli. These include: the dilaton field; and the possibility of charged moduli. Furthermore, the class of vacua under consideration here are supersymmetric and there may exist moduli associated with the supersymmetric flat directions. Moreover, in string theory, each continuous coupling naively implies the existence of a moduli field, and therefore prior to fixing all the couplings in a given model, it seems premature to claim that the entire issue of moduli has been addressed. The untwisted moduli fields are, however, those that parameterize the shape and size of the underlying six-dimensional compactified space. In regard to the dilaton field, the models investigated here are perturbative heterotic strings. Following the string duality advances, we now know that the heterotic string is a perturbative limit of the more fundamental quantum M-theory. At present we have no knowledge about this quantum theory, aside from the existence of its effective limits. The heterotic limit is an expansion in vanishing string coupling, and therefore one would not expect to be able to fix the dilaton VEV in this limit (although one may entertain some nonperturbative possibilities [32]). In this paper it was found that fixation of the other untwisted moduli is achieved by an intrinsic string mechanism, and hence at the perturbative quantum gravity level. However, at present the quantum M-theory is not available. The lesson from the current paper is that the quantum M-theory may allow further possibilities to the problem of dilaton stabilization, which are not readily gleaned in the effective low energy point quantum field theory description. Additional flat direction moduli and charged moduli may, in general, exist in the string models. Their existence, or absence, in the string model is more model dependent and requires a model by model analysis. However, the structure of the three generation free fermionic models suggests 106 A.E. Faraggi / Nuclear Physics B 728 (2005) 83–108 that flat, or charged, moduli are not interchanged with untwisted moduli, and hence an underlying continuous geometrical manifold is not restored. Finally, it should be emphasized that whether or not extra dimensions play a physical role in nature would, of course, require further study and investigation. From the discussion in Sec- tions 2 and 3 it is found that the untwisted sector of the NAHE-based free fermionic models produces three complex and three Kähler structure moduli. This outcome remains valid in any free fermionic model which is left–right symmetric. As discussed in Section 2 there do exist three generation free fermionic models that are left–right symmetric. Therefore in these models the entire set of moduli fields exist in the effective low energy field theory and the moduli remain unfixed. However, the important point is that the free fermionic models also allow for boundary conditions which are left–right asymmetric. Naturally, the space of models is vast and we can construct models in which asymmetric boundary conditions are assigned on one, two or three of the complex planes. The remarkable fact is the existence of three generation models in which asymmetric boundary conditions are assigned to all three complex planes. In this set of models the entire set of untwisted moduli are projected out by the GSO projections and the untwisted moduli are fixed. The results of this paper indicate the existence of quasi-realistic string vacua in which the extra dimensions do not possess a classical physical realization. On the other hand there are also three generation models in which the moduli are retained and the geometrical description is maintained, and models in which some of the extra dimensions are frozen whereas others are undetermined. Which, if any, of these possibilities is relevant to nature, would, of course, remain, for the time being, a hotly contested issue. Moreover, the models may of course still contain additional moduli and the role of these moduli requires more detailed investigation. We should also remember that we are discussing here perturbative heterotic string models. In this limit the dilaton remains unfixed, and it may be that nonperturbative effects may give rise to additional moduli. Nevertheless, it is extremely intriguing that in the class of three generation free fermionic models, that are constructed in the vicinity of the self-dual point under the T-duality transformations, one finds that the models possess the intrinsic mechanism to fix all the untwisted geometrical moduli. One would anticipate that the self-dual point, being the symmetry point under the T-duality transformations, plays a vital role in the vacuum selection [33]. The three generation free fermionic models then highlight the class of Z2 × Z2 orbifolds as the naturally relevant one. The availability, in this class, to act asymmetrically on all the six real compactified dimensions, then affords the possibility of fixing all the untwisted Kähler and complex structure moduli in these models.

Acknowledgements

This work is supported in part by PPARC. I would like to thank the anonymous referee for valuable comments.

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Dirac quasinormal frequencies of the Kerr–Newman black hole

Jiliang Jing, Qiyuan Pan

Institute of Physics and Department of Physics, Hunan Normal University, Changsha, Hunan 410081, PR China Received 27 June 2005; accepted 22 August 2005 Available online 12 September 2005

Abstract The Dirac quasinormal modes (QNMs) of the Kerr–Newman black hole are investigated using continued fraction approach. It is shown that the quasinormal frequencies in the complex ω plane move counterclockwise as the charge or angular momentum per unit mass of the black hole increases. They get a spiral-like shape, moving out of their Schwarzschild or Reissner–Nordström values and “looping in” towards some limiting frequencies as the charge and angular momentum per unit mass tend to their extremal values. The number of the spirals increases as the overtone number increases but decreases as the angular quantum number increases. It is also found that both the real and imaginary parts are oscillatory functions of the angular momentum per unit mass, and the oscillation becomes faster as the overtone number increases but slower as the angular quantum number increases.  2005 Elsevier B.V. All rights reserved.

PACS: 04.70.Dy; 04.30.Db; 04.70.Bw; 97.60.Lf

1. Introduction

It is well known that QNMs possess discrete spectra of complex characteristic frequencies which are entirely ﬁxed by the structure of the background spacetime and irrelevant of the initial perturbations [1,2]. Thus, it is believed that one can directly identify the existence of a black hole by comparing QNMs with the gravitational waves observed in the universe, as well as test the stability of the event horizon against small perturbations. Meanwhile, it is generally believed that the study of QNMs may lead to a deeper understanding of the thermodynamic properties of black holes in loop quantum gravity [3,4] since the real part of quasinormal frequencies with a

E-mail address: [email protected] (J. Jing).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.08.038 110 J. Jing, Q. Pan / Nuclear Physics B 728 (2005) 109–120 large imaginary part for the scalar field in the Schwarzschild black hole is equal to the Barbero– Immirzi parameter [3–6], a factor introduced by hand in order that the loop quantum gravity reproduces correct entropy of the black hole. It was also argued that the QNMs of anti-de Sitter (AdS) black holes have a direct interpretation in terms of the dual conformal field theory (CFT) [7–9]. According to the AdS/CFT correspondence, a large static black hole in asymptotically AdS spacetime corresponds to a (approximately) thermal state in CFT, and the decay of the test field in the black-hole spacetime corresponds to the decay of the perturbed state in CFT. The dynamical timescale for the return to the thermal equilibrium can be done in the AdS spacetime, and then translated into the CFT using the AdS/CFT correspondence. Thus, QNMs for various black holes have been studied extensively [10–30]. Although many authors studied the QNMs of scalar, electromagnetic and gravitational perturbations, the investigation of the Dirac QNMs is very limited [31–35]. Furthermore, in these papers the study of the Dirac QNMs was limited to use Pöshl–Teller potential or WKB approximative methods. The Pöshl–Teller potential approximative may be a valuable tool in certain situations, but in general results obtained from the approximating potentials should not be expected to be accurate. The WKB method can be generalized to deal with the Reissner–Nordström and Kerr black holes. In the latter case, unfortunately, the method is not very simple to apply and does not yield very accurate results. It is well known that the Leaver’s continued fraction technique is the best workhorse method to compute highly damped modes and is very reliable. However, the method cannot be directly used to study the QNMs of the Dirac fields if we take the standard wave equations. The reason to obstruct the study of the Dirac QNMs using continued fraction technique is that for static black holes the standard wave equation 2 d 2 + ω Z± = V±Z± 2 (1.1) dr∗ possesses a special potential √ 2 ∆ d ∆ V± = λ ± λ (1.2) r4 dr∗ r2 √ which is the function of ∆ [31,35,36], where ∆ is a function related to the metric of the spacetimes, say ∆ = r2 − 2Mr for the Schwarzschild black hole. We cannot investigate the Dirac QNMs using the continued fraction approaches√ because we have to expand the potential as a series at the event horizon but the factor ∆ does not permit us to do that. Recently, we [37,38] found that the wave function and the potential of the Dirac fields can be expressed as new forms. Starting from the new wave function and potential, we [37,38] studied the Dirac QNMs of the Schwarzschild black hole using the continued fraction [39] and Hill- determinant approaches [40,41] and the Dirac QNMs of the Schwarzschild–anti-de Sitter and Reissner–Nordström–anti-de Sitter black holes using Horowitz–Hubeny approach [11].Themain purpose of this paper is to extend the investigation to the Kerr–Newman black hole because this black hole is the only asymptotically flat electro-vacuum solution of the Einstein–Maxwell system [42] and it is the most general black hole among classical black holes. The organization of this paper is as follows. In Section 2 the decoupled Dirac equations and corresponding wave equations in the Kerr–Newman spacetime are obtained using Newman– Penrose formalism. In Section 3 the angular and radial continued fraction equations are introduced. In Section 4 the numerical results for the Dirac QNMs in the Reissner–Nordström, Kerr and Kerr–Newman black holes are presented. Section 5 is devoted to a summary. J. Jing, Q. Pan / Nuclear Physics B 728 (2005) 109–120 111

2. Dirac equations in the Kerr–Newman spacetime

In a curve spacetime the Dirac equations [43] can be expressed as √ ∇ B + ¯ = √2 BB P iµQB 0, B ¯ 2 ∇BB Q + iµPB = 0, (2.1) B B where ∇BB is covariant differentiation, P and Q are the two-component spinors representing ¯ the wave functions, PB is the complex conjugate of PB , and µ is the mass of the Dirac particle. In the Newman–Penrose formalism [44] the equations become − (D + − ρ)P0 + (δ¯ + π − α)P 1 = 2 1/2iµQ¯ 1 , − (∆ + µ − γ)P1 + (δ + β − τ)P0 =−2 1/2iµQ¯ 0 , − (D +¯ −¯ρ)Q¯ 0 + (δ +¯π −¯α)Q¯ 1 =−2 1/2iµP1, − (∆ +¯µ −¯γ)Q¯ 1 + (δ¯ + β¯ −¯τ)Q¯ 0 = 2 1/2iµP0. (2.2) The null tetrad for the Kerr–Newman black hole can be taken as r2 + a2 a lµ = , 1, 0, , ∆ ∆ 1 nµ = r2 + a2 , −∆, ,a , ¯ 0 2ρρ 1 i mµ = √ iasin θ,0, 1, , (2.3) 2ρ¯ sin θ with ρ = r − iacos θ, 2 2 2 ∆ = r − 2Mr + a + Q = (r − r+)(r − r−), (2.4) where M, Q and a represent the mass, charge and angular momentum per unit mass of the Kerr– Newman black hole respectively, and r± = M ± M2 − a2 − Q2.AsQ → 0 the Kerr–Newman metric reduces to the rotating Kerr metric, and as a → 0 it reduces to the charged Reissner– Nordström metric. Then the wave functions can be taken as

0 1 −i(ωt−mϕ) P = R− (r)S− (θ)e , ρ 1/2 1/2 1 −i(ωt−mϕ) P = R+1/2(r)S+1/2(θ)e , ¯ 1 −i(ωt−mϕ) Q = R+1/2(r)S−1/2(θ)e , ¯ 0 1 −i(ωt−mϕ) Q =− R− (r)S+ (θ)e , (2.5) ρ¯ 1/2 1/2 where ω and m are the energy and angular momentum of the Dirac particle. After the tedious calculation Eq. (2.2) can be simpliﬁed as √ √ ∆ D R− = (λ + iµr) ∆ R+ , (2.6) √ 0√1/2 1/2 D† R = − R ∆ 0 ∆ +1/2 (λ iµr) −1/2, (2.7) L1/2S+1/2 =−(λ − aµcos θ)S−1/2, (2.8) 112 J. Jing, Q. Pan / Nuclear Physics B 728 (2005) 109–120

L† = + 1/2S−1/2 (λ aµcos θ)S+1/2, (2.9) with ∂ iK n d∆ D = − + , n ∂r ∆ ∆ dr ∂ iK n d∆ D† = + + , n ∂r ∆ ∆ dr ∂ m L = − aωsin θ + + n cot θ, n ∂θ sin θ L† = ∂ + − m + n aωsin θ n cot θ, ∂θ sin θ K = r2 + a2 ω − ma. (2.10)

We can eliminate S+1/2 (or S−1/2) from Eqs. (2.8) and (2.9). Defining u = cos θ, we find that the angular equation can be expressed as + 2 d 2 dSs 2 (m su) 1 − u + (aωu) − 2aωsu + s + Alm − Ss = 0, (2.11) du du 1 − u2 where Alm is the angular separation constant. It is interesting to note that the angular equation (2.11) is the same as in the Kerr case. We will focus√ our attention on the massless Dirac field in this paper. Therefore, we can elimi- R R nate√ −1/2 (or ∆ +1/2) from Eqs. (2.6) and (2.7) to obtain a radial decoupled Dirac equation for ∆ R+1/2 (or R−1/2). Then, introducing an usual tortoise coordinate

r2 + a2 dr∗ = dr, (2.12) ∆ and resolving the equation in the form

∆−s/2 Rs = √ Ψs, (2.13) r2 + a2 we obtain the wave equation

d2Ψ s + ω2 − V Ψ = , 2 s s 0 (2.14) dr∗ with

dZ 2 Vs =− + Z dr∗ ∆ 2maω(r2 + a2) − m2a2 isK d∆ + + − 4isωr + λ2 , (2.15) (r2 + a2)2 ∆ ∆ dr where Z =− s d∆ − r∆ and λ2 = A − 2maω + (aω)2. We will see that we can eas- 2(r2+a2) dr (r2+a2)2 lm ily work out the Dirac quasinormal frequencies of the Kerr–Newman black hole from Eqs. (2.11) and (2.14) using the continued fraction approach. J. Jing, Q. Pan / Nuclear Physics B 728 (2005) 109–120 113

3. Angular and radial continued fraction equations

For the Kerr–Newman black hole, we have to solve Eq. (2.11) numerically following Leaver [39]. Boundary conditions for Eq. (2.11) are that Ss is regular at the regular singular points u =±1 where the indices are given by ±(m + s)/2atu = 1 and ±(m − s)/2atu =−1. So a solution to Eq. (2.11) may be expressed as ∞ = aωu + k1 − k2 θ + n Ss(u) e (1 u) (1 u) an(1 u) , (3.1) n=0 where k1 =|m − s|/2, k2 =|m + s|/2, and the superscript θ denotes the association with the angular equation. The series coefﬁcients are related by a three term recurrence relation and the boundary condition at u =+1 is satisﬁed only by its minimal solution sequence. The three-term recurrence relation can be written as θ θ + θ θ = α0 a1 β0 a0 0, (3.2) θ θ + θ θ + θ θ = αnan+1 βn an γn an−1 0 (n 1), (3.3) where θ =− + + + αn 2(n 1)(n 2k1 1), βθ = n(n − 1) + 2n(k + k + 1 − 2aω) − 2aω(2k + s + 1) n 1 2 1 2 2 + (k1 + k2)(k1 + k2 + 1) − a ω + s(s + 1) + Alm , θ = + + + γn 2aω(n k1 k2 s). (3.4)

We will obtain the minimal solution if the angular separation constant Alm is a root of the continued fraction equation αθ γ θ αθ γ θ αθ γ θ αθ γ θ 0 = βθ − 0 1 1 2 2 3 3 4 ···. (3.5) 0 θ − θ − θ − θ − β1 β2 β3 β4 The QNMs of the Kerr–Newman black hole are defined to be the modes with purely ingoing waves at the event horizon and purely outgoing waves at infinity [1]. Then, the boundary conditions on wave function Ψs at the horizon (r = r+) and infinity (r →+∞) can be expressed as

− s −iσ+ (r − r+) 2 ,r→ r+, Ψs ∼ (3.6) r−s+iωeiωr,r→+∞, 1 ± = [ 2 + 2 − ] where σ r+−r− (r± a )ω ma . A solution to Eq. (2.14) which satisﬁes the desired behavior at the boundary can be written in the form s s 2 2 − −iσ+ −1− +2iω+iσ− iω(r−r−) Ψs = r + a (r − r+) 2 (r − r−) 2 e ∞ n r − r+ × an . (3.7) r − r− n=0

If we take r+ + r− = 1 and b = r+ − r−, the sequence of the expansion coefﬁcients {an: n = 1, 2,...} is determined by a three-term recurrence relation staring with a0 = 1: 114 J. Jing, Q. Pan / Nuclear Physics B 728 (2005) 109–120

α0a1 + β0a0 = 0,

αnan+1 + βnan + γnan−1 = 0 (n = 1, 2,...). (3.8)

The recurrence coefﬁcient αn, βn and γn are given in terms of n and the black hole parameters by

2 αn = n + (C0 + 1)n + C0, 2 βn =−2n + (C1 + 2)n + C3, 2 γn = n + (C2 − 3)n + C4 − C2 + 2, (3.9) and the intermediate constants Cn are defined by 2 2 2i r+ + r− C = 1 − s − iω − + a2 ω − ma , 0 b 2 2 2 4i r+ + r− C =−4 + 2iω(2 + b) + + a2 ω − ma , 1 b 2 2 2 2i r+ + r− C = s + 3 − 3iω − + a2 ω − ma , 2 b 2 2 2 C3 = ω 4 + 2b − 4r+r− + 3a − 2maω − s − 1 + (2 + b)iω − Alm 2 2 2(2ω + i) r+ + r− + + a2 ω − ma , b 2 2 2 2(2ω + i) r+ + r− C = s + 1 − 2ω2 − (2s + 3)iω − + a2 ω − ma . (3.10) 4 b 2 It is interesting to note that the three-term recursion relation is obtained form the wave equation (2.14) directly. The calculation is simpler than other cases. For example, the coefficients of the expansion for the electromagnetic and gravitational fields in the Reissner–Nordström black hole are determined by a four-term recursion relation which should be reduced to a three-term relation using a Gaussian eliminated step (see [41] for details). Our results are also more concise than that for the electromagnetic and gravitational perturbations in the Kerr–Newman black-hole spacetime obtained by Berti and Kokkotas [26]. The series (3.7) converges and the r =+∞boundary condition (3.6) is satisfied if, for a given s, a, m and Alm, the frequency ω is a root of the continued fraction equation αn−1γn αn−2γn−1 α0γ1 αnγn+1 αn+1γn+2 αn+2γn+3 βn − ··· = ··· βn−1− βn−2− β0 βn+1− βn+2− βn+3− (n = 1, 2,...). (3.11) The solution to the radial equation (3.11) is strictly bounded up with the determination of the eigenvalue Alm of the angular equation (3.5), which appears explicitly in the coefficients of the recurrence relation associated with the radial equation. We could attempt to solve for both ω and Alm simultaneously, but this would be a time-consuming procedure. Instead, we first fix the value of a, l, m and ω and find the angular separation constant Alm by looking for the zero of the angular continued fraction (3.5). Then, we use the corresponding eigenvalue to look for the zeros of the radial continued fraction. The nth quasinormal frequency is (numerically) the most stable root of the nth inversion of the continued fraction relation (3.11). J. Jing, Q. Pan / Nuclear Physics B 728 (2005) 109–120 115

4. Numerical results

In this section we present the numerical results obtained by using the numerical procedure just outlined in the previous section. The results will be organized into three subsections: Dirac QNMs of the Reissner–Nordström black hole, Dirac QNMs of the Kerr black hole and Dirac QNMs of the Kerr–Newman black hole.

4.1. Dirac QNMs of the Reissner–Nordström black hole

As the angular momentum per unit mass tends to zero (a → 0) the Kerr–Newman metric reduces to the Reissner–Nordström metric. In this limit the angular separation constant Alm = 2 = + 1 2 λ (l 2 ) [37] (l is the quantum number characterizing the angular distribution). Thus, the QNMs in this static spacetime can be obtained easily. The Dirac quasinormal frequencies of the Reissner–Nordström black hole for n = 0, 1, 2, 3 and λ = 1 are listed in Table 1 and that for n = 0, 1, 2, 3 and λ = 2 are listed in Table 2. From the tables we find that both the real and imaginary parts of the quasinormal frequencies decrease as the overtone number increases for the fixed charge and angular quantum number (i.e., Q = const and l = const). We also find that both the real and imaginary parts of the quasinormal frequencies increases as the angular quantum number increases (except for the cases Q = 0.45 and n = 0) for the fixed charge and overtone number.

Table 1 Dirac quasinormal frequencies of the Reissner–Nordström black hole for λ = 1 Qω(n = 0) ω (n = 1) ω (n = 2) ω (n = 3) 0.00 0.365926 − 0.193965i 0.295644 − 0.633857i 0.240740 − 1.12845i 0.208512−1.63397i 0.05 0.366591 − 0.194066i 0.296432 − 0.634048i 0.241590 − 1.12866i 0.209400−1.63420i 0.10 0.368615 − 0.194366i 0.298836 − 0.634601i 0.244190 − 1.12923i 0.212121−1.63480i 0.15 0.372096 − 0.194851i 0.302988 − 0.635442i 0.248696 − 1.13001i 0.216843−1.63555i 0.20 0.377208 − 0.195488i 0.309132 − 0.636419i 0.255389 − 1.13068i 0.223871−1.63594i 0.25 0.384241 − 0.196213i 0.317661 − 0.637242i 0.264718 − 1.13064i 0.233661−1.63507i 0.30 0.393651 − 0.196891i 0.329196 − 0.637331i 0.277339 − 1.12871i 0.246805−1.63118i 0.35 0.406184 − 0.197220i 0.344692 − 0.635457i 0.294100 − 1.12248i 0.263748−1.62065i 0.40 0.423108 − 0.196443i 0.365519 − 0.628639i 0.315375 − 1.10611i 0.282749−1.59477i 0.45 0.446635 − 0.192202i 0.391957 − 0.607831i 0.333021 − 1.06224i 0.277400−1.52872i

Table 2 Dirac quasinormal frequencies of the Reissner–Nordström black hole for λ = 2 Qω(n = 0) ω (n = 1) ω (n = 2) ω (n = 3) 0.00 0.760074 − 0.192810i 0.711666 − 0.595995i 0.638523 − 1.03691i 0.569867−1.51490i 0.05 0.761372 − 0.192916i 0.713058 − 0.595276i 0.640042 − 1.03728i 0.571477−1.51529i 0.10 0.765327 − 0.193230i 0.717299 − 0.596107i 0.644677 − 1.03836i 0.576395−1.51643i 0.15 0.772119 − 0.193738i 0.724597 − 0.597440i 0.652672 − 1.04004i 0.584893−1.51813i 0.20 0.782084 − 0.194414i 0.735335 − 0.599172i 0.664476 − 1.04210i 0.597473−1.52001i 0.25 0.795772 − 0.195199i 0.750138 − 0.601099i 0.680818 − 1.04411i 0.614939−1.52137i 0.30 0.814058 − 0.195973i 0.770003 − 0.602799i 0.702848 − 1.04523i 0.638525−1.52082i 0.35 0.838381 − 0.196464i 0.796544 − 0.603349i 0.732368 − 1.04363i 0.670048−1.51544i 0.40 0.871259 − 0.195997i 0.832478 − 0.600485i 0.772116 − 1.03491i 0.711672−1.49828i 0.45 0.918762 − 0.192368i 0.882263 − 0.587382i 0.824382 − 1.00617i 0.760227−1.44942i 116 J. Jing, Q. Pan / Nuclear Physics B 728 (2005) 109–120

Table 3 Dirac quasinormal frequencies of the Kerr black hole for λ = 1 aω(n = 0) ω (n = 1) ω (n = 2) ω (n = 3) 0.00 0.365926 − 0.193965i 0.295644 − 0.633857i 0.240740 − 1.12845i 0.208512−1.63397i 0.05 0.366218 − 0.193805i 0.296135 − 0.633141i 0.241239 − 1.12700i 0.208875−1.63180i 0.10 0.367099 − 0.193316i 0.297605 − 0.630951i 0.242707 − 1.12257i 0.209884−1.62514i 0.15 0.368589 − 0.192461i 0.300054 − 0.627144i 0.245045 − 1.11486i 0.211268−1.61355i 0.20 0.370717 − 0.191175i 0.303460 − 0.621458i 0.248029 − 1.10334i 0.212433−1.59615i 0.25 0.373525 − 0.189350i 0.307751 − 0.613449i 0.251166 − 1.08710i 0.212101−1.57147i 0.30 0.377060 − 0.186806i 0.312708 − 0.602389i 0.253306 − 1.06460i 0.207277−1.53691i 0.35 0.381352 − 0.183245i 0.317679 − 0.587057i 0.251404 − 1.03327i 0.189183−1.48777i 0.40 0.386329 − 0.178141i 0.320563 − 0.565425i 0.234871 − 0.98938i 0.106512−1.41213i 0.45 0.391413 − 0.170595i 0.313069 − 0.536419i 0.178522 − 0.99702i 0.067565−1.44066i

Table 4 Dirac quasinormal frequencies of the Kerr black hole for λ = 2 aω(n = 0) ω (n = 1) ω (n = 2) ω (n = 3) 0.00 0.760074 − 0.192810i 0.711666 − 0.594995i 0.638523 − 1.03691i 0.569867−1.51490i 0.05 0.760593 − 0.192662i 0.712351 − 0.594475i 0.639418 − 1.03584i 0.570876−1.51315i 0.10 0.762165 − 0.192207i 0.714416 − 0.592884i 0.642106 − 1.03257i 0.573887−1.50779i 0.15 0.764829 − 0.191416i 0.717897 − 0.590117i 0.646595 − 1.02688i 0.578836−1.49851i 0.20 0.768656 − 0.190231i 0.722849 − 0.585981i 0.652877 − 1.01841i 0.585563−1.48468i 0.25 0.773753 − 0.188558i 0.729342 − 0.580157i 0.660882 − 1.00652i 0.593690−1.46530i 0.30 0.780274 − 0.186246i 0.737432 − 0.572132i 0.670362 − 0.99020i 0.602309−1.43876i 0.35 0.788430 − 0.183045i 0.747081 − 0.561063i 0.680552 − 0.96784i 0.609060−1.40255i 0.40 0.798494 − 0.178518i 0.757886 − 0.545547i 0.689090 − 0.93700i 0.607082−1.35381i 0.45 0.810750 − 0.171857i 0.768079 − 0.523427i 0.688731 − 0.89635i 0.580342−1.30276i

4.2. Dirac QNMs of the Kerr black hole

The Dirac quasinormal frequencies of the Kerr black hole for n = 0, 1, 2, 3 and λ = 1are listed in Table 3 and that for n = 0, 1, 2, 3 and λ = 2 are listed in Table 4. We find that both the real and imaginary parts of the quasinormal frequencies decrease as the overtone number increases for the fixed angular momentum per unit mass and angular quantum number. We also find that both the real and imaginary parts of the quasinormal frequencies increases as the angular quantum number increases (except for the cases a = 0.4, 0.45 and n = 0) for the fixed angular momentum per unit mass and overtone number. By comparing the results of the Reissner–Nordström and Kerr black holes, it is interesting to note that the real part of the quasinormal frequencies of the Reissner–Nordström black hole is greater than that of the Kerr black hole, but the imaginary part of the quasinormal frequencies of the Reissner–Nordström black hole is less than that of the Kerr black hole if we take the same values of the charge and the angular momentum per unit mass. That is to say, the decay of the Dirac fields in the charged black-hole spacetime is faster than that in the rotating black-hole spacetime if the values of the charge and angular momentum per unit mass are the same.

4.3. Dirac QNMs of the Kerr–Newman black hole

The Dirac quasinormal frequencies of the Kerr–Newman black hole for n = 0, 1, 2 and λ = 1 are shown by Fig. 1 and those for n = 5, 6, 7 and λ = 2areshownbyFig. 2. The left columns J. Jing, Q. Pan / Nuclear Physics B 728 (2005) 109–120 117 a and the a 0. In each panel, the = m 2and / 1 = l of the quasinormal frequencies versus (ω) and Im (ω) from zero to the extremal limit and the three thin curves are obtained by Q 2, both the real and imaginary parts are oscillatory functions of n 4 (left to right). These curves show that the frequencies generally move counterclockwise . 0 , 2 . 0 , 0 = Q 0) which is drawn by increasing = plane of the first three Dirac QNMs of the Kerr–Newman black hole for a 4. These panels tell us that, for . ω 0 , 2 . 0 , 0 = Q from zero to the extremal limit for fixed values of the charge a ) increases and the number of spirals increases as the overtone number increases. The other panels draw Re Q (or a Fig. 1. The left three panels show trajectories in the complex thick curve corresponds to theincreasing Reissner–Nordström black hole ( as in which the curves from bottom to top refer to oscillations become faster as the overtone number increases. 118 J. Jing, Q. Pan / Nuclear Physics B 728 (2005) 109–120 ω from zero to the a in which the curves 0inthecomplex a = m 2and / and the oscillations begin earlier 3 a = l 7, , 6 , 5 = n of the quasinormal frequencies versus (ω) and Im 0 and the three thin curves are obtained by increasing = (ω) a 5, both the real and imaginary parts are oscillatory functions of n from zero to the extremal limit for Q 4 (left to right). The other panels draw Re . 0 , 2 . 0 , 0 = Q 4. These panels tell us that, for . 0 , 2 . 0 , 0 = Q Fig. 2. The left three panels describe the behavior of the Dirac quasinormal frequencies of the Kerr–Newman black hole for plane. In each panel, the thick curve is drawn by increasing extremal limit for fixed values of the charge from bottom to top refer to as the charge increases. J. Jing, Q. Pan / Nuclear Physics B 728 (2005) 109–120 119 in Figs. 1 and 2 describe the behavior of quasinormal frequencies in the complex ω plane which show that the frequencies generally move counterclockwise as the charge (the thick curves) or the angular momentum per unit mass (the thin curves) increases. They get a spiral-like shape, moving out of the Schwarzschild (Q = 0 and a = 0) or Reissner–Nordström (0

5. Summary

The wave equations for the Dirac fields in the Kerr–Newman black-hole spacetime are obtained by means of the Newman–Penrose formalism. The expansion coefficients of the wave equation which satisfy appropriate boundary conditions are directly determined by a three-term recurrence relation. Then, the Dirac quasinormal frequencies of the Kerr–Newman black hole are evaluated using continued fraction approach and the results are presented by tables and figures. By comparing the results of the Reissner–Nordström and Kerr black holes, it is interesting to note that the real part of the quasinormal frequencies of the Reissner–Nordström black hole is greater than that of the Kerr black hole, but the imaginary part is less than that of the Kerr black hole if we take the same values of the charge and the angular momentum per unit mass. That is to say, the decay of the Dirac fields in the charged black-hole spacetime is faster than that in the rotating black-hole spacetime if the values of the charge and angular momentum per unit mass are the same. From figures we find that the frequencies in the complex ω plane generally move counterclockwise as the charge or the angular momentum per unit mass increases. They get a spiral-like shape, moving out of their Schwarzschild or Reissner–Nordström values and “looping in” towards some limiting frequencies as the charge and angular momentum per unit mass tend to their extremal values. For a given angular quantum number λ, we observe that the number of spirals increases as the overtone number increases. However, for a given overtone number, the increasing λ has the effect of “unwinding” the spirals. We also find that the real and imaginary parts of the quasinormal frequencies are oscillatory functions of the angular momentum per unit mass, and the oscillating behavior starts earlier and earlier as the overtone number grows for a fixed λ (or as the charge Q increases for fixed n and λ), but it begins later and later as λ increases for a fixed overtone number.

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No. 10473004; the FANEDD under Grant No. 200317; and the SRFDP under Grant No. 120 J. Jing, Q. Pan / Nuclear Physics B 728 (2005) 109–120

20040542003; the Hunan Provincial Natural Science Foundation of China under Grant No. 05JJ0001; and the National Basic Research program of China under Grant No. 2003CB716300.

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Minimal ﬂavor violation in the lepton sector

Vincenzo Cirigliano a, Benjamín Grinstein b, Gino Isidori c, Mark B. Wise a

a California Institute of Technology, Pasadena, CA 91125, USA b Department of Physics, University of California at San Diego, La Jolla, CA 92093, USA c INFN, Laboratori Nazionali di Frascati, Via E. Fermi 40, I-00044 Frascati, Italy Received 14 July 2005; accepted 11 August 2005 Available online 8 September 2005

Abstract We extend the notion of minimal flavor violation to the lepton sector. We introduce a symmetry principle which allows us to express lepton flavor violation in the charged lepton sector in terms of neutrino masses and mixing angles. We explore the dependence of the rates for flavor changing radiative charged lepton decays (i → j γ )andµ-to-e conversion in nuclei on the scales for total lepton number violation, lepton flavor violation and the neutrino masses and mixing angles. Measurable rates are obtained when the scale for total lepton number violation is much larger than the scale for lepton flavor violation.  2005 Elsevier B.V. All rights reserved.

1. Introduction

With the discovery of neutrino masses and mixing it has been clearly established that lepton flavor is not conserved. The smallness of neutrino masses also provides a strong indication in favor of the non-conservation of total lepton number, although this information cannot be directly extracted from data yet. In analogy to what happens in the quark sector, the non-conservation of lepton flavor points toward the existence of lepton flavor violating (LFV) processes with charged leptons, which however have not been observed so far. Within the Standard Model (SM), the flavor violation in the quark sector is induced by Yukawa interactions. These do not break the baryon number and, to a good approximation, leave the neutral currents flavor diagonal. This fact puts very stringent constraints on the structure of possible new degrees of freedom. New, flavor-dependent interactions beyond the SM can readily be ruled

E-mail address: [email protected] (G. Isidori).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.08.037 122 V. Cirigliano et al. / Nuclear Physics B 728 (2005) 121–134 out if they contribute significantly to flavor changing neutral currents (FCNC). One way to evade this constraint is to assume that all the new degrees of freedom, or at least all the new particles carrying flavor quantum numbers, are very heavy. This solution is theoretically unappealing and largely non-testable. A more appealing scenario is to assume that the new flavor-changing couplings appearing in SM extensions are suppressed by some symmetry principle. The most re- strictive and predictive symmetry principle of this type is the so-called minimal flavor violation (MFV) hypothesis [1–3]: the assumption that the SM Yukawa couplings are the only sources of quark-flavor symmetry breaking. In this case all the cancellations that render FCNC auto- matically small in the SM apply just as well to the new degrees of freedom, allowing for very reasonable mass scales of the new particles. Interestingly, the MFV hypothesis can be formulated in a very general way in terms of an effective field theory [3], without the need of specifying the nature of the new degrees of freedom. In a theory where the new degrees of freedom also carry lepton flavor quantum numbers, it is natural to expect that a similar mechanism occurs also in the lepton sector. In this paper we extend the notion of MFV to the lepton sector. In other words, we define and analyze a consistent class of SM extensions where the sources of LFV are linked in a minimal way to the known structure of the neutrino and charged-lepton mass matrices. This allows us to address in a general way several interesting questions. In particular, we shall analyze the general requirements about the scale of new physics under which we can expect observable effects in low- energy rare LFV processes, such as µ → eγ and µ-to-e conversion in nuclei, without requiring the existence of new uncontrollable sources of lepton flavor mixing. We shall also identify some model-independent relations among different LFV observables which could allow to falsify this general hypothesis about the flavor structure of physics beyond the SM. The large difference between charged lepton and neutrino masses is naturally attributed to the breaking of total lepton number. This assumption has very important consequences to estimate the overall size of the LFV terms. As we shall show, only by decoupling the mechanisms of lepton flavor mixing and lepton number violation can we generate sizable LFV amplitudes in the charged-lepton sector. Since lepton flavor and lepton number correspond to two independent symmetry groups, this decoupling can naturally be implemented in an effective field theory approach with the minimal particle content, namely without introducing right-handed neutrino fields. However, in most explicit SM extensions this result is achieved by means of the see-saw mechanism with heavy right-handed neutrinos. For this reason, we shall consider two main possibilities in order to define the minimal sources of flavor symmetry breaking in the lepton sector: (i) a scenario without right-handed neutrinos, where the (left-handed) Majorana mass matrix is the only irreducible source of flavor symmetry breaking; (ii) a scenario with right-handed neutrinos, where the Yukawa couplings define the irreducible sources of flavor symmetry breaking and the (right-handed) Majorana mass matrix has a trivial flavor structure.

2. Minimal breaking of the lepton ﬂavor symmetry

In the absence of Yukawa couplings, the flavor symmetry of the quark sector of the SM would × × i i i be SU(3)Q SU(3)U SU(3)D corresponding to individual rotations of the QL, uR and dR fields (the left-handed quark doublet and the two right-handed quark singlets) for i = 1, 2, 3. Models with MFV have only two independent sources of breaking of this group, namely the two Yukawa couplings λU and λD. Each of them breaks the symmetry in a specific way: in the ¯ ¯ spurion sense, λU transforms as a (3, 3, 1) while λD as a (3, 1, 3). In MFV models any higher dimension operator that describes long distance remnants of very short distance physics must V. Cirigliano et al. / Nuclear Physics B 728 (2005) 121–134 123 be invariant under the full flavor symmetry group when the couplings λU and λD are taken to transform as spurions as above [3]. In order to define a similar minimal flavor violating structure for the leptons, we first need to specify the field content of the theory in the lepton sector. As anticipated, we shall consider two cases:

i (1) Minimal field content: three left-handed lepton doublets LL and three right-handed charged i lepton singlets eR (SM field content). In this case the lepton flavor symmetry group is

GLF = SU(3)L × SU(3)E. (1) The lepton sector is also invariant under two U(1) symmetries, which can be identified with total lepton number, U(1)LN, and the weak hypercharge. i (2) Extended field content: three right-handed neutrinos, νR, in addition to the SM fields. In this case the field content of the lepton sector is very similar to that of the quark sector, with a × maximal flavor group GLF SU(3)νR .

In the following we shall deﬁne separately the assumptions of minimal lepton ﬂavor violation (MLFV) in these two cases.

2.1. Minimal ﬁeld content

In this case the minimal choice for the neutrino mass matrix is a left-handed Majorana mass term transforming as (6, 1) under GLF. Because of the SU(2)L gauge symmetry, this mass term cannot be generated by renormalizable interactions. Moreover, the absence of right-handed neutrino ﬁelds requires the breaking of total lepton number. We deﬁne the MLFV hypothesis in this case as follows:

(1) The breaking of the U(1)LN is independent from the breaking of the lepton flavor symmetry (GLF) and is associated to a very high scale ΛLN. ij ij (2) There are only two irreducible sources of lepton-flavor symmetry breaking, λe and gν , defined by1 L =− ij ¯i † j − 1 ij ¯ ci T j + Sym.Br. λe eR H LL gν LL τ2H H τ2LL h.c. 2ΛLN 2 →− ij ¯i j − v ij ¯ ci j + vλe eReL gν νL νL h.c. (2) 2ΛLN ij The smallness of the neutrino mass is attributed to the smallness of v/ΛLN, while gν can have entries of O(1) as in the standard see-saw mechanism.

The transformation properties of the lepton ﬁeld under GLF are

LL → VLLL,eR → VReR. (3)

∗ 1 Throughout this paper we use four-component spinor ﬁelds, and ψc =−iγ2ψ denotes the charge conjugate of the ﬁeld ψ.Wealsousev =H 0174 GeV. 124 V. Cirigliano et al. / Nuclear Physics B 728 (2005) 121–134

ij ij Thus the Lagrangian (2) is formally invariant under this symmetry if the matrices λe and gν are taken as spurions transforming as → † → ∗ † λe VRλeVL,gν VL gνVL. (4)

Since we are interested in LFV processes with external charged leptons, we can use the GLF invariance and rotate the fields in the basis where λe is flavor diagonal. In such basis m 1 λ = = diag(m ,m ,m ), e v v e µ τ Λ ∗ Λ ∗ g = LN Uˆ m Uˆ † = LN Uˆ diag(m ,m ,m )Uˆ †, (5) ν v2 ν v2 ν1 ν2 ν3 where Uˆ is the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) mixing matrix. The latter can be ˆ = † written as U UeL UνL in terms of the unitary matrices which connect a generic basis of the lepton fields to the mass-eigenstate basis (denoted by a prime): = = = eL UeL eL,eR UeR eR,νL UνL νL. (6)

In the basis deﬁned by (5) the simplest spurion combination transforming as (8, 1) under GLF, or the coupling which controls the amount of LFV in the charged-lepton sector, is2 Λ2 ∆| = g†g = LN Umˆ 2Uˆ †. (7) minimal ν ν v4 ν

2.2. Extended ﬁeld content

The second scenario we consider has three right-handed neutrinos in addition to the SM fields, × with a maximal flavor group GLF SU(3)νR . There is a large freedom in deciding how to break this group in order to generate the observed masses and mixing. In addition to the standard Yukawa coupling for the charged leptons, in principle we can introduce neutrino mass terms transforming as (6, 1, 1), (1, 1, 6), and (3¯, 1, 3). Since we are interested in a minimal scenario, with unambiguous links between the irreducible sources of flavor-symmetry breaking and the observable couplings in the neutrino mass matrix, we must choose only one of these possibilities. In order to distinguish this scenario from the previous one, and guided by the structure of explicit models with see-saw mechanism (see e.g. Ref. [4]), we make the following assumptions:

(1) The right-handed neutrino mass term breaks SU(3)νR to O(3)νR , namely is proportional to the identity matrix in ﬂavor space:

1 ij ci j ij ij Lν -mass =− M ν¯ ν + h.c. with M = Mνδ . (8) R 2 ν R R ν

(2) The right-handed neutrino mass is the only source of U(1)LN breaking and its scale is large compared to the electroweak symmetry breaking scale: |Mν|v. ij (3) The remaining lepton-ﬂavor symmetry is broken only by two irreducible sources, λe and ij λν , deﬁned by

2 † gν gν also contains a (1, 1) piece under GLF. However, it does not contribute to lepton ﬂavor violation. Note also that if CP were an exact symmetry, VL in Eq. (4) would be required to be real, and therefore ∆|minimal = gν . V. Cirigliano et al. / Nuclear Physics B 728 (2005) 121–134 125 L =− ij ¯i † j + ij ¯ i T j + Sym.Br. λe eR H LL iλν νR H τ2LL h.c. →− ij ¯i j − ij ¯ i j + vλe eReL vλν νRνL h.c. (9) × The theory is thus formally invariant under the group GLF O(3)νR , with the following transformation properties for ﬁelds and spurions:

LL → VLLL,eR → VReR,νR → OννR, → † → † λe VRλeVL,λν OνλνVL. (10) At low energies the heavy right-handed neutrinos are integrated out, generating an effective left- handed Majorana mass matrix,

2 = v T mν λν λν, (11) Mν as in the minimal scenario. Once we identify Mν with ΛLN, the two scenarios are perfectly equivalent as far as ﬁeld content and operator structure are concerned. However, in the extended case the effective left-handed neutrino mass matrix is not an irreducible source of ﬂavor breaking, since it can be expressed in terms of the neutrino Yukawa coupling, λν :

2 2 v gν [ ]↔ v T [ ] minimal case λν λν extended case . (12) ΛLN Mν This has important consequences for LFV processes: the simplest spurion combination trans- † † forming as (8, 1) under GLF is now λνλν and is not quadratic in the neutrino masses as gν gν in † T Eq. (7). Since λνλν and λν λν are not necessarily diagonalized by the same orthogonal transformation, this also implies that the connection between LFV processes and the observables in the neutrino sector is not completely unambiguous in this case. We can overcome this difﬁculty by † neglecting CP violation in the neutrino mass matrix. With this additional assumption, λνλν and T λν λν are the same (real) combination diagonalized by the (real) PMNS matrix:

CP limit M ∆| = λ†λ −→ ν Umˆ Uˆ †. (13) extended ν ν v2 ν

3. Operator analysis for LFV processes

In addition to the SM leptons, we assume that at some scale ΛLFV above the electroweak scale and well below ΛLN (or Mν ) there are new degrees of freedom carrying lepton flavor quantum numbers. Integrating them out, their effect will show up at low energies as a series of non-renormalizable operators suppressed by inverse powers of ΛLFV. According to the MLFV hypothesis, these operators must be constructed in terms of SM fields and the spurions λe and gν (or λν ), and must be invariant under GLF when the spurions transform as in Eq. (4) or Eq. (10). We are interested in those operators of dimension five and six that could lead to LFV process with charged leptons. These operators must conserve total lepton number, otherwise they would be suppressed by the large U(1)LN breaking scale. As a consequence, no dimension-five term turns out to be relevant. For processes involving only two lepton fields, such as µ → eγ and µ- to-e conversion, the basic building blocks are the bilinears L¯ i ΓLj , e¯i ΓLj and e¯i Γej . Their L L R L R R ¯ indexes must be contracted with spurion combinations transforming under GLF as (8, 1), (3, 3) 126 V. Cirigliano et al. / Nuclear Physics B 728 (2005) 121–134 and (1, 8), respectively. Combinations of this type are

† 2 † (8, 1)∆,λeλe,∆ ,λeλe∆,..., (14) ¯ † (3, 3)λe,λe∆,λeλeλe,..., (15) † † (1, 8)λeλe,λe∆λe,..., (16) where ∆ is defined in Eq. (7) or Eq. (13) for the two scenarios. Given the smallness of λe (which is unambiguously fixed by charged lepton masses), we can safely neglect terms which are of second order in λe. We shall also assume that the entries of ∆ are perturbative, retaining only linear terms in this effective coupling. In this limit the only relevant LFV couplings are ∆ and λe∆. Moreover, we work only to linear order in the quark Yukawa couplings, λU and λD. The resulting dimension-six operators bilinear in the lepton fields can be written as

(1) = ¯ µ † (1) = † ¯ µν OLL LLγ ∆LLH iDµH, ORL g H eRσ λe∆LLBµν, (2) = ¯ µ a † a (2) = † ¯ µν a a OLL LLγ τ ∆LL H τ iDµH, ORL gH eRσ τ λe∆LLWµν, (3) = ¯ µ ¯ (3) = † ¯ OLL LLγ ∆LLQLγµQL,ORL (DµH) eRλe∆DµLL, (4d) = ¯ µ ¯ (4) =¯ ¯ OLL LLγ ∆LLdRγµdR,ORL eRλe∆LLQLλDdR, (4u) = ¯ µ ¯ (5) =¯ µν ¯ OLL LLγ ∆LLuRγµuR,ORL eRσ λe∆LLQLσµνλDdR, (5) = ¯ µ a ¯ a (6) =¯ ¯ † 2 OLL LLγ τ ∆LLQLγµτ QL,ORL eRλe∆LLuRλU iτ QL, (7) =¯ µν ¯ † 2 ORL eRσ λe∆LLuRσµνλU iτ QL. (17) † † We have omitted operators of the type H e¯RλeLLH H , which correct the charged lepton mass matrix but produce no FCNC interactions. (3) → The operator ORL does not contribute to the radiative lepton flavor changing decays i j γ , 2 and its contribution to µ–e conversion is suppressed by memµ/v .TheMFVassumptioninthe quark sector requires the RL operators with a quark current to contain at least one power of the quark Yukawa couplings λD or λU . Only the top-quark Yukawa is non-negligible, and hence, for (4) (7) the low energy processes we consider ORL –ORL can be neglected. Since the top quark Yukawa is order one, in principle, operators involving higher orders in λU could be important. They induce non-negligible FCNC currents in the down-quark sector of ∗ ¯i µ j the type VCKMtiVCKMtj dLγ dL [3].Forµ-to-e conversion only the coupling to light quarks is 2 relevant, and this additional contribution is suppressed by |VCKMtd| 1. In this paper we shall analyze the phenomenological consequences of the MLFV hypothesis only in processes involving two lepton fields, for which significant prospects of experimental improvements are foreseen in the near future [5,6]. However, one can in principle apply it also to four-lepton processes, such as µ → 3e. In this case one needs to extend the operator basis (17) (3−5) including the generalization of OLL , namely ¯ µ ¯ ¯ µ ¯ µ a ¯ a LLγ ∆LLLLγµLL, LLγ ∆LLe¯RγµeR, LLγ τ ∆LLLLγµτ LL, (18) and also new structures of the type

¯ c ¯ † c ¯ c T ¯ † ∗ c LLgνLLLLgν LL or LLλν λνLLLLλνλνLL. (19) V. Cirigliano et al. / Nuclear Physics B 728 (2005) 121–134 127

3.1. Explicit structure of the LFV couplings

Given the structure of operators in Eq. (17), it is clear that the strength of LFV processes is determined by the entries of the matrix ∆ in the mass-eigenstate basis of charged leptons. These are listed below for the two scenarios we are considering, and for the two allowed structures (normal and inverted hierarchy) of the neutrino mass matrix:

(1) Minimal ﬁeld content. According to Eq. (7),wehave 2 Λ ∗ ∗ ∆ = LN m2 δ + Uˆ Uˆ m2 ± Uˆ Uˆ m2 , (20) ij v4 ν1 ij i2 j2 sol i3 j3 atm 2 2 where matm and msol denote the squared mass differences deduced from atmospheric and solar neutrino data, respectively. The plus sign corresponds to normal hierarchy (mν1 <

mν2 mν3 ), while the minus one to the inverted case (mν3 mν1

while in the inverted hierarchy case (ν3 is the lightest neutrino) 2 mν →0 m mν →0 − −→3 sol − −→3 − 2 mν2 mν1 ,mν3 mν1 matm. (25) 2 2 matm

After using input from oscillation experiments, the couplings bij still depend on the spectrum ordering, the lightest neutrino mass, and the value of s13 (the dependence from δ has disappeared because of the assumption of CP conservation).

4. Phenomenology

We are now ready to analyze the phenomenological implications of the new LFV operators. In particular, we are interested in answering the following questions: (i) under which conditions on the new physics scales ΛLN (or Mν ) and ΛLFV can we expect observable effects in low energy reactions and therefore positive signals in forthcoming experiments? (ii) Is there a specific pattern in the decay rates predicted by MLFV? Can we use it to falsify the assumption of minimal flavor violation? → A ≡ In order to address these issues, we will study the rates for µ e conversion in nuclei Γconv − + → − + A ≡ Γ(µ A(N, Z) e A(N, Z)), experimentally normalized to the capture rate Γcapt − Γ(µ + A(Z, N) → νµ + A(Z − 1,N + 1)), and the radiative decays µ → eγ , τ → µγ , τ → eγ . Throughout, we will use normalized branching fractions defined as: A → A Γconv Γ(i j γ) B ≡ ,B→ ≡ . (26) µ→e A i j γ → ¯ Γcapt Γ(i j νiνj )

The starting point of our analysis is the effective Lagrangian generated at a scale ΛLFV 1 5 1 2 L = c(i) O(i) + c(j)O(j) + h.c. . (27) Λ2 LL LL Λ2 RL RL LFV i=1 LFV j=1 In principle one should evolve this Lagrangian down to the mass of the decaying particles. How- ever, for the purpose of the present work we shall neglect the effect of electroweak corrections and treat the c(i) as effective Wilson coefﬁcients renormalized at the low scale. In terms of these we ﬁnd (in the limit mj mi ) 4 2 2 v 2 (2) (1) 2 B → = π e |∆ | c − c i j γ 384 4 ij RL RL (28) ΛLFV and

2 5 4 A 32GF mµ v 2 1 2 (p) 1 (n) (1) (2) B → = |∆µe| − s V − V c + c µ e Γ A Λ4 4 w 4 LL LL capt LFV 3 1 1 + V (p) + V (n) c(3) + V (p) + V (n) c(4u) + V (p) + V (n) c(4d) 2 LL 2 LL 2 LL

2 1 (p) (n) (5) eD (2) (1) ∗ + −V + V c − c − c , (29) 2 LL 4 RL RL where we use the notation of Ref. [7] for the dimensionless nucleus-dependent overlap integrals (n) (p) V , V , D and we denote by sw = sin θw = 0.23 the weak mixing angle. V. Cirigliano et al. / Nuclear Physics B 728 (2005) 121–134 129

4.1. Minimal ﬁeld content

Let us now consider the scenario with minimal ﬁeld content. By making in Eqs. (28) and (29) the replacement v4 Λ4 |∆ |2 → LN |a |2 4 ij 4 ij (30) ΛLFV ΛLFV one can see that all LFV rates have the following structure

4 = −50 ΛLN ; (i) Bi →j (γ ) 10 Ri →j (γ ) s13,δ c . (31) ΛLFV −50 O The overall numerical factor 10 is chosen such that the Ri →j (γ ) have a natural size of (1). | |2 2 2 2 ≈ −52 Its value can easily be understood by noting that aij ( matm/v ) 10 . A glance at the explicit structure of the aij in Eq. (21) shows that their size is maximized for = max 2 2 s13 s13 (in both normal and inverted hierarchy), due to matm msol. In order to derive order-of-magnitude conditions on the ratio ΛLN/ΛLFV, we consider the reference case defined = max = by s13 s13 , δ 0, and the reference values quoted in Table 1. Then setting all the Wilson (2) = (3) = coefficients to zero but for cRL cLL 1, and using the overlap integrals and capture rates reported in Ref. [7] (Table 1 of [7]), we find

4 − ΛLN 6.6 × 10 50 for Al, Bµ→e = Λ 19.6 × 10−50 for Au, LFV 4 −50 ΛLN Bµ→eγ = 8.3 × 10 . (32) ΛLFV

Despite the strong dependence of the numerical coefﬁcients in Eq. (32) on s13, illustrated in Fig. 1, these results allow us to draw several interesting conclusions.

• If there is no large hierarchy between the scales of lepton-number and lepton-ﬂavor violation, there is no hope to observe LFV signals in charged-lepton processes. On the other hand, if ΛLFV is not far from the TeV scale (as expected in many realistic scenarios), it is natural 13 to expect visible LFV processes for a wide range of ΛLN: from 10 GeV up to the GUT −13 scale. For instance a Bµ→e = O(10 ), within reach of the MECO experiment, is natu- 9 13 rally obtained for ΛLN ∼ 10 ΛLFV, which for ΛLFV ∼ 10 TeV implies ΛLN ∼ 10 GeV. −13 Such a ratio of scales would also imply Bµ→eγ = O(10 ), within the reach of the MEG experiment. Note that the requirement of “perturbative” treatment of the couplings gν , together with 2 upper limits on the light neutrino masses, implies upper limits on the scale ΛLN v gν/mν . 13 By loosely requiring |gν| < 1 one obtains ΛLN 3 × 10 (1eV/mν) GeV. This means that we cannot make the ratio ΛLN/ΛLFV arbitrarily large.

Table 1 Reference values of neutrino mixing parameters used in the phenomenological analysis (for a detailed discussion see e.g. [8]) 2 2 max msol matm θsol s13 − − ◦ 8.0 × 10 5 eV2 2.5 × 10 3 eV2 33 0.25 130 V. Cirigliano et al. / Nuclear Physics B 728 (2005) 121–134

• Interestingly, µ → e conversion and µ → eγ have a quite different sensitivity on the type of operators involved. In particular, while µ → eγ is sensitive only to the LR operators, the µ → e conversion is more sensitive to the LL terms: ( ) 2 = (i) = −3 Bµ→e(cRL 1, other c 0) 3 × 10 for Al, = − (33) (3) = (i) = 1.5 × 10 3 for Au, Bµ→e(cLL 1, other c 0) (2) (i) Bµ→e(c = 1, other c = 0) 0.47 for Al, RL = (34) (1) = (i) = 0.17 for Au. Bµ→e(cLL 1, other c 0) • The comparison of the various Bli →lj γ rates is a useful tool to illustrate the predictive power of the MLFV (and eventually to rule it out from data). In Fig. 2 we report the ratios Bµ→eγ /Bτ→µγ and Bµ→eγ /Bτ→eγ as a function of s13 for three different values of the CP violating phase δ in the normal hierarchy case (the inverted case is obtained by replacing δ with π − δ). One observes the clear pattern Bτ→µγ Bτ→eγ ∼ Bµ→eγ , with hierarchy increasing as s13 → 0. Observation of deviations from this pattern could in the future falsify the hypothesis of minimal ﬂavor violation in the lepton sector. • There is a window in parameter space where we can expect observable effects in τ decays. As illustrated in Fig. 3, Bτ→µγ does not depend on s13, while Bµe does. In the normal hierarchy

(3) = (2) = (i) = Fig. 1. Ratios Ri deﬁned in Eq. (31) as a function of s13 for cLL cRL 1 and all other c 0. The shaded bands correspond to variation of the phase δ between 0 and π.

Fig. 2. Ratios Bµ→eγ /Bτ→µγ (left) and Bµ→eγ /Bτ→eγ (right) as a function of s13 for different values of the CP violating phase δ in the normal hierarchy case. The uncertainty due to the ﬁrst 3 entries in Table 1 is not shown. V. Cirigliano et al. / Nuclear Physics B 728 (2005) 121–134 131

= 10 (2) − (1) = Fig. 3. Bτ→µγ and Bµ→eγ as a function of s13,forΛLN/ΛLFV 10 and cRL cRL 1. The shading corresponds to different values of the phase δ and the normal/inverted spectrum. The uncertainty due to the ﬁrst 3 entries in Table 1 is not shown.

= → 2 2 → case, for δ π and s13 sc msol/ matm, one has Bµ→eγ 0. Therefore, close to this region of parameter space one can have a sizable Bτ→µγ while Bµ→eγ can be kept below 10 −8 the present experimental limits. In particular, for ΛLN ∼ 10 ΛLFV we ﬁnd Bτ→µγ ∼ 10 , which implies a branching ratio for τ → µγ above 10−9 possibly observable at (super) B factories. Note that a change in the ratio ΛLN/ΛLFV would only result into a shift of the vertical scale in Fig. 3, without affecting the relative distance between the Bµ→eγ and Bτ→µγ bands.

4.2. Extended ﬁeld content

The discussion of the extended model proceeds in a very similar way, by replacing the di- 4 2 2 mensionless couplings aij with the bij , and (ΛLN/ΛLFV) with (vMν/ΛLFV) . The analog of Eq. (31) reads

2 −25 vMν ˆ lightest (i) B → = R → s ,m ; c . i j (γ ) 10 2 i j (γ ) 13 ν (35) ΛLFV ˆ As illustrated in Fig. 4, the dimensionless functions Ri →j (γ ) depend on both s13 and mνlightest , = max → and the maximal values are obtained for s13 s13 and mνlightest 0. In order to explore the 2 = max sensitivity to the scale ratio vMν/ΛLFV, we again pick a favorable reference point (s13 s13 = (2) = (3) = and mνlightest 0) and set all the Wilson coefﬁcients to zero except for cRL cLL 1. In the normal hierarchy case we then ﬁnd

2 −24 2 vMν 1.3 × 10 for Al, −24 vMν B → = B → = 1.6 × 10 , (36) µ e 2 × −24 µ eγ 2 ΛLFV 3.7 10 for Au, ΛLFV while for the inverted case:

2 −25 2 vMν 6.7 × 10 for Al, −25 vMν B → = B → = 8 × 10 . (37) µ e 2 × −24 µ eγ 2 ΛLFV 2.0 10 for Au, ΛLFV 132 V. Cirigliano et al. / Nuclear Physics B 728 (2005) 121–134

ˆ (3) = (2) = (i) = Fig. 4. Ratios Ri deﬁned in Eq. (35) as a function of s13 for cLL cRL 1, all other c 0, normal spectrum and δ = 0.

2 = × 7 (2) − (1) = Fig. 5. Bτ→µγ and Bµ→eγ as a function of s13,for(vMν )/ΛLFV 5 10 and cRL cRL 1. The shading corresponds to different values of the lightest neutrino mass, ranging between 0 and 0.02 eV. The two choices of δ correspond to the ± sign in Eq. (23).

The general conclusions we can infer from this scenario are the following:

• As was the case with minimal ﬁeld content, a large hierarchy between Mν and ΛLFV is −13 required to obtain observable effects. For example, fractions Bµ→e(γ ) = O(10 ) are ob- ∼ × 5 2 ∼ ∼ × 11 tained for Mν 3 10 ΛLFV/v, which for ΛLFV 10 TeV gives Mν 2 10 GeV. • Comparing the values of Mν in this scenario versus the corresponding values of ΛLN in the minimal case, we ﬁnd that the same effect in a given LFV process is typically obtained for ΛLN >Mν . This can be understood by noting that in the minimal case the LFV am- 2 2 4 plitudes are proportional to combinations of the type mνΛLN /v , while in the extended 2 case this factor is replaced by mνMν/v . The two scales of lepton number violation are indeed equivalent when this factor is of O(1), namely for the maximal value allowed by the 2 15 “perturbative condition” on the Yukawa couplings: Mν ∼ ΛLN ∼ v / mν ∼ 10 GeV. • Much like in the previous scenario, in this case the ratios of the various LFV rates are unambiguously determined in terms of neutrino masses and mixing angles; however, the results are potentially different than in the minimal case because of the different relation between V. Cirigliano et al. / Nuclear Physics B 728 (2005) 121–134 133

LFV parameters and neutrino mass matrix. As illustrated in Fig. 5, one still observes the pattern Bτ→µγ Bµ→eγ (∼ Bτ→eγ ). For a given choice of δ = 0orπ, the strength of the µ → e suppression is very sensitive to whether the hierarchy is normal or inverted. For δ = 0 the present experimental limit on Bµ→eγ allows large values of Bτ→µγ only for the inverted hierarchy, whereas for δ = π, a large region with a sizable Bτ→µγ is allowed only for the normal hierarchy. Note that the overall vertical scale of Fig. 5 depends on both the 2 ratio (vMν)/ΛLFV—as implied by Eq. (35)—and the value of the lightest neutrino mass (as illustrated by Fig. 4).

5. Conclusions

In this paper we extend the notion of MFV to the lepton sector. We define a symmetry principle and an effective field theory which allows us to relate, in a general way, lepton-flavor mixing in the neutrino sector to lepton-flavor violation in the charged lepton sector. The construction of such effective theory allow us to address fundamental questions about the flavor structure of the lepton sector in a very general way, insensitive to many details about the physics beyond the SM. We find two ways in which we can define the sources of flavor symmetry breaking in the lepton sector in a minimal and thus very predictive way: (i) a scenario where the left-handed Majorana mass matrix is the only irreducible source of flavor symmetry breaking; and (ii) a scenario with heavy right-handed neutrinos, where the Yukawa couplings define the irreducible sources of flavor symmetry breaking and the right-handed Majorana mass matrix has a trivial flavor structure. We find that visible LFV effects in the charged lepton sector are generated when there is a large hierarchy between the scales of lepton flavor mixing (ΛLFV) and total lepton number violation (ΛLN). This condition is indeed realized within the explicit extensions of the SM widely discussed in the literature which predict sizable LFV effects in charged leptons [4,10] (for an updated discussion see Ref. [11] and references therein). Within our general framework, we find that the new generation of experiments on LFV would naturally probe the existence of new degrees of freedom carrying lepton flavor numbers up to energy scales of the order of 103 TeV, if the scale ΛLN is close to the GUT scale. The two scenarios we have considered are highly predictive and possibly testable by future experiments. While the rates for LFV effects strongly depend on the (unknown) scales ΛLN and ΛLFV, the ratio of different LFV rates are unambiguously predicted in terms of neutrino masses and mixing angles. At present, the uncertainty in our predictions for such ratios arises mainly from the poorly constrained value of s13 and, to a lesser extent, from the neutrino spectrum ordering and the CP violating phase δ. One of the clearest consequences from the phenomenological point of view is that if s13 0.1 there is no hope to observe τ → µγ at future accelerators. On the other hand, µ → eγ and µ- to-e conversion are within reach of future experiments for reasonable values of the symmetry breaking scales.

Acknowledgements

We thank Curtis Callan for valuable discussions. V.C. was supported by Caltech through the Sherman Fairchild fund, and acknowledges the hospitality of the LNF Spring Institute 2005. This work was supported by the US Department of Energy under grants DE-FG03-97ER40546 and DE-FG03-92ER40701. 134 V. Cirigliano et al. / Nuclear Physics B 728 (2005) 121–134

Table 2 Reference values of nuclear overlap integrals and capture rates used in the phenomenological analysis [7]. The overlap integrals are those computed using “Method I” in Ref. [7] (p) (n) 6 −1 V V DΓcapt (10 s ) 27 13Al 0.0161 0.0173 0.0362 0.7054 197 79Au 0.0974 0.146 0.189 13.07

Appendix A

We use the results of Kitano et al. [7] on the rate of µ-to-e conversion in various nuclei for the numerical results in this paper. In this appendix we summarize the results from Ref. [7] that ˜(p,n) we use in this paper. In terms of effective couplings gLV , AR, and of nuclear overlap integrals D, V (p,n), the branching ratio for the radiative decay is

2 2 Bµ→eγ = 384π |AR| (A.1) and the branching ratio for conversion in nuclei is 2G2 m5 = F µ ∗ +˜(p) (p) +˜(n) (n) 2 Bµ→e ARD gLV V gLV V . (A.2) Γcapt The numerical values of the nuclear overlap integrals used in this paper are given in Table 2. ˜(p,n) The relations between the effective couplings gLV , AR, and the coefﬁcients in the effective Lagrangian, Eq. (27),are: 2 4v ∗ 1 3 1 1 g˜(p) =− ∆ − s2 c(1) + c(2) + c(3) + c(4u) + c(4d) − c(5) , (A.3) LV Λ2 µe 4 w LL LL 2 LL LL 2 LL 2 LL LFV 2 4v ∗ 1 3 1 1 g˜(n) =− ∆ − c(1) + c(2) + c(3) + c(4u) + c(4d) + c(5) , LV 2 µe LL LL LL LL LL LL (A.4) ΛLFV 4 2 2 2 ev2 A = ∆ −c(1) + c(2) . R 2 µe RL RL (A.5) ΛLFV

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A stringy test of ﬂux-induced isometry gauging

Amir-Kian Kashani-Poor a,b, Alessandro Tomasiello b

a SLAC, 2575 Sand Hill Road, Menlo Park, CA 94025, USA b ITP, Stanford University, Stanford, CA 94305-4060, USA Received 17 June 2005; accepted 29 August 2005 Available online 15 September 2005

Abstract Supergravity analysis suggests that the effect of fluxes in string theory compactifications is to gauge isometries of the scalar manifold. However, isometries are generically broken by brane instanton effects. Here we demonstrate how fluxes protect exactly those isometries from quantum corrections which are gauged according to the classical supergravity analysis. We also argue that all other isometries are generically broken.  2005 Elsevier B.V. All rights reserved.

1. Introduction

Dimensionally reducing 10d supergravity on a Calabi–Yau manifold in a background of fluxes gives rise to gauged supergravity in four dimensions [1]. The ‘gauged’ refers to the fact that the hypermultiplets are charged under some of the vector multiplets. Since the hypers coordinatize a complicated scalar manifold, they can only acquire charges if the metric on this manifold exhibits isometries, which can then be gauged. This is analogous to gauging the shift symmetry of a free scalar field, which we can think of as gauging the translational isometry of the Euclidean line. The scalar manifold coordinatized by hypermultiplets in N = 2 supergravity is a quaternionic manifold [2]. When the supergravity arises upon CY compactifications of string theory (we will be considering IIA in this paper), the scalar fields in the hypermultiplets arise from complex structure moduli of the CY, as well as from the RR 3-form potential. Isometries of the scalar manifold arise (roughly) due to the shift symmetry of the RR potential in the 10d action. The 10d supergravity action in turn has this property because it captures the low energy interaction of strings, and these are not charged under the RR fields. The non-perturbative spectrum of string

E-mail address: [email protected] (A.-K. Kashani-Poor).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.08.040 136 A.-K. Kashani-Poor, A. Tomasiello / Nuclear Physics B 728 (2005) 135–147 theory, on the other hand, does contain objects, D-branes, that are charged under these fields. We hence expect D-brane instantons to break isometries of the quaternionic metric by introducing explicit RR potential dependence into the action. How must the fluxes-gauge-isometries prescription be modified to incorporate the lifting of these isometries? To give the punch line away, not at all. Our analysis shows that permissible instantons are correlated with fluxes in such a way to protect those isometries which are to be gauged from being removed by quantum corrections. This does not mean that the only way the flux modifies the action is by gauging. Indeed, in the second part of this paper we develop the case that other isometries of the hypermultiplet moduli space are lifted by instanton corrections which are themselves flux dependent. With the current state of brane instanton technology, it is a daunting task to show that these corrections preserve the quaternionic structure of the metric. The underlying assumption however is that fluxes break supersymmetry spontaneously in 4 dimensions, in which case this is guaranteed.

2. Isometries protected by ﬂuxes

In this section we show how NS flux protects certain isometries. In the first two subsections we briefly remind the reader how the gauging arises upon dimensional reduction, and how brane instantons enter the calculation. In Section 2.3, we discuss how quantum corrections seem to invalidate the fluxes-gauge-isometries picture, by removing the isometries. Tadpole considerations come to the rescue constraining allowed instantons so that the isometries to be gauged are preserved. In this last subsection, we also introduce some instanton configurations which will occupy us in later sections, when trying to assess how the other isometries are affected by brane instanton corrections.

2.1. Review of ﬂuxes gauge isometries

We consider type IIA compactified on a Calabi–Yau X. Each hypermultiplet, aside from the universal one, contains two scalars za which stem from complex structure moduli of X, and two A ˜ 3 A ˜ A scalars ξ , ξA from the coefficients of C3 in an expansion in a basis for H : C3 = ξ αA + ξAβ . 2,1 2,1 0 ˜ Here a = 1,...,h , A = 0,...,h . ξ and ξ0 together with the dilaton φ and the NSNS axion a are the scalars of the universal hypermultiplet. Via dimensional reduction, these scalars enter into the metric of the hypermultiplet scalar manifold as follows [3,4]: 4φ 2 2 a b¯ e A A A A ds = dφ + g ¯ dz dz¯ + da + ξ˜ dξ − ξ dξ˜ da + ξ˜ dξ − ξ dξ˜ ab 4 A A A A 2φ e − − Im M 1 AB dξ˜ + M dξC dξ˜ + M¯ dξD . 2 A AC B BD i A A i 2,2 We consider turning on the fluxes F4 = eiω˜ and H = p αA +qAβ , with ω˜ a basis for H (X) (this essentially encompasses the most general choice of fluxes consistent with locality of the four-dimensional action [5,6]). Turning on F4 leads to the gauging of the isometry

(kF )i =−2ei∂a (2.1) of the scalar manifold by the ith vector multiplet, while H leads to the gauging of A ˜ A A k = p ξ − q ξ ∂ + p ∂ A + q ∂˜ (2.2) H A A a ξ A ξA by the graviphoton. A.-K. Kashani-Poor, A. Tomasiello / Nuclear Physics B 728 (2005) 135–147 137

In the following, the symplectic structure on H 3 will not be relevant for our considerations, = i { i} 3 hence we do not indicate it in our notation, writing C3 ξiγ ,for γ a basis of H , and likewise H = p γ i . We also introduce the dual basis in homology {Γ }, such that γ j = δj . i i Γi i 2.2. Brane instantons

Brane instantons are Euclidean branes which wrap cycles in the internal manifold at a point in time. The rules of incorporating brane instanton corrections to 4d effective actions are not completely understood. Arguing along the lines of [7–9], we use the E-brane action to derive vertex operators which describe the coupling between spacetime ﬁelds and brane ﬁelds. To calculate instanton contributions to a spacetime correlator, we assemble the appropriate vertex operators and then perform the path integral over the brane degrees of freedom. In addition, any contribution in the instanton sector will contain a factor from the classical part of the brane instanton action, − O = e Sinst,class ··· inst − + = e vol i C ···. (2.3)

2.3. Isometries lost and isometries regained

The isometries kF and kH we are considering depend on modes descending from the NSNS B field and the RR potential C3. da derives from the spacetime components of B. Since these components of B, in the absence of sources and non-trivial spacetime topology, are pure gauge and can hence be set to zero in the brane action, neither a contribution to Sinst,class depending on a nor a vertex operator coupling a to degrees of freedom on the brane arise.1 We hence do not worry about brane instanton corrections lifting kF , and in the case of kH , we can focus on the 2 shift symmetry part of the isometry in ξi . For E2 branes (the instantonic version of D2 branes), the classical contribution already appears to generically break any isometry involving a shift symmetry in the fields ξi . Consider an instanton configuration consisting of E2 branes wrapping various cycles; let the sum of these cycles in homology be i Γinst = c Γi. (2.4) i

This conﬁguration contributes a ξi dependence i C3 = c ξi (2.5) Γ i to the effective action. Acting with (2.2) on (2.5) results in i kH C3 = c pi, (2.6) Γ i

1 The underlying assumption here is that going off-shell does not give rise to additional vertex operators. 2 Because the isometry in question involves a ﬁeld dependent shift in a in addition to a constant shift in the ξi ,the shift symmetry of RR potentials does not prove that it is preserved by perturbative stringy corrections. Nevertheless, this is believed to be the case [10–12]. We feel the question of non-perturbative lifting is more pressing, as brane instantons pose an immediate threat to the isometry’s survival. 138 A.-K. Kashani-Poor, A. Tomasiello / Nuclear Physics B 728 (2005) 135–147

Fig. 1. A permissible instanton configuration. The ellipses denote E2 branes wrapping homologically different 3 cycles, the connecting lines are E0 branes. The numbers inside the ellipses indicate the quanta of H flux through the respective cycle. which generally does not vanish. Hence the fluxes-gauge-isometries picture does not appear to survive quantum corrections, since the isometries to be gauged do not. We now proceed to show why this is not the case. The key point is that H flux induces a magnetic charge for the gauge field on the brane. One way to see this is to note that B always appears in the combination F + B. The Bianchi condition thus becomes dF = 0 → dF + H = 0 (in general, this has a refinement in integral cohomology which gains a contribution from the third Stiefel–Whitney class; however, for three-cycles, this is always zero; see for example [7]). Hence, an E2 brane wrapping a compact cycle threaded by H flux violates Gauss’ law on its worldvolume, unless H is cohomologically trivial. On i an instanton wrapping a cycle c Γi , in the notation introduced above, this condition reads i = i c pi 0. The LHS of this equation is exactly the potential violation of the isometry which we obtained in (2.6) ! We conclude that the instantons which would break the isometry kH , i.e., i = those for which i c pi 0, are ruled out by Gauss’ law. One could think of more elaborate strategies to produce instantons which break the isometry. We can add E0 branes ending on the E2 brane, as these are magnetic sources for the worldvolume gauge field F . A configuration of E2 and E0 branes which is consistent with Maxwell’s law is depicted in Fig. 1 (this configuration is a close relative of ones which have appeared in [13]). However, each E0 is a charge on the E2 on which it starts, but a charge of opposite sign on the E2 on which it terminates. Thus we can quickly conclude that also for this more general configuration i c pi = 0. (2.7) i After having saved the isometries we needed from quantum corrections, we will now show how non-academic the worry was: in general, all the isometries which are not needed for gauging get lifted by quantum corrections.

3. Isometries lifted by instantons

In the previous section, we demonstrated that those isometries of the form A ˜ A A k = p ξ − q ξ ∂ + p ∂ A + q ∂˜ (3.1) H A A a ξ A ξA that are to be gauged upon turning on H ﬂuxes are protected. More speciﬁcally, we argued that the part of the isometries at stake is the shift symmetry of the instanton contribution in ξi = A ˜ { } (i 1,...,b3, ξi collectively denoting ξ and ξA). These shift symmetries ∂ξi span an integral A.-K. Kashani-Poor, A. Tomasiello / Nuclear Physics B 728 (2005) 135–147 139

S lattice b3 of dimension b3 (the coefﬁcients are integer). The H -ﬂux represents a vector in this lattice, and we showed that symmetries in the direction of this vector are protected by Gauss’ law. In this section, we wish to show that the symmetries in the (b3 − 1)-dimensional orthogonal S complement b3−1 are lifted.

3.1. Which instantons are present?

In the previous section, we considered instantons only at the crude level of homology. It is far from obvious that choices of representatives of the homology cycles appearing in the instanton configurations of the previous section can be made that give rise to a local minimum of the action. Since we were able to rule out the lifting of the isometries of interest by all such candidate instantons, the possible over-counting was inconsequential. In this section, we wish to argue that all isometries that are not protected by flux are lifted. We must hence consider the question of what type of brane configurations qualify as instantons more carefully. In the absence of flux, if we can saturate the supersymmetric BPS bound, we can minimize the action globally; geometrically, this gets translated into the condition that there exist a special Lagrangian (SLag) submanifold representative of the cycle with a flat bundle over it, as first obtained in [7]. The presence of a B field modifies this result in two ways [14].First,inthe absence of E0 branes, H has to be cohomologically trivial on the submanifold, as we have already seen. Second, the flatness condition now becomes, not surprisingly, F + B = 0. In presence of B the bundle on a brane should not be seen as a true bundle anymore, but as a projective bundle determined by B (see for example [15]); hence this condition can be satisfied. We have pointed out before that the integral refinements to the tadpole condition are not relevant for three- dimensional SLag’s. S To lift all elements of b3−1 by E2 instantons wrapping SLags would require a basis of the S subspace of 3-homology dual to b3−1 for which each element had a SLag representative somewhere in moduli space. The final qualification is possible because the existence of isometries is a global question on moduli space; it is sufficient that the vector field not be Killing at some point to declare the isometry broken. In spite of this qualification, this is a rather strong requirement, S since by appropriate choice of the flux H , the corresponding b3−1 can be dual to any codimen- sion 1 subspace of the third homology vector space H3; and it is not clear that all cycles in H3 (not just a basis3) have a SLag representative. We therefore argue next for the existence of a class S of instantons, of the type depicted in Fig. 1, which, to lift all elements of b3−1 for any choice of flux H , requires only that a basis of the whole of H3 exists such that each element has a SLag representative somewhere in moduli space. To argue for the existence of such instantons, let us first consider the fluxless case. We start by a brane wrapping a SLag cycle at a certain point in moduli space and follow its fate as we move in moduli space [17,18]. There are stability walls beyond which a SLag which represents a cycle C = A + B can decay to two separate but intersecting SLag’s, one on the cycle A and one on the cycle B, which are separately BPS, but not mutually BPS. Conversely, on the other side of the wall, A and B put together decay to the SLag representing A + B. We can think of the process as a formation of a bound state, via condensation of a (string) world-sheet tachyon.

3 That such a basis exists can be seen by going to the mirror. There, we have to look for a basis of K-theory all elements of which are Π-stable [16]. Thanks to the Lefschetz theorem, for any integral element in H 1,1 and in H 2,2 there exists a dual analytic cycle. These cycles are all stable in the large volume limit of the mirror. Hence their duals are SLag at least at one point of the complex structure moduli space of the original Calabi–Yau. 140 A.-K. Kashani-Poor, A. Tomasiello / Nuclear Physics B 728 (2005) 135–147

Now, in our case in which H is also present, after the SLag representing A + B decays, we are = left with two intersecting SLag’s representing A and B, such that, if we previously had A+B H 0, now A,B H need not be zero. E0’s must then be added to the configuration to rescue Gauss’ law. We have thus almost arrived in the situation in Fig. 1; except that the ellipses here intersect. The E0’s can all shrink to the intersection point, or be stuck at some geodesics between the cycles which is a local minimum of the path-minimization problem. That these configurations contribute is required by the fact that amplitudes to which instantons contribute cannot abruptly jump on any stability wall, so when a SLag wrapped by an E2 brane disappears, something else must take its place. Encouraged by this, we can now ask whether action minimizing E2– E0 configurations also arise for E2’s wrapping individual SLag cycles Ai that do not intersect, hence cycles the homology class of the sum C = Ai of which does not necessarily have a SLag representative anywhere on moduli space. Again, in the intersecting case, connecting the configuration continuously to a BPS configuration was circumstantial evidence for this being the case. In the non-intersecting case we have to work harder. Each of the E2’s we are taking is wrapping a SLag, and each E0, now of finite length, lies on a geodesic connecting them, so each constituent separately minimizes the action; but can there be a deformation of the whole system that lowers the action? We do not have a definite argument to rule this out, but we can note the 3 3 following. Given a typical size R of the Calabi–Yau, the E2’s have action of order R /(gsls ), whereas the E0’s have action ∼ R/(gsls). To speak of geometric objects in the first place, we have to take R ls ; hence the E2’s contribute much more action than the E0’s do. Hence it makes sense to first settle the E2’s in such a way as to minimize the action, and then insert the E0’s to accommodate them. This intuition will also guide us in the zero mode analysis which we perform below. Of course we do not have full control over this new kind of instanton; we will try to analyze the contribution of such instantons together with that of the more conventional instantons in the rest of this section. We will argue that they do contribute; at the very least, it seems that the conditions for them to contribute have little to do with the conditions that a cycle admit a SLag representative.

3.2. Ingredients of the instanton computation

I J K L As in [7], we shall take contributions to the 4 hyperino coupling [19] RIJKLχ¯+χ+χ¯− χ− in the instanton background as our benchmark of whether instanton conﬁgurations are deforming 2,1 the hypermultiplet metric (RIJKL is the Sp(2h ) piece of the quaternionic curvature, ± indicate I J K L the chirality of the spinors). For this purpose, we consider the correlator ¯χ+χ+χ¯− χ− in an instanton background. As we noted in Section 2.2, any contribution to this correlator from a given = + i instanton sector will involve a factor of the classical instanton action, Sinst,class vol i ciξ , i and hence break isometries of the quaternionic metric that are not preserved by ciξ . As we mentioned in that section, the rules for performing brane instanton calculations are not well understood. Results for the exact form of instanton corrections to the hypermultiplet metric can be found in the literature [20–27]; they are usually obtained via indirect arguments, or in 4d SUGRA. Our modest goal is to determine whether the contribution to the correlator in an instanton background vanishes or not; hence, any non-vanishing contribution will do. When arguing for the presence of corrections to correlators in non-trivial instanton backgrounds, special attention must be devoted to two issues: fermionic zero modes, and integration of non-trivial determinants. The issue of zero modes will be the topic of the following subsection. The second issue we consider here. A.-K. Kashani-Poor, A. Tomasiello / Nuclear Physics B 728 (2005) 135–147 141

After having integrated out massive modes, one gets an effective action to be integrated on the manifold of bosonic zero modes, and which is a function of them only. In many cases with appropriately high supersymmetry, this function can simply be 1, but in our case supersymmetry is spontaneously broken (a fact which will play an important role later). So one should worry that integration of this effective action gives zero. This worry does not arise for instantons that are isolated, such that the integral over the bosonic moduli space is simply a spacetime integral. For this reason we will restrict to the case in which there exists a basis {Γi} of H3 consisting of homology spheres, which are isolated. This turns out to be the case whenever we have a Landau– Ginzburg point in the moduli space, though not in general. Here we are also implicitly using that zero modes of the combined system E2–E0 can be obtained by considering the E2’s alone, a fact for which we will argue in Section 3.2.2.

3.2.1. Zero modes In this section, we discuss the notion of exact zero modes, to argue that in contradistinction to the fluxless case, any brane instanton configuration can contribute to corrections to the hypermultiplet metric. In a nutshell, this is because the fluxes break the supersymmetry of the background. In the latter part of this section, we introduce the notion of approximate zero modes as a useful tool for organizing our perturbative calculation. A fermionic path integral is formally defined by expanding the fermion ψ in modes ψi , ψ = ξiψi , and performing Berezin integration over the Grassmann coefficients ξi .Theψi are, as we will discuss, the eigenmodes of a suitably chosen operator M. If the integrand does not depend on one of the ξi , the integral vanishes. Assume the action is quadratic in ψ, I = ψD(φ)ψ, with φ generically denoting bosonic fields in the theory. If the classical equations of motions have a solution (φ0,ψ0), such that in particular ψ0 is a non-trivial element of the kernel of the operator D(φ0), then we call ψ0 an exact fermionic zero mode. Such exact zero modes will generically not exist. In supersymmetric theories, they can be generated by acting on bosonic instanton solutions with the supercharges which are broken in the presence of instantons. Expanding ψ in modes of D(φ0), i.e., choosing M = D(φ0), we see that the action evaluated at φ = φ0 is independent of the exact zero modes. The restriction φ = φ0 implies that we can conclude that without insertions to soak up the exact zero modes, the path integral vanishes at tree level in all bosonic fields.Zero mode dependence does arise away from φ = φ0, i.e., when quantum modes of the bosonic fields are taken into account. To decide whether these quantum modes are sufficient to lift the zero modes is tricky. ADS argue that aside from those zero modes generated by supersymmetry, this should be possible.4 Even without performing a technically daunting detailed analysis for our instantons, we hence feel doubly assured that they will generically contribute to the hypermultiplet

4 A more modern localization argument for the supersymmetric zero modes not being lifted relies, roughly speaking, on introducing a different mode expansion of the fermion over every point of the bosonic field space. This procedure can be realized in the presence of a fermionic symmetry F acting on field space E. If the action is free, i.e., is without fixed points, bosonic field space is given by E/F, and the fermionic modes over each point of E/F are obtained by acting by F. As long as the integrand is invariant under the action of F, it will be independent of these modes, and the fermionic integral over each point of E/F vanishes. If however there are loci on which the action is not zero, i.e., supersymmetric field configurations, then these must be excluded from E before the quotient can be taken. These loci give rise to non-trivial contributions to the path integral. BPS instantons are field configurations which are fixed points of some but not all of the supercharges. Such configurations can contribute to the expectation value of operators which are not invariant under those supercharges which do not fix the instanton. For example, if we want to determine whether a superpotential is generated in a four-dimensional theory, we can 142 A.-K. Kashani-Poor, A. Tomasiello / Nuclear Physics B 728 (2005) 135–147 metric: exact fermionic zero modes will generically not exist, since supersymmetry is broken by the fluxes; if they do exist, they will be lifted by quantum interactions. Since we have now convinced ourselves that our amplitude does not vanish due to known non-perturbative arguments, we can go about organizing our perturbation theory. In the large volume limit, form fields are suppressed by powers of the size of the Calabi–Yau R. The flux of p a p-form field Fp, which is an integral over a cycle, is of order FpR , and has to be an integer. 3 4 So H is of order 1/R and F4 of order 1/R . In the limit in which R is large, fluxes can hence be treated as perturbations. We decompose the quadratic operator D introduced in the previous section as D0 + δD0, where all of the flux dependence is incorporated into δD0, which we treat as an interaction term. Zero modes of D0 (which are present due to the supersymmetry in the absence of fluxes) are called approximate zero modes. By expanding fermions in modes of D0, i.e., choosing M = D0, we need to soak up these approximate zero modes in each term in our perturbative expansion, by insertions or by pulling down ψδD0ψ interactions. We flesh out this picture in the remaining parts of this section.

3.2.2. Instanton vertices E2 instantons For E2 instantons wrapping SLags, the fermion that featured prominently in the above is the world volume fermion θ on the E2. It has 4 approximate zero modes, as in the absence of flux, the instanton breaks half of the 8 supersymmetries of the background. Any correlator of spacetime fields hence needs 4 insertions of θ so as not to vanish in this instanton background. These insertions are provided by bringing down interaction vertices of the worldvolume action involving θ (these will be gravitino-θ couplings and flux induced mass terms for θ, as we discuss below). Let us label these interaction vertices by V(x)i . x here denotes the position of the instanton in spacetime. It is a collective coordinate the integration over which is included in the path integral measure. To soak up the correct number of zero modes, we hence V end up with a factor dx α iα (x) in the path integral. As far as Feynman rules are concerned, such a factor behaves just like an ordinary interaction vertex in the action.

E2–E0 instantons We wish to argue that the above considerations are also sufficient to deal with the E2–E0 instantons. For this purpose, we want to consider, in addition to the flux, the E0’s as a perturbation as well. This seems consistent, since, as we have seen above (Section 3.2), the E0’s are much lighter than the E2’s. We hence want to think of the E0 branes as solitonic solutions in the E2 brane world volume theory. This has been worked out in various situations dual to the E2–E0 system of interest here in the literature (see e.g. [28–30]). The E0 branes introduce magnetic charges for the gauge field on the world volume. Solutions to the classical equations of motion for the bosonic fields Xi which encode the transverse position of the brane in the presence of these charges will, far away from the E0 sources, simply be Xi = const, while close to the sources, Xi will spike out to build a bridge to the other E2 brane. In this picture, the E0 brane degrees of freedom arise as fluctuations around the bridge part of the configuration of the world volume fields of the E2. These could be studied by replacing this contribution to the E2 brane action by a one-dimensional E0 brane action. calculate the correlation function of two fermions. The two fermionic insertions will break two of the supercharges. The other two supercharges localize the path integral on the zero locus of the two associated fermionic symmetries. Since the BPS instanton is contained in this zero locus, the generation of a superpotential due to this instanton is possible. A similar reasoning can be applied to dθ4 terms in N = 2 theories. A.-K. Kashani-Poor, A. Tomasiello / Nuclear Physics B 728 (2005) 135–147 143

With this picture in place, we want to perform a zero mode analysis of the E2–E0 system. First consider bosonic fluctuations. These should be well approximated by independent fluctuations of the E2 branes, with the E0’s simply adjusting themselves to accommodate these fluctuations: the E0’s, being geodesics, will generically be rigid far away from the junction points with the E2’s. The fluctuations hence have support mainly on the individual E2’s. The same is hence true for the approximate fermionic zero modes related to these via supersymmetry. We will assume that these are the only zero modes present. This point of view instructs us to consider action minimizing configuration where each E2 separately, i.e., ignoring the E0 branes and by consistency therefore also the potentially cohomologically non-trivial H -flux, minimizes the action, e.g., by wrapping a SLag, which is the case we will consider. Then, while they generically will break (in the absence of flux) different supersymmetries, each should exhibit a world volume fermion θ with four approximate zero modes (arising from the action of different supercharges on the E2’s). Again, these zero modes are the ones that would be exact if we were considering the E2’s only in a background without flux, and become approximate once we add the perturbation of the flux and the E0’s. Aside from allowing us to settle the E2 constituents on SLags, the above analysis suggests that the approximate zero modes of the E2–E0 system can be thought of as being supported on the E2’s only. We will hence only need to determine interaction terms of the E2 world volume fermion θ.

Couplings of the hyperinos to brane fermions Following [7], we obtain the coupling between the worldvolume spinor and the hyperinos from the corresponding interaction term in the brane action. For simplicity, we derive this coupling first in the context of reduction from eleven dimensions to five, that is, by considering the worldvolume action of the M2 brane. We will then reduce the result from five to four dimensions. In the CY reduction of 11d SUGRA to 5d N = 2 SUGRA, the hyperinos descend from the 11d gravitino. In the M2 action there is a coupling [7,31–33] between the gravitino ψM and the worldsheet spinor θ: i ψ† ∂aXM ∂ XN γ − abc∂ XM ∂ XN ∂ XP γ θ = ψ†γ a(1 − Γ)θ. (3.2) M a N 2 a b c NP a √ The first term stems from expanding out the superspace counterpart of g, the second term from the WZ term for C3 on the M2 brane. We have also rewritten the vertex in a more compact form, anticipating some notation that will be useful later; quantities with worldsheet index a are pulled M ≡ M ≡ i abc back using the ∂aX , as usual, for example ψa ∂aX ψM . The matrix Γ 3! γabc plays a prominent role in kappa-symmetric actions, and it will appear again in flux-induced vertex operators below. To extract the hyperino-θ coupling from the gravitino-θ vertex, we need to consider the mode expansion of the gravitino. In compactifying M-theory on a CY, the massless hyperinos χI , I = 2,...,2h2,1 + 2, arise as coefficients of the expansion of the internal directions of the 11d gravitino (as well as the gauginos ηΘ , Θ = 1,...,h1,1 − 1; recall that, in contrast to Calabi– Yau reduction to four dimensions, one obtains h1,1 − 1 vector multiplets; an additional vector multiplet arises upon reduction of the gravity multiplet from five to four dimensions). The har- 0,p monic spinors in the internal directions are well known to be given by ⊕pH (CY ). A harmonic 0,p 2,p vector-spinor whose vector index is holomorphic is given by ⊕pH (CY, T ) =⊕pH (CY ); 144 A.-K. Kashani-Poor, A. Tomasiello / Nuclear Physics B 728 (2005) 135–147 we have used the Dolbeault theorem and triviality of the canonical bundle. In formulas,

h2,1 h1,1 Λ j¯k¯ Θ j¯ ψ = χ ⊗ d γ + + η ⊗ d γ + i Λ ij¯k¯ Θ ij¯ Λ=1 Θ=1 h2,1 h1,1−1 Λ j¯k¯ h1,1 l¯ jk Θ ij¯ = χ ⊗ d γ + + χ ⊗ d Ω γ − + η ⊗ d γ + , (3.3) Λ ij¯k¯ 0 il¯ jk Θ ij¯ Λ=1 Θ=1 2 h1,1 ∼ = where we have chosen the basis of H such that d J and set χ0 ηh1,1 .Here± are the 5 two Weyl covariantly constant spinors on the Calabi–Yau. We can likewise reduce ψi¯, getting 1,1 ¯ 1,1 another set of 5d fermions {˜χ }. Then, by deﬁning d0 ≡ dh Ωl and d0 ≡ dh Ωl ¯¯ we can Λ ij k il¯ jk i¯j¯k¯ il¯ jk assemble everything in the form [7], = ⊗ I NP + +··· ΨM χI dMNP γ (+ −) , (3.4) = 2,1 + ={ ˜ } I ={ Λ Λ } ··· I 1,...,2h 2, where χI χΛ, χΛ , d d3,0,d2,1,d1,2,d0,3 , and denotes the gaugino part. The zero modes of the world volume spinor θ are of the form ξ ⊗ , where ξ is a iφ −iφ 4d spinor and = e Γ + − e Γ − [7]; φΓ here denotes the phase of the SLag. Inserting the expansion (3.4) into the gravitino-θ vertex yields the following hyperino-θ vertex operator [7]. I† ±iφΓ Vχθ = 8iχ ξ e dI , (3.5) Γ with the positive sign for I = 1,...,h2,1 + 1, and the negative sign for the remaining indices. For notational convenience, we redeﬁne the basis of three cocycles to absorb the φΓ phase. When = + + − we reduce this expression to four dimensions, it retains the same form, χI χI χI being understood now as a four-dimensional Dirac spinor.

The flux induced mass term The quadratic term in the world volume fermion in the M2 action is given by [32–34] as 1 θ † γ aD + G γ MNPQ + 8δ M γ NPQ γ a (1 − Γ)θ. (3.6) a 288 MNPQ a a As above, a,b,... are coordinates along the world volume of the brane, M,N,... are target = i abc space indices, and Γ 3! γabc. The gamma matrix combination in the bracket is familiar from the gravitino supersymmetry variation of 11-dimensional supergravity. We now perform the computation of the instanton vertex induced by flux in the following way. While staying in eleven dimensions, we take G = F + H ∧ dx5; this gives rise to a vertex for five-dimensional spinors which we then reduce along the fifth direction to four-dimensional 11 = 5d ⊗ 6d Dirac spinors. Since γ γ5 γ7 arises in the part of calculation involving H ,theresult depends on the chirality of the 4d part of the world volume spinor zero modes. The final result

5 In Minkowski signature, the gravitino is Majorana, which leads to a relation between ψi and ψi¯. However, in Euclid- ean signature, which is appropriate to instanton computations, the Majorana condition is not possible in 11d; so we will always work with a complexiﬁcation of the fermionic degrees of freedom. This subtlety is present in many instanton computations, also in four dimensions. For an analysis of rigid N = 2 SUSY in Euclidean signature, see [35]. A.-K. Kashani-Poor, A. Tomasiello / Nuclear Physics B 728 (2005) 135–147 145 reads ± † 1 2 1 3,0 2,1 V = ξ±ξ± − vol F J ∓ Im 2H + 6H . (3.7) θθ 2 3 3

3.3. Contributions to the curvature-4-fermion coupling

We now assemble the pieces from the above analysis to calculate the leading instanton correction to the 4 chiralino term I J K L χ¯+χ+χ¯− χ− (3.8) in the 4d action. In the case of an E2 brane wrapping a SLag, we have the same number of approximate zero modes as [7], namely 4. The leading instanton contribution to (3.8) is hence given by four insertions of Vχθ, yielding + ¯ I J ¯ K L ∼ vol(Γ ) i Γ C3 χ+χ+χ− χ− inst e dI dJ dK dL (3.9) Γ Γ Γ Γ as in [7]. We will have additional zero modes for the E2–E0 conﬁgurations. Let us for deﬁniteness consider a case in which there are two SLag’s Σ1, Σ2 connected by some E0’s, and that there are only the eight zero modes coming from broken supercharges. In addition to the vertex operators from the previous case, we now need two Vθθ vertices to soak up the approximate zero modes of θ. The result is vol(Γ1+Γ2)+i + C3 + − ¯ I J ¯ K L ∼ Γ1 Γ2 V V χ+χ+χ− χ− inst e θθ(Γi) θθ(Γj ) ∈Z i,j 2

× dI dJ dK dL (3.10)

Γi+1 Γi+1 Γj+1 Γj+1 V± where θθ are given by (3.7). The conclusion of this section is that E2–E0 instantons give contributions which generically break all isometries not protected by ﬂux (due to the C3 dependence coming from the exponent). Also note that the Vθθ vertices introduce explicit ﬂux dependence in the quantum corrected quaternionic metric.

Acknowledgements

We would like to thank L. Anguelova, P. Aspinwall, A. Ceresole, G. Dall’Agata, S. Kachru, Juan Maldacena, J. McGreevy, G. Moore, M. Rocek,ˇ A. Strominger, S. Vandoren for useful discussions and correspondence. This work was supported by NSF grant 0244728. The work of A.K. was also supported by the US Department of Energy under contract number DE-AC02- 76SF00515.

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Hagedorn transition and chronology protection in string theory

Miguel S. Costa, Carlos A.R. Herdeiro, J. Penedones, N. Sousa

Departamento de Física e Centro de Física do Porto, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre, 687 4169-007 Porto, Portugal Received 13 July 2005; accepted 15 August 2005 Available online 2 September 2005

Abstract We conjecture chronology is protected in string theory due to the condensation of light winding strings near closed null curves. This condensation triggers a Hagedorn phase transition, whose end-point target space geometry should be chronological. Contrary to conventional arguments, chronology is protected by an infrared effect. We support this conjecture by studying strings in the O-plane orbifold, where we show that some winding string states are unstable and condense in the non-causal region of spacetime. The one- loop string partition function has infrared divergences associated to the condensation of these states.  2005 Elsevier B.V. All rights reserved.

1. Introduction

The possibility of time travel has been one of the theoretical physicists’ favorite puzzles. Of particular interest is time traveling to the past, since it raises many causality paradoxes. Although our experience tells us such voyages are unlikely to be achievable, there is no rigorous proof of their impossibility. In an attempt to address this problem, Hawking put forward a chronology protection conjecture [1], extending earlier results by Tipler [2]. His argument was based on the behavior of quantum ﬁeld theory in the presence of closed null curves, since in a spacetime with a Cauchy surface any future causality problems must be preceded by the formation of one such curve. Hawking argued that the one-loop energy–momentum tensor becomes very large near a

E-mail addresses: [email protected] (M.S. Costa), [email protected] (C.A.R. Herdeiro), [email protected] (J. Penedones), [email protected] (N. Sousa).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.08.022 M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178 149 closed null curve, therefore causing a significant backreaction in the geometry. The outcome would be a naked singularity or a gravitational evolution that prevents formation of the closed null curve itself. Either way, the mechanism that protects chronology is an ultraviolet effect. Since the ultraviolet behavior of string theory is different from that of quantum field theories, it is questionable that the above mechanism will protect chronology. Theories with extended objects provide privileged probes of spacetimes with closed causal curves since the extended objects can wrap around them. As mentioned above, closed null curves are particularly important because causality problems first arise when such curves are created. In the framework of string theory, there are winding string states that can become light just before they wrap a closed null curve, when their proper length is of the string scale. A phase transition can then occur, analogous to the Hagedorn phase transition where winding string states become massless for a compactification circle with radius of the string scale [3,4]. It is therefore plausible that light winding states play a role in protecting chronology. In this work we establish this analogy in detail by studying a toy model, where we are able to show that such phase transition does in fact occur and it is associated to the condensation of string fields in the causally problematic region. We expect this mechanism to hold in general, which leads us to the following conjecture: closed null curves do not form in string theory because light winding states condense, causing a phase transition whose end-point target space geometry is chronological. This mechanism for protecting chronology is truly stringy and is an infrared effect, in sharp contrast with Hawking’s proposal. The toy model we shall consider is an orbifold of three-dimensional Minkowski space introduced in [5], called the O-plane orbifold. The orbifold quotient space is a good laboratory to study the issue of chronology protection because it is free of singularities and it develops closed causal curves. There is, however, a truncation of spacetime that is free of closed causal curves. It is the main point of this work to show that some winding string states, which condense in the pathological region of spacetime, give a possible mechanism for the protection of chronology. For other works of strings on orbifolds with closed causal curves and on closed string backgrounds with bad chronology see [6–47]. Since the orbifold closed causal curves are topological, it is questionable whether the conjectured mechanism is generic, in the sense that it might not work for non-topological closed causal curves dictated by the Einstein equations [48]. However, when fermions obey anti-periodic boundary conditions around the closed curves, we believe the mechanism is general because the appearance of unstable winding states relies only on the existence of closed curves with very small proper length. We start our discussion in Section 2 by giving a detailed review of the O-plane orbifold. In particular, we analyze its causal structure, wave functions and quantum field theory divergences. We find infrared divergences associated to the condensation of some particle states and show that the theory is ultraviolet divergent because of the presence of the non-causal region. In Section 3 we consider strings in the O-plane orbifold. Again we analyze the string wave functions and compute the one-loop partition function, which when appropriately interpreted has only infrared divergences. These divergences are associated to the presence of unstable string states, whose condensation is conjectured to protect chronology in Section 4. We give some concluding remarks in Section 5. Appendix A contains some technical results regarding wave functions and corresponding Hilbert space measure. In Appendix B we give an alternative derivation of the particle and string partition functions using the path integral formalism. 150 M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178

2. Review of O-plane orbifold

In this section we shall review some basic facts about the O-plane orbifold introduced in [5]. This is an orbifold of three-dimensional Minkowski spacetime. To deﬁne it consider the light-cone coordinates (x−,x,x+), normalized such that the ﬂat metric is given by ds2 = −2 dx+ dx− + dx2. The orbifold group element Ω = eκ is generated by the Killing vector

κ = 2πi(RP− + ∆J ), where ∂ ∂ ∂ iJ = x+ − x ,iP− = , ∂x ∂x+ ∂x− are, respectively, the generators of a null boost and a null translation and R and ∆ are constants. The orbits of this Killing vector field can be found in Fig. 1. The orbifold identification can be explicitly written as x− x− + 2πR x ≡ x ∼ x − 2π∆x− − 2π 2R∆ . + + − + 2 2 − + 4 3 2 x x 2π∆x 2π ∆ x 3 π R∆ Since the generator of the O-plane orbifold involves the time direction one might wonder about the causal structure of quotient space, and in particular about the existence of closed causal curves. For this purpose it is convenient to introduce a new set of coordinates y = (y−,y,y+), in which the orbifold action becomes trivial. These are defined by

− − E − x = y ,x= y − (y )2, 2 2 + + − E − x = y − Eyy + (y )3, 6 where E ≡ ∆/R. Now the orbifold identiﬁcation becomes simply − − y ∼ y + 2πR

− Fig. 1. Orbits of the Killing vector ﬁeld κ projected in the XX -plane of the covering space. The surface κ2 = 0divides spacetime in a good region and a bad region. The dashed straight lines are timelike geodesics connecting image points, − projected in the XX -plane. M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178 151 and the metric − + − ds2 =−2 dy dy + 2Ey(dy )2 + dy2. The surface y = 0 divides spacetime in two regions, one with κ spacelike (y>0)—the good region—and the other with κ timelike (y<0)—the bad region. The region with y<0 is expected to be pathological. Indeed, although there are closed timelike curves passing through every point in this space, these curves always enter the y<0 region as shown in Fig. 1. Moreover, in the covering space there are points which are light-like separated from their image. These points lie on the so-called polarization surfaces [49]. Null curves joining a point and its image become closed null curves in quotient space and are dangerous in quantum ﬁeld theory because they are responsible for extra divergences, as we shall review in Section 2.3.

2.1. Particle dynamics

To understand dynamics in the O-plane orbifold, we start by analyzing the motion of point particles. Clearly, particle trajectories are straight lines in covering space. However, we are interested in studying the problem from the view point of an observer using y-coordinates, which reduces to considering the following Lagrangian + − − L =−2y˙ y˙ + 2Ey(y˙ )2 +˙y2. Dots are derivatives with respect to an affine parameter. We now introduce the canonical conjugate momenta (p−,p,p+), of which the first and third components are conserved. This is because the quotient space preserves only two of the six isometries of three-dimensional Minkowski + − space, namely, those generated by ∂+ ≡ ∂/∂y and ∂− ≡ ∂/∂y . The problem then simplifies to the motion of the particle along direction y, with an effective Hamiltonian given by 2 2 H = p + V(y), V(y)≡−2Ep+y − 2p+p−. (1) The motion is thus determined by a linear potential. The constant force associated to this potential is clearly an inertial force, since the y-coordinates describe an accelerating frame. Now comes an important point. The Hamiltonian, which is a constant of motion, is the momentum squared of the particle µ 2 H = pµp =−M , (2) and is therefore fixed by the particle’s mass. The classical turning point y0 for a particle of light- cone energy p+ and Kaluza–Klein momentum p− is then found by setting p = 0 in the previous equation 2 2 M = 2p+p− + 2Ey0p+. (3) Consider first a particle of M2 > 0 and p− = 0. It is clear from Fig. 2 that such a particle will never enter the bad region y<0. Since the slope of the potential gets steeper as p+ increases, the higher the light-cone energy of the particle, the closer it gets to y = 0. In the low energy limit p+ → 0, the particle is pushed away to y =+∞. On the other hand, particles of M2 < 0 and p− = 0 always cross into the bad region and, for low energies, reach very far out into this region as can be seen from Fig. 2. For particles with p− = 0 the potential is displaced along y, so it no longer goes through y = 0. However, the above qualitative behavior is the same: particles with M2 > 0 are repelled away from the bad region and for low energies are pushed away to y =+∞; particles with M2 < 0 are also repelled but for low energies reach far out into the bad region. 152 M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178

Fig. 2. Effective potential V(y)and conserved energy −M2 for particle motion along direction y. Shown are the cases 2 2 2 of M > 0(left)andM < 0 (right) for p− = 0. The turning point y0 is deﬁned by V(y0) =−M . Particle states with 2 p+ < 0 are allowed quantum mechanically.

2.2. Single particle wave functions

To quantize the particle, we follow the usual prescription of replacing pµ →−iDµ in the mass-shell condition (2). The single particle wave functions φ(y−,y,y+) obey the Klein– Gordon equation − − 2 + 2 = 2 2∂+∂− 2Ey∂+ ∂y φ M φ. To find the solutions to this differential equation we first consider a complete basis of functions in the orbifold. The details are presented in Appendix A.1; here we state the necessary results. In addition to the orbifold periodicity 0 y− 2πR, we consider a large box defined + by 0 y L+ and −L/2 y L/2. The wave functions

1/3 |K(p+)| ++ m − = i(p+y R y ) φp+,y0,m(y) Ai(z)e , (4) 2πRL+ρ(y0,p+) form a complete normalized basis and are eigenfunctions of the Laplacian in the O-plane orbifold. The Kaluza–Klein momentum is quantized as p− = m/R and the other quantum numbers are the light-cone energy p+ and the classical turning point y0. The function Ai(z) is an Airy 3 3 function with argument given by z = K(y0 − y) and 1/2 2 1 L K = K(p+) = 2Ep+,ρ(y,p+) = K(p+) sgn(K) − y . (5) 0 π 2 0 A y-plot of a wave function is given in Fig. 3. To sum over states in the Hilbert space the appropriate integration measure is given by

∞ L ∞ 2 L+ dp+ dy ρ(y ,p+). 2π 0 0 m=−∞ −∞ − L 2 On-shell wave functions are eigenfunctions of the Laplacian with eigenvalue M2 and therefore their quantum numbers satisfy the mass-shell condition (3). To deﬁne the evolution of a given M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178 153

Fig. 3. Proﬁle of a wave function in the y-direction.

Fig. 4. Dispersion relation for particles with M2 > 0andm = 0.

field configuration, we shall consider an initial value problem defined on a surface of constant y+. The normal n˜ to this surface has norm given by n˜2 =−2Ey. Thus, in the good region y>0 this surface is space-like and the initial value problem is standard. However, for y<0weare specifying the “initial” data on a time-like surface. We shall interpret this “initial” data as a boundary condition. This is satisfactory to understand the evolution of perturbations in the good region. In Appendix A.2 we construct a basis of functions on a surface of constant y+. The light- cone evolution of each mode will be determined by its light-cone energy fixed by the on-shell relation. In the appendix we show that, in order to be normalizable on a surface of constant y+, the modes must have K ∈ R. In particular, for K<0, we have unstable modes with p+ purely imaginary and y0 complex. To understand the subtleties of matter propagation in the O-plane orbifold, we consider the cases of massive particles (M2 > 0) and tachyons (M2 < 0) separately, both for m = 0. Massless particles can be included in the case of massive particles by introducing an infrared regulator. For massive particles, the behavior of the wave function can be easily understood from the corresponding classical effective potential in (1). The turning point y0 is now the point where the wave function turns from oscillatory to evanescent (see Fig. 2). For on-shell particles, the mass formula (3), with p− = 0, defines the light-cone energy for a given y0. This dispersion relation is plotted in Fig. 4. As expected from the classical behavior, in the low energy limit we have y0 →+∞and the wave function vanishes everywhere. In Appendix A.2 we show that the set of on-shell wave functions with y0 ∈ R is a complete basis of normalizable functions + on a surface of y constant. Thus, from Fig. 4, we see that to include modes with y0 < 0we 154 M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178

Fig. 5. Dispersion relation for particles with M2 < 0andm = 0. must allow for imaginary p+. Then we are effectively dealing with the classical potential V(y) 2 with p+ < 0 and the wave functions oscillate in the bad region. Since the light-cone energy is imaginary, these modes can grow in time. A generic field configuration will then grow in light- cone time y+, mainly in the bad region. We conclude that massive particles have unstable modes in this spacetime. If the bad region were to be excised, we would expect to be left only with well-behaved modes. Next consider a tachyon, also with m = 0. As shown in Fig. 5,forp+ real the turning point lies at y0 < 0 and the wave function oscillates according to the classical trajectory in Fig. 2.To allow for states with y0 > 0 one must consider imaginary p+. These unstable modes oscillate in space for −∞ , 2 0 2 0 (6) R (y0) 2 2 where R (y0) = 2Ey0R is the proper radius squared of the orbifold compact direction, computed at the turning point y0. When this condition is not satisfied, p+ becomes complex and the associated wave functions are not normalizable on a surface of constant y+. However, in Appen- dix A.2, we show that there are other unstable on-shell states, with p+ purely imaginary and y0 complex, which are normalizable.

2.3. Quantum ﬁeld theory and ultraviolet catastrophe

To compare with the string theory results presented later on in this paper, we study the usual quantum ﬁeld theory pathologies in the O-plane orbifold due to the presence of closed causal curves. These are the ultraviolet problems that led Hawking to conjecture that a strong gravitational backreaction would protect chronology. Consider ﬁrst the existence of polarization surfaces, arising from light-like separated images. Since the covering space is Minkowski space, it is simple to compute the relativistic interval between image points E ∆2 ≡x −x 2 = 8E(πnR)2 y − (πnR)2 . n n 6 M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178 155

In particular, points located at E y = y ≡ (πnR)2, n 6 are null separated from their nth image. It is important to note that the corresponding closed null curve intersects the y = 0 surface, so it is possible that cutting off the y<0 region cures any pathology arising from the polarization surfaces, as proposed in [13,18]. For a scalar field φ of mass M the two-point function can be written in Euclidean Schwinger parameterization as +∞ −M∆w 2 e dl 1 l ∆ φ(x)φ( x ) = = exp − M2 − w , 4π∆ 2 ( πl)3/2 2 2l w w w 2 0 2 =− 2 −∞ +∞ where ∆w x xw and the trajectory winding number w runs from to . When x lies on the nth polarization surface, the two-point function has extra divergences as x →x, coming from the w =±n terms in the sum. These are ultraviolet divergences that are not renormalizable [49]. Likewise, the expectation value of the energy–momentum tensor diverges at the polarization surfaces. Again, these problems could be cured if one excises the bad region. It is the main point of this paper to suggest that a truly stringy phenomena in fact excises such region, hence protecting chronology. Next consider the one-loop contribution to the vacuum energy, written as +∞ dl l Z = Tr exp −i M2 + p pµ , 2l 2 µ 0 where the trace is over off-shell single particle states. This is written in first quantized language so as to compare it later with the string torus amplitude. Choosing the basis |p+,y0,m to perform the trace and Wick rotating l →−il, p+ → ip+, we arrive at the expression +∞ dl L+ l 2 m 2 Z = i dp+ dy ρ(y ,ip+) exp − M − 2ip+ + 2Ey p+ . 2l 2π 0 0 2 R 0 m 0 One could question the validity of this Wick rotation since the final expression clearly has large l divergences that were absent in the original one. The alternative would be to choose a state 2 µ dependent Wick rotation depending on the sign of (M + pµp ). However, in string theory the requirement of modular invariance forces us to treat all modes equally [14]. The chosen Wick rotation is physically reasonable because massive particle states that propagate in the good region, i.e., states with y0 > 0, give a finite contribution to the partition function. On the other hand, states with y0 < 0 will originate infrared divergences provided p+ is large enough, for both massive and tachyonic particles. Restricting to states with y0 > 0 and for large l,thep+ integral can be done using the saddle point approximation. The l integral is then dominated by an exponential with the following argument l m2 − M2 + . 2 2 R (y0) 156 M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178

Hence, the condition for this integral to be infrared convergent is equivalent to the condition for the existence of on-shell stable states with y0 positive, as can be seen from (6). For massive particles, this condition holds for all on-shell states that oscillate in the good region, which have y0 > 0. On the other hand, for tachyonic particles, the above condition is not always satisfied and therefore the partition function becomes infrared divergent. This is related to the existence of on-shell unstable states. These states oscillate in both good and bad regions and therefore their condensation would change the whole of spacetime. To analyze the ultraviolet behavior of the partition function it is useful to consider the expression obtained from the path integral formalism. We report on this computation in Appendix B and on its derivation starting from the above canonical result in Appendix A.1. Here we simply give the final result ∞ 2 dl 1 l 2 (2πwR(y)) Z = iL+(2πR) dy exp − M − , 2l ( πl)3/2 2 2l w 2 0 where y is now the particle’s average position along this direction and w is the winding number of closed trajectories. The w = 0 term gives the usual renormalizable ultraviolet divergence from the region l → 0. The other terms are ultraviolet finite for y>0, diverge for y → 0 and acquire, by analytic continuation, an imaginary part for y<0. These are extra ultraviolet divergences that are absent in string theory.

3. Strings in the O-plane orbifold

We now move on from particles to strings. We work with the bosonic string in critical dimension, adding the transverse space R23 to the O-plane orbifold. Since these directions are mere spectators we include their contribution explicitly only when necessary. We work in units where α = 1. When considering strings the novelty is the appearance of twisted sectors, which correspond to strings winding around the y−-direction. As we shall see, these winding states play a crucial role in protecting chronology. Start with the string Lagrangian ˙ 2 2 1 + − − Y + − − Y L = −Y˙ Y˙ + EY(Y˙ )2 + + (Y ) (Y ) − EY(Y ) 2 − , (7) 2π 2 2 where dots and primes denote derivatives of the worldsheet ﬁelds (Y −,Y,Y+) with respect to the worldsheet coordinates τ and σ , respectively. The coordinate σ runs from 0 to π. Consider the center of mass motion of a closed string that winds w times around the compact direction. The embedding functions have the form − − + + Y (τ, σ ) = 2Rwσ + y (τ), Y (τ, σ ) = y(τ), Y (τ, σ ) = y (τ), (8) from which we derive the string center of mass conjugate momenta

1 + − 1 1 − p− =− (y˙ − 2Eyy˙ ), p = y,˙ p+ =− y˙ . (9) 2 2 2 Again, only p+ and p− are conserved. As for the particle case, the motion along y is determined by the effective Hamiltonian 2 2 2 H = p + V(y), V(y)≡−2E p+ − (wR) y − 2p+p−. (10) M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178 157

We see that there is an additional linear contribution to the potential from the winding of the string. Its interpretation is simple. Since the proper length of the compact direction increases with y, it costs energy for a winding string to increase its value of y. Defining the conserved quantity λ =−H, the string center of mass turning point y0, where p = 0, satisfies 2 2 λ = 2p+p− + 2Ey0 p+ − (wR) . (11) For the particular solution (8) the Hamiltonian in (10) generates translations along the worldsheet ˜ time and it is related to the Virasoro zero-modes by H = 2(L0 + L0). This is zero by the classical ˜ Virasoro constraints. As we shall see in Section 3.2, in the quantum theory 2(L0 + L0) = H + 2(N +N)˜ = 4, where N and N˜ are the usual number operators. For each string state it is therefore necessary to consider both negative and positive values of λ = 2(n +ñ − 2). The corresponding center of mass classical dynamics reduces to the motion of a particle of constant energy in the potential in (10). We analyze it below, together with the associated wave functions.

3.1. Winding states dynamics and wave functions

We have just seen that on-shell classical string states satisfy the relation λ =−H, which using (10) yields

µ 2 −pµp = λ + 2Ey(wR) . (12) This constraint deﬁnes the wave equation for the string center of mass − − 2 + 2 = 2 2∂+∂− 2Ey∂+ ∂y φ M (y)φ, where

M2(y) = λ + 2Ey(wR)2. (13) This Klein–Gordon equation can be derived from the Lagrangian L = ∂φ∂φ¯ + M2(y)φφ¯ for a complex scalar field associated with strings of winding number ±w. The major difference to the particle case is that this string mass depends on the y-coordinate and therefore it is not conserved. From a pure field theoretical view point one would then expect that something special happens for M(y)= 0. We shall come back to this point in Section 4. Solutions to the above differential equation have the same form as for the particle case. The only difference is that K in the expression (4) for the basis of functions and in the corresponding definitions (5) is now given by 2 2 K(p+) = 2E p+ − (wR) . On-shell states satisfy the string dispersion relation (11). As for the particle case, on-shell wave functions, which are normalizable on a surface of constant y+,haveK ∈ R.ForK<0this corresponds to unstable modes with p+ purely imaginary and y0 complex. Let us first focus on the case λ<0 and, for simplicity, m = 0. The classical trajectories and behavior of wave functions can be immediately understood from Fig. 6. There are three different 2 2 kinematic regimes, determined by the light-cone energy p+.For0 p+ <(wR) the string comes from y =−∞until it reaches a turning point y0 > 0, where it bounces back. The wave function oscillates up to the turning point and becomes evanescent thereafter. In the low energy 158 M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178

Fig. 6. Effective potential V(y)and conserved energy −λ for string motion along direction y. Shown is the case of λ<0 2 2 2 2 2 2 and m = 0, for p+ <(wR) (left) and p+ >(wR) (right). For p+ = (wR) the potential becomes ﬂat. In contrast with the particle case, when p+ = 0 the potential is tilted.

Fig. 7. Dispersion relation for string states with λ<0andm = 0.

limit p+ = 0, the slope of potential remains positive and the wave function oscillates for y(wR) the string comes from y =+∞and turns around at y0 < 0. Its wave function becomes evanescent beyond that point. For large light-cone energies, the potential becomes infinitely steep, as it was the case for particles. In Fig. 7 we plot the dispersion relation for states with λ<0 and m = 0. As shown in Ap- pendix A.2, to determine the evolution of generic perturbations defined on a surface of constant + y , one must include all real values of y0 in the on-shell basis. Hence, we must consider states with imaginary p+, which grow in time. The turning point y0 for these unstable states satisfies 0

Fig. 8. Effective potential V(y)and conserved energy −λ for string motion along direction y. Shown is the case of λ>0 2 2 2 2 2 2 and m = 0, for p+ <(wR) (left) and p+ >(wR) (right). For p+ = (wR) there is no classical trajectory nor wave function. In contrast with the particle case, when p+ = 0 the effective potential is tilted.

Fig. 9. Dispersion relation for string states with λ>0andm = 0.

One then expects that the condensation of these modes will drastically change spacetime in the region y0 case, and again set m = 0. Trajectories and wave functions can be 2 2 understood looking at Fig. 8.For0 p+ <(wR) the string comes from negative y and turns back at y0 < 0. The low energy limit p+ = 0 is similar to the above case but with y0 < 0. As the 2 2 potential tilts and one reaches p+ = (wR) , the turning point y0 is pushed to −∞ and the wave 2 2 function disappears. For p+ >(wR) the string comes in from positive y and bounces back at y0 > 0. The high energy limit is the same as before. In Fig. 9 we plot the dispersion relation for λ>0 and m = 0, both for p+ real and imaginary. The unstable modes will grow considerably for y , 2 0 2 0 0 (14) R (y0) 2 2 where R (y0) = 2Ey0R is again the proper radius squared of the orbifold compact direction, computed at the turning point y0. On-shell unstable states, which are normalizable on a surface of + constant y , occur for p+ purely imaginary and y0 complex ﬁxed by the dispersion relation (11). 160 M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178

3.2. Canonical quantization

We now quantize the string using y-coordinates and in light-cone gauge. The worldsheet theory with the Lagrangian (7) is not a free field theory. However, the worldsheet field Y − obeys the free wave equation. Thus the light-cone gauge can be implemented by demanding that Y − does not have oscillator terms. In this gauge, the solutions of the wave equation for the worldsheet fields have the following expansion − − Y (τ, σ ) = 2Rwσ + y (τ), i 1 + − Y(τ,σ)= y(τ) + √ a˜ e2in(τ σ) + a e2in(τ σ) , n n n 2 n=0 + + i 1 + + + − Y (τ, σ ) = y (τ) + √ a˜ (τ)e2in(τ σ) + a (τ)e2in(τ σ) . n n n 2 n=0 ˜+ + As usual in light-cone gauge, the coefficients an (τ) and an (τ) are not independent degrees of ˜ freedom because they are fixed by the Virasoro constraints Ln = Ln = 0forn = 0. From the string Lagrangian (7) we obtain the conjugate momenta

1 ˙ + ˙ − 1 ˙ 1 ˙ − 1 − P− =− (Y − 2EYY ), P = Y, P+ =− Y =− y˙ , 2π 2π 2π 2π which can be integrated along σ to yield the string center of mass momentum pµ derived in (9). µ The worldsheet energy–momentum tensor T±± = ∂±Y ∂±Yµ enables us to compute the Virasoro zero-modes π 1 1 µ 1 2 mw L = T−− dσ = p p + Ey(wR) + + N, 0 4π 4 µ 2 2 0 π ˜ 1 1 µ 1 2 mw ˜ L = T++ dσ = p p + Ey(wR) − + N, 0 4π 4 µ 2 2 0 where N and N˜ are the usual light-cone oscillator contributions. The Virasoro zero-modes sum and difference are 1 L + L˜ = p pµ + Ey(wR)2 + N + N,˜ 0 0 2 µ ˜ ˜ L0 − L0 = mw + N − N. Given any N and N˜ , studying the classical dynamics of the string center of mass reduces to the effective Hamiltonian problem described in Section 3.2. The corresponding conserved quantity ˜ λ is ﬁxed by setting L0 = L0 = 0. Canonical quantization of the string is done by imposing equal-time commutation relations. This gives the non-vanishing commutators for the string center of mass − + y (τ), p−(τ) = i, y(τ),p(τ) = i, y (τ), p+(τ) = i and for the oscillators

[an,am]=nδn+m, [˜an, a˜m]=nδn+m. M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178 161

Since the oscillators decouple from the zero modes the normal ordering for the Virasoro zero- ˜ modes is the usual one. For the bosonic string one has L0 = L0 = 1 and physical states satisfy the on-shell operator relation 2 µ 2 ˜ M (y) ≡−pµp = 2Ey(wR) + 2(N + N − 2), and the level matching condition N˜ − N = mw.

The quantum numbers for the string center of mass wave functions φp+,y0,m will then obey the on-shell relation m 2 2 λ = 2p+ + 2Ey p+ − (wR) = 2(n +˜n − 2). R 0 We conclude that for the bosonic string both positive and negative values of λ are allowed. The corresponding behavior of the string center of mass wave functions was studied in the previous subsection, where we saw that states with negative values of λ grow signiﬁcantly in the bad region, as well as a bit inside the good region.

3.3. Partition function

We now compute the bosonic string partition function and ﬁnd divergences associated with the condensation of physical states. We shall see these are all infrared divergences. In the canonical formalism one takes the trace over the Hilbert space of states 2 d τ − ˜ − Z = Tr qL0 1q¯L0 1 , τ2 F 2πiτ where q = e and τ = τ1 + iτ2 is the modular parameter of the Euclidean torus, restricted to the fundamental domain F. Since we are computing the trace in light-cone gauge, one inte- grates over all the string zero-modes but keeps only the physical transverse oscillators. Using the explicit form of the Virasoro zero-modes it follows that 2 d τ ˜ Z = Tr exp 2πiτ1(N − N − mw) τ2 F 1 + 2πτ − p pµ − Ey(wR)2 − N − N˜ + 2 . 2 2 µ In the expression for the Virasoro zero-modes the string center of mass momentum and the number operators commute. Therefore, one can perform the trace over the oscillators to obtain the same result as for Minkowski space. Integrating over momenta in the spectator directions and choosing the basis |p+,y0,m to perform the trace over the remaining zero-modes we arrive at the expression d2τ dp+ Z = L+V23 Z0(τ) dy0 ρ(y0,p+) exp(−2πiτ1mw + πτ2λ), (15) τ2 2π F m,w √ −23 −48 where Z0 = (2π τ2 ) |η(τ)| and we recall that on the basis |p+,y0,m the operator H deﬁned in (10) has eigenvalue −λ given by (11). 162 M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178

3.3.1. Infrared divergences We want to understand the divergences in the partition function coming from the infrared region τ2 →+∞. In particular, we are interested in investigating the contribution of each string state to such divergence. For this purpose we cut the ultraviolet region of modular integration by setting τ2 1, in order to integrate over τ1 to enforce the level matching condition. Recalling the expansion of the Dedekind eta function η(τ) we have √ ∞ −23 −1 n n˜ Z0 = 2π τ2 (qq)¯ dndn˜ q q¯ n,n˜=0 where dn and dn˜ represent the degeneracy of left and right moving states. As for the particle case we define the analytic continuation of the p+ integral as p+ → ip+. The infrared behavior of the partition function can then be studied from +∞ L+V23 dp+ dτ2 Z ∼ i d d˜ dy ρ(y ,p+) √ 23 n n 0 0 25 (2π) 2π ( τ2 ) m,w,n,n˜ 1 m 2 2 × exp −2πτ −ip+ + Ey p+ + (wR) + n +ñ − 2 , 2 R 0 where the sum is restricted by the level matching condition for physical states. The infrared divergences come from states in the Hilbert space such that the real part of the argument in the above exponential is positive. Hence, given (m,w,n,n)˜ and negative y0, the integral is infrared divergent for p+ large enough. Let us then focus on the region of integration with y0 positive and look for further infrared divergences. Using the saddle point approximation we obtain the large τ2 behavior of the p+ integral. Then one is left with a τ2 integral dominated by an exponential with argument m2 −πτ (n +ñ − ) + + w2R2(y ) . 2 2 2 2 0 R (y0) As for the particle case, we see that the condition for the integral to be infrared convergent is equivalent to the condition for the existence of on-shell stable states with y0 positive, as can be seen from (14).Forλ>0, this condition holds for all on-shell states that oscillate in the good region, which have y0 > 0. String states with λ<0, do not always satisfy the above condition and therefore give rise to infrared divergences. In particular, the state with m = 0, w = 1 and 2 n =ñ = 0 renders the partition function divergent for y0 < 2/(ER ). Excluding the usual closed string tachyon, this is the state that condenses the furthest into the y>0 region.

3.3.2. Ultraviolet finiteness Up to now we have cut the ultraviolet region of modular integration to understand the infrared origin of the divergences. In string theory one expects no further divergences arising from the ultraviolet. Let us analyze the partition function without cutting off the ultraviolet region. The divergences we will find correspond to those found in the previous analysis. Consider the expression (15) for the partition function. Performing the Wick rotation, the sum in Kaluza–Klein momentum can be Poisson resummed using τ p+ τ p+ exp 2πim τ w − 2 = δ τ w − 2 − w , 1 R 1 R m w M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178 163 such that the p+ integral becomes trivial. As explained in Appendix A.1, we can then use the integral representation of the measure to interchange the integral over y0 with the integral over the string center of mass y, arriving at the final expression L+(2πR)V d2τ − π Z = i 23 η(τ) 48 dy − R2(y)TT¯ , 26 14 exp (2π) τ τ2 F 2 w,w where T = wτ − w .InAppendix B we derive independently this result using the path integral formalism. It is easy to check that the integrand is modular invariant. Unlike the particle case, for fixed (w, w ) and y → 0 there are no ultraviolet divergences because the τ2 integration is restricted to the fundamental domain F.

3.3.3. Hagedorn behavior The term in the partition function with w = w = 0 gives the usual Minkowski space contribution. The remaining terms in the sum are the contribution from the winding states and Kaluza–Klein states in their dual guise. From now on we consider only these terms since they are the ones which contain the orbifold information. Using the usual trick of replacing the funda- F ={ − 1 1 } = mental region by the strip Γ τ: 2 <τ1 < 2 ,τ2 > 0 , while simultaneously setting w 0 and w > 0inthesum[50], we can write 2 ∞ L+(2πR)V d τ − π Z = i 23 η(τ) 48 dy − w 2R2(y) . 26 14 exp (2π) τ2 = τ2 Γ w 1 If one sets R(y) = R this expression would be that of the partition function for bosonic strings in Minkowski space with one compact direction of radius R, which for a critical value of the moduli R has a divergence associated to a Hagedorn phase transition [3]. Similarly, in the O-plane orbifold we expect to obtain a divergence for a critical value of the string center of mass coordinate y. To see this expand the Dedekind eta function η(τ) in a series and perform the τ1 integration to obtain ∞ ∞ L+(2πR)V dτ π Z = i 23 d2 dy 2 − π(n− )τ − w2R2(y) . 26 n 14 exp 4 1 2 (2π) = = τ2 τ2 w 1,n 0 0

Let us analyze the τ2 integral for specific values of n.Forn = 0, the integral diverges exponentially as τ2 →∞. This is the usual infrared tachyonic divergence of the bosonic closed string. Note that the y dependence is irrelevant in what concerns this divergence. For n 1 fixed, the τ2 integral diverges for y → 0. It diverges in the region τ2 → 0, so one might think that this is an ultraviolet divergence. However, this is not the case because the small τ2 region of the strip Γ does not result only from a modular map of the fundamental domain ultraviolet region. We expect these divergences to be infrared divergences. An instructive analogy is to compare with the case of bosonic string compactified on a circle in the limit of vanishing radius. In this limit there are winding states that become massless and the divergence should be thought in the T-dual picture as a large volume effect. For positive y,theτ2 integral converges and the partition function can be explicitly written as a Hankel function of first kind ∞ 13/2 L+(2πR)V 4(n − 1) Z =−i 23 πd2 dy H (1) 4πi (n − 1)w2R2(y) . (2π)26 n w2R2(y) 13 w=1,n=0 164 M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178

Using the asymptotic expansions for the degeneracy of states and for the Hankel function √ − |z|→∞ 2 − − d ∼ n 27/4e4π n,H(1)(z) ∼ ei(z νπ/2 π/4), n ν πz we can see that the large n expansion of the integrand in the partition function is dominated by the exponential term √ √ − 2 2 e4π n 2 w R (y) . We conclude that, for each winding number w,thesuminn diverges when 2 y

4. Hagedorn transition—the chronology warden

Having found unstable states and related infrared divergences in the string partition function that resemble what happens in the Hagedorn phase transition, we now suggest how this can protect chronology. Let us start by recalling the behavior of a tachyon field in Minkowski space. The condensation of the field occurs via the growth of low momentum modes, as can be seen from the dispersion relation plotted in Fig. 10. In the case of the bosonic string one expects that the closed string tachyon condensation will change the spacetime structure. For the O-plane orbifold a similar phenomenon occurs. In particular, there are winding string states that play an important role because the associated field has unstable modes that propagate not only in the bad region, but also a bit inside the good region. The growth of these modes will then induce a large backreaction in the region where they propagate, triggering a phase transition. It is therefore natural to conjecture that this phase transition protects chronology. For simplicity, let us focus on the bosonic string and we shall comment on the extension to the superstring afterwards. As usual, we neglect the closed string tachyon. Consider the state with m = 0, w = 1 and n =ñ = 0. From Eq. (13) we can compute the position where the mass M(y) of this field vanishes 2 y = y = . (16) c ER2

Fig. 10. Dispersion relation for a tachyon ﬁeld in Minkowski space. M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178 165

For y

4.1. Extension to the superstring

It is straightforward to extend the previous results to the superstring, where one needs to specify the orbifold spin structure. For both spin structures, we need to recall that, in the NS–NS sector, the normal ordering for the Virasoro zero-modes gives the on-shell relation λ = 2(N + N˜ − 1). Consider ﬁrst the supersymmetry breaking case where spacetime fermions have anti-periodic boundary conditions [51]. In this case states with even winding number have the usual supersymmetric GSO projection, while states with odd winding number have reversed GSO projection. The physical picture is very similar to the bosonic string just considered, but one does not have the usual bosonic string tachyon. There are winding states that condense and whose wave function also oscillates inside the region y>0. These states have λ =−2, m = 0 and odd winding 2 number w.Forw =±1themassM(y) vanishes for y = yc = 1/(ER ) and it becomes negative for y

5. Concluding remarks

We believe the results here presented for the O-plane orbifold with supersymmetry breaking boundary conditions are general and provide a truly stringy mechanism to protect chronology. Suppose we have an arbitrary spacetime with a partial Cauchy surface. Suppose also that, as the geometry evolves, we are about to create a closed null curve in the future of such surface. Then, when there are closed spacelike curves with proper radius of order of the string length, winding string fields become massless and start condensing. There will be a Hagedorn phase transition, which protects chronology. In other words, after the phase transition the new vacuum will be chronological. Hawking’s chronology protection mechanism is based on an ultraviolet effect. He advocates that strong curvature effects due to a large one loop energy–momentum tensor for the matter fields will create a singularity or will prevent spacetime to curve such that non-causal curves do not form. In contrast, the string theory mechanism for chronology protection is an infrared effect, as it is usually the case for phase transitions. Other situations in string theory where the appearance of light winding states triggers a phase transition are well known. An interesting follow up of our proposal is to find the exact end-point of the condensation. Our conjecture is that this phase transition will remove the bad region from spacetime. Indeed, phase transitions associated with light winding string states, that produce such a drastic effect in spacetime and that lead to a change of topology, are expected to occur in the context of the AdS/Gauge-theory duality at finite temperature [53], or in the context of circle compactifications with supersymmetry breaking boundary conditions [54]. The authors of the latter work argued that, after condensation of the winding string field, the worldsheet theory will flow to an infrared trivial fixed point with no target space geometric interpretation. We believe something similar will happen that excludes the formation of non-causal curves. In order to check the generality of the mechanism presented herein, interesting laboratories include other time-dependent orbifolds and other closed string backgrounds, which have already been studied in relation to closed causal curves in string theory.

Acknowledgements

The authors wish to thank L. Cornalba for many enlightening discussions. We would also like to thank F. Correia and S. Hirano for useful discussions. C.H., J.P. and N.S. are funded by the Fundação para a Ciência e Tecnologia fellowships SFRH/BPD/5544/2001, M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178 167

SFRH/BD/9248/2002 and SFRH/BPD/13357/2003, respectively. This work was also supported by Fundação Calouste Gulbenkian through Programa de Estímulo à Investigação,bytheMarie Curie research grant MERG-CT-2004-511309 and by the FCT grants POCTI/FNU/38004/2001 and POCTI/FNU/50161/2003. Centro de Física do Porto is partially funded by FCT through POCTI programme.

Appendix A. Wave functions in the O-plane orbifold

In this appendix we prove several results used in the main text regarding wave functions. In Appendix A.1 we construct a complete basis of states in the orbifold geometry and deﬁne a measure for these states. In Appendix A.2 we consider on-shell wave functions normalizable on a surface of constant y+.

A.1. Off-shell wave functions and Hilbert space measure

To define a measure in the Hilbert space we shall regularize the wave functions by working in a box of volume V = 2πRLL+. More concretely, we impose the boundary conditions − + − + φ(y ,y,y ) = φ(y + 2πR,y,y ), − + = − + + φ(y ,y,y ) φ(y ,y,y L+), − L + − L + φ y , − ,y = φ y , ,y = 0. 2 2 Furthermore, it is convenient to work with a basis of functions formed by eigenfunctions of the operator H = + 2 + 2 − 2 2∂+∂− 2Ey ∂+ (wR) ∂y . These wave functions can be written in the form ++ − φ = f(z)ei(p+y p−y ), where p+ = 2πr/L+ and p− = m/R,forr and m integers. For fixed winding number w,we 3 3 define z = K(y0 − y) , with the constant K depending on the light-cone energy as 2 2 K = K(p+) = 2E p+ − (wR) and with y0 related to the eigenvalue −λ of the operator H by

λ = 2p+p− + y0K(p+). The function f(z)satisﬁes the Airy differential equation 2 − = ∂z z f(z) 0, subject to Dirichlet boundary conditions at the points 3 3 L z± = K y ± η , 0 2 where η = sgn(K) and we conveniently chose z+ >z−. With these boundary conditions the Airy differential equation has solutions only for discrete values of y0, which we denote as y0 = y0(s), with s a positive integer. The spectrum of the operator λ is now discrete. 168 M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178

To deﬁne the Hilbert space measure in the large volume limit we need to understand the behavior of the function y0(s) for large L. Since the function f(z)is a linear combination of the Airy functions Ai(z) and Bi(z), the boundary condition f(z+) = 0gives

f(z)∝ Bi(z+) Ai(z) − Ai(z+) Bi(z).

To quantize y0 one imposes the other boundary condition

Bi(z+) Ai(z−) = Ai(z+) Bi(z−).

The density of states around a ﬁxed value of y0 can be computed by observing that in the limit of large L one has z+ 1. Then the above equation reduces to Ai(z−) = 0 because of the exponential growth of the function Bi(z) for positive real values of z. Denoting the zeros of the Airy function Ai by as , the function y0(s) is implicitly deﬁned by L 3 a3 = K y (s) − η . s 0 2 Again in the limit of large L, one can use the asymptotic expansion for the zeros of the Airy function 2/3 3π(4s − 1) − a =− 1 + O s 2 s 8 to determine y0(s). Two consecutive zeros will then give the following difference between discrete values of y0 − 3π(4s − 1) 1/3 δy ≡ y (s + 1) − y (s) −π K , 0 0 0 8 which can be written as a function of y0 − L 1/2 δy −π K η − y . 0 2 0

Since δy0 → 0forL →+∞we can move from sums to integrals L 2 F y0(s) → dy0 ρ(y0)F (y0), s − L 2 = − | | with measure ρ s δ(y0 y0(s)) 1/ δy0 given by 1 L 1/2 ρ(y ,p+) K(p+) η − y . 0 π 2 0

Notice that there may be other states with y0 outside the above region of integration, but these are negligible for large L. The normalized basis of functions in the large volume limit is

1/3 |K(p+)| ++ m − = i(p+y R y ) φp+,y0,m(y) Ai(z)e , 2πRL+ρ(y0,p+) M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178 169 satisfying the orthogonality and completeness conditions

2π 1 | = − − p+,y0,m p+,y0,m δ(p+ p+) δ(y0 y0)δm,m , L+ ρ(y0,p+) ∞ L 2 L+ ∗ dp+ dy ρ(y ,p+)φ (y )φ (y) = δ(y −y ). 2π 0 0 p+,m,y0 p+,m,y0 m −∞ − L 2 It is important to have an independent check of the above measure. In the reminder of this appendix we provide that check. We shall start from the canonical partition function for a particle and derive the result obtained independently in the next appendix using the path integral formalism. This involves a Poisson summation of the Kaluza–Klein quantum number m and a rewriting of the measure ρ(y0,p+) as an integral over a variable y. From the path integral computation one sees that this integration variable should be interpreted as the particle average position along the y-direction. A similar calculation can be done for the string partition function. We start with the canonical particle partition function written as

∞ ∞ L 2 dl L+ l 2 m 2 Z = i dp+ dy ρ(y ,ip+) exp − M − 2ip+ + 2Ep+y 2l 2π 0 0 2 R 0 m 0 −∞ − L 2 and write the measure ρ(y0,ip+) in the integral form

L 1/2 2 |K(ip+)| Θ[η(y − y0)] ρ(y0,ip+) = dy √ , 2π η(y − y0) − L 2 where Θ is the usual Heaviside step function. The integral over y0 reduces to a half Gaussian integral in the L →∞limit

L L +ηy 2 2 1/2 Θ[η(y − y )] l du l 2π l 0 Ky0 (Ky−|K|u) Ky dy0 √ e 2 = √ e 2 → e 2 . η(y − y0) u l|K| − L 0 2 Using this result the partition function becomes

∞ ∞ L 2 dl L+ dy l 2 m 2 Z = i dp+ √ exp − M − 2ip+ + 2Ep+y . 2l 2π 2 R m 2πl 0 −∞ − L 2 Then, performing the Gaussian integral over the light-cone energy we have

L ∞ 2 2 dl 1 R l 2 m Z = iL+ dy exp − M + , 2l 2πl R(y) 2 R2(y) m − L 0 2 170 M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178 where R2(y) = 2EyR2 as deﬁned in the main text. The Poisson summation formula − 2 1 − 2 −1 e πAm = √ e πw A , m A w can be used to obtain the expression for the partition function derived from the path integral formalism L ∞ 2 2 dl 1 l 2 (2πwR(y)) Z = iL+(2πR) dy exp − M − . 2l ( πl)3/2 2 2l w 2 − L 0 2

A.2. On-shell wave functions

We shall consider the light-cone evolution of a generic field configuration defined on a surface of constant y+. Then, given the field and its first derivative on this surface, the evolution is determined by the equations of motion. We shall address this problem by considering a basis of normalizable functions on the surface of constant y+ with simple on-shell evolution. + The basis can be constructed using the orbifold modes φp+,y0,m(y), restricted to a constant y surface and with the quantum numbers satisfying the on-shell condition 2 2 λ = 2p+p− + 2Ey0 p+ − (wR) , − for λ fixed. We shall call these functions φy0,m(y ,y). Discarding an irrelevant normalization constant, we define these functions by − m − = 1/3 1/3 − i R y φy0,m(y, y ) K Ai K (y0 y) e . Their on-shell evolution to the orbifold spacetime is determined by the on-shell value of the light-cone energy p+. From the asymptotic behavior of the Airy functions, it follows that 2 2 K = 2E p+ − (wR) must be real, otherwise the function Ai would grow exponentially for y →+∞or for y →−∞. Notice that, one cannot include the other Airy function Bi, because it always grows exponentially − at least in one direction. We conclude that the functions φy0,m(y ,y)are normalizable provided both y0 and p+ are real, or y0 is complex and p+ is purely imaginary. The set of solutions to the on-shell condition with K real can be parameterized as1 βλ wR = u, βy = u ± α u2 + 2u, p+ =∓ u2 + 2u, K 0 u where u ∈ R and we defined α = 2mw/λ and β = E(2wR)2/λ. In this parameterization the − ± − unstable modes appear for 2

1 The case of w = 0 can be included by setting u = w2t and then taking the limit w → 0 with ﬁxed t. In this limit, the ellipse in Fig. 11, associated to complex values of y0, degenerates to a parabola. M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178 171

Fig. 11. Contours in the βy0 complex plane for |α| < 1. The lines outside the real axis belong to an ellipse with foci at Γ± =−1 ± 1 − α2.For|α|=1 the ellipse becomes a circle and the contours start, instead, at the center of the circle. For α = 0 the ellipse degenerates to the real axis. For |α| > 1 the foci of the ellipse move to the complex plane, the left contour starts and ﬁnishes at βy0 →+∞and the right contour at βy0 →−∞.

± − coalesce to the real line and, for each value of u, the two functions φu,m(y ,y) are equal with symmetric values of the light-cone energy. Using the above parameterization, the initial value problem reduces to finding the functions F˜ ±(u, m) such that − = ˜ + + − + ˜ − − − F(y,y ) du F (u, m)φu,m(y, y ) F (u, m)φu,m(y, y ) , m − =− −1 2 + ˜ + + − F (y, y ) iwR duu u 2u F (u, m)φu,m(y, y ) m − ˜ − − − F (u, m)φu,m(y, y ) , for any given function F(y,y−) and its first derivative F (y, y−). We start by decomposing the function and its derivative in Fourier modes − iqy+i m y− F(y,y ) = dq e R Gm(q), m − + m − = iqy i R y F (y, y ) dq e Gm(q). m Then, by linearity, we can break the problem in two cases: F generic and F = 0; F generic and F = 0. Moreover, we only need to consider each Fourier mode independently. Firstly, we consider an initial configuration with zero derivative

− iqy+i m y− − F(y,y ) = e R ,F(y, y ) = 0. Using the integral representation of the Airy function2 +∞ 2 ± − i m y− ds 3 β u φ (y, y ) = βe R exp is βy − u ∓ α u2 + 2u − is , u,m 2π 3λ −∞

2 This integral representation is valid for complex values of y0(u), provided one deforms the real axis integration contour as s → s + i sgn(u). 172 M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178

˜ ± we ﬁnd that the corresponding coefﬁcients Fq (u, m) in the expansion have to satisfy √ √ 3 2 du −iu(s+ s β ) −isα u2+2u ˜ + isα u2+2u ˜ − δ(q − βs) = e 3λ e F (u, m) + e F (u, m) , 2π q q √ √ 3 2 − − + s β − 2+ + 2+ − = 1 2 + iu s 3λ isα u 2u ˜ − isα u 2u ˜ 0 duu u 2ue e Fq (u, m) e Fq (u, m) .

Secondly, we choose the initial condition

− − iqy+i m y− F(y,y ) = 0,F(y, y ) = e R , ˜ ± and ﬁnd the following equations for the corresponding Fq (u, m) √ √ 3 2 − + s β − 2+ + 2+ − = iu s 3λ isα u 2u ˜ + isα u 2u ˜ 0 due e Fq (u, m) e Fq (u, m) , 3 2 du −1 −iu s+ s β δ(q − βs) =−iwR u u2 + 2ue 3λ 2π √ √ × −isα u2+2u ˜ + − isα u2+2u ˜ − e Fq (u, m) e Fq (u, m) . Finally, the expansion for the generic function F with derivative F has coefﬁcients ˜ ± = ˜ ± + ˜ ± F (u, m) dq Gm(q)Fq (u, m) Gm(q)Fq (u, m) .

Unfortunately, we do not know if the basis of functions is complete because we were not able ˜ ± ˜ ± to compute the functions Fq (u, m) and F q (u, m) in the general case. However, in the case of functions independent of y−, so that we only need to consider wave functions with m = 0, we showed that the basis is complete for λ>0 and that it spans very generic functions for λ<0, as explained below. The latter case is the most relevant case for the results of this paper, since it includes the winding states that were argued to protect chronology in the O-plane orbifold. For m = 0 the on-shell relation becomes λ = y0K, and therefore reality of K implies that y0 is also real. In the u parameterization this corresponds to setting α = 0 and the above formulas simplify considerably. In particular, deﬁning

+ − 1 F˜ (u, 0) = F˜ (u, 0) = F˜ (u), q q 2 q ˜ + =−˜ − = √iu ˜ F q (u, 0) Fq (u, 0) Fq (u), 2wR u2 + 2u ˜ one is left to ﬁnd Fq such that 3 2 du −iu s+ s β ˜ δ(q − βs) = e 3λ F (u). 2π q For λ>0 this equation has solution 2 3 1 q i u q+ q F˜ (u) = 1 + e β 3λ , q β λ showing the completeness of the basis for λ>0 (and λ = 0 by adding√ a mass regulator). When λ<0, the function s +β2s3/3λ is not injective in√ the interval |βs| √< 2 −λ. Therefore, the above ˜ explicit solution for Fq is only valid for |q| > 2 −λ.For|q| < 2 −λ, the integral equation for M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178 173

˜ Fq does not have a solution, showing that the basis is not complete. The subspace of functions generated by this basis is the set of functions whose Fourier transform G0(q) is√ a function of the particular combination q + q3/3λ. Hence, the Fourier coefﬁcients with |q| > −3λ can be ﬁxed arbitrarily, but also determine the remaining low momenta Fourier modes. We conclude that we can construct well localized perturbations, with some constraint on their long range decay properties (this can be deduced from the constraints on the pole structure of the Fourier transform). An interesting question is if one can complete the basis or, on the other hand, if the above restrictions are necessary for consistent evolution in the presence of closed causal curves.

Appendix B. Path integral formalism

In this appendix we compute the particle and string partition functions using the path integral formalism. The expressions we obtain can also be derived from the canonical formalism using the method described in the previous appendix. They serve as a check of the measure in the Hilbert space and provide a physical interpretation for the integration variable y. Before we start the computations it is convenient to write the orbifold identiﬁcations more compactly as

− J Ω(R,∆)x ≡ e 2π∆ x + 2πRT(π∆), where we deﬁned the column vector T(a) and the nilpotent matrix J as   1 000 T(a) ≡  −a  , J ≡ 100. 2 2 010 3 a A straightforward computation shows that the action of the orbifold group on a vector x satisﬁes

n xn ≡ Ω (R, ∆)x = Ω(nR,n∆)x.

B.1. Particle partition function

The one-loop vacuum energy for a ﬁeld of mass M is given by the particle partition function

∞ 1 dl l 1 Z = DX exp − M2 − X˙ 2 dτ , 2l 2 2l 0 0 where the functional integral is done over periodic trajectories. Working in covering space, these trajectories are periodic up to the orbifold identiﬁcation

− J X(τ + l) = e 2πw∆ X(τ) + 2πwRT(πw∆), where w is the trajectory winding number. A general path can then be expanded in a Fourier series as −2πw∆τJ 2πinτ X(τ) = e Xne + 2πwRτT(πw∆τ), n 174 M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178

= ∗ where Xn X−n so that X is real. Replacing this expansion in the action the contributions from zero-modes and from quantum fluctuations decouple, so the partition function has the form ∞ dl l Z = Z Z exp − M2 . 2l qu cl 2 w 0 The contribution from quantum fluctuations is given by3 ∞ ∞ ∞ 3 Z = 3 ∗ − † = (2π) qu 2 dXn dXn exp XnMnXn , det Mn n=1 n=1 n=1 where the matrix (Mn)µν is   w∆ 2 − 2w∆i −1 (2πn)2 n n  w∆  Mn = 2 i 10. l n −100 The determinant is therefore equal to that in the Minkowski space computation and one obtains −3/2 after zeta function regularization Zqu = i(2πl) . Next we consider the contribution from the zero-modes 2 (2πwR) E E − Z = dX exp − X + (X )2 , cl 0 l 0 2 0 F where the integration is over the orbifold fundamental domain. Moving to the y-coordinates this reduces to 2 − + (2πwR(y)) Z = dy dy dy exp − . cl 2l Integrating over the y+ and y− directions, we obtain the final answer for the particle partition function ∞ ∞ 2 dl 1 l 2 (2πwR(y)) Z = iL+(2πR) dy exp − M − . 2l ( πl)3/2 2 2l w 2 −∞ 0

B.2. String partition function

We now compute the bosonic string partition function − [ ] Z = DX Dγe S X,γ , where the Euclidean worldsheet action is 1 √ S = d2σ γγab∂ X · ∂ X. 4π a b

√ 3 3 The√ factor of 2 in the measure appears because the coefﬁcients associated to real normalized modes are 2Re(Xn) and 2Im(Xn). M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178 175

The dot denotes the inner product A · B = AT GB, with G the target space metric. Using the 2 2 symmetries of the action we fix the metric on the torus γab to be given by ds =|dσ1 + τdσ2| , with modular parameter τ = τ1 +iτ2. This gauge fixing introduces ghost contributions which will be added in the end. With this choice of metric, and following the notation introduced above, the orbifold identifications defining the (w1,w2)-twisted sector are

−2πw1∆J X(σ1 + π,σ2) = e X(σ1,σ2) + 2πRw1T(πw1∆), −2πw2∆J X(σ1,σ2 + π)= e X(σ1,σ2) + 2πRw2T(πw2∆). Functions that obey these conditions can be written as a Fourier series − J + = 2∆v 2i(n1σ1 n2σ2) + X e Xn1,n2 e 2RvT(∆v), n1,n2 = + = ∗ where v w1σ1 w2σ2 and reality of X requires X−n1,−n2 Xn ,n .TheX0,0 are zero-modes 1 2 while the other Xn1,n2 describe quantum fluctuations. The gauge fixed partition function without the ghost contribution has the explicit form 2 1 2 1 2 Z = d τ DX exp d σ |τ| ∂1X · ∂1X − 2τ1∂1X · ∂2X + ∂2X · ∂2X , 4π τ2 F where the τ integration is over the fundamental domain of the torus F. To evaluate the action of the above Fourier expansion of the fields it is convenient to consider the zero-modes and quantum fluctuations separately. We include the term 2RvT(∆v) in the zero-mode contribution to the partition function. Start with the quantum fluctuations, for which we follow the analysis of [11,16]. In this case the action has contributions of the type − J − J + · = 2∆v − J 2∆v 2i(n1σ1 n2σ2) ∂aX ∂bX 2inae 2wa∆ e Xn1,n2 e n ,n =0,0 1 2 − J − J + · 2∆v − J 2∆v 2i(n1σ1 n2σ2) 2inbe 2wb∆ e Xn1,n2 e . n1,n2=0,0

The structure of the various terms is such that after integrating over σ1 and σ2 we are left only with terms which, for each mode (n1,n2), can be written as 2 X † C J r T GJ s X . n1,n2 rs n1,n2 r,s=0 The path integral requires us to do the integration over the Fourier modes. These are Gaussian integrals and for each mode can be obtained from the determinant of a matrix (Mn1,n2 )µν of the form 2 M−− M−x M−+ = J r T J s = Mn1,n2 Crs G Mx− Mxx 0 . r,s=0 M+− 00

It is then easy to see that the terms that give a contribution to the determinant of Mn1,n2 have r = s = 0. These terms only appear for w1 = w2 = 0, so quantum fluctuations are insensitive to the winding of the string. The same happened in the computation of the partition functions for the null-boost and null-brane orbifolds presented in [11,16]. Including the contribution of ghosts, 176 M.S. Costa et al. / Nuclear Physics B 728 (2005) 148–178 the result is then the same as in Minkowski space, namely √ i −3 −2 Zqu(τ) = 2π τ2 η(τ) . τ2 This is just the partition function of three free bosons plus ghosts. Next we evaluate the zero-mode contribution to the partition function. When n1 = n2 = 0, X simplifies to − J X = e 2∆v x + 2RvT(∆v), where we defined X0,0 ≡x. The string center of mass X does not coincide with the zero-mode x. However, changing to the y-coordinates, one can check that the y component of the zero-mode coincides with the string center of mass component Y . This is an important point since, in the main text, we interpreted the integration variable y as the string center of mass. A straightforward computation shows that E − ∂ X · ∂ X = 8ER2w w x + (x )2 = 8ER2w w y, a b a b 2 a b which yields an action 2 2πER ¯ S = yTT, T = w1τ − w2. τ2 Adding the oscillators contribution from the transverse directions, the full partition function is then ∞ L+(2πR)V d2τ − π Z = i 23 η(τ) 48 dy − R2(y)TT¯ . 26 14 exp (2π) τ τ2 F 2 w1,w2 −∞

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Quantum corrections to spin effects in general relativity

G.G. Kirilin 1

Budker Institute of Nuclear Physics SB RAS, 630090 Novosibirsk, Russia Received 28 July 2005; accepted 30 August 2005 Available online 13 September 2005

Abstract Quantum power corrections to the gravitational spin–orbit and spin–spin interactions, as well as to the Lense–Thirring effect, were considered for particles of spin 1/2. These corrections arise from diagrams of second order in Newton gravitational constant G with two massless particles in the unitary cut in the t-channel. The corrections obtained differ from the previous calculation of the corrections to spin effects for rotating compound bodies with spinless constituents.  2005 Elsevier B.V. All rights reserved.

1. Introduction

It is common knowledge that the scattering amplitude of two gravitating bodies M =1(p1 − 2 q),2(p2 + q)|1(p1), 2(p2) contains terms logarithmic in the momentum transfer log |q | in the q2 → 0 limit. The terms arise from diagrams with two massless particles in the unitary cut in the t-channel. In the nonrelativistic limit these terms generate quantum power corrections to the classical Hamiltonian [8]. Thus quantum corrections to the Newton potential take the form [1–20]

41 G2hm¯ m 4 G2hm¯ m N G2hm¯ m U =− 1 2 − 1 2 − ν 1 2 . (1) 10π c3r3 15π c3r3 15π c3r3 The ﬁrst term in the expression (1) corresponds to the contribution of gravitons, the second term corresponds to the contribution of photons and the last one is the contribution of neutrinos, where Nν is the number of massless two-component neutrinos (we put c = 1, h¯ = 1 below).

E-mail addresses: [email protected], [email protected] (G.G. Kirilin). 1 Partly supported by Institute für Theoretische Physik-II Ruhr-Universität, Germany.

In the case of interaction of scalar particles, all contributions logarithmic in q2 can be separated into two parts. The first one is the contributions in which the coefficient before log|q2| is a 2 function of s = (p1 + p2) with cuts in the complex plane. These contributions can be generated by “box” diagrams only (see Fig. 5(c), (d)) that have unitary cut in the s-oru-channel. These contributions, called in [17] irreducible, can be calculated by the double dispersion relation in variables s and t = q2. The distinctive characteristic of the contributions is that they are infrared divergent (see expression (A.8)) and, correspondingly, contain logarithms where q2 plays a role of an ultraviolet cutoff log |q2|/λ2. The second part is the contributions that are entire rational functions of s. They arise from the diagrams that have no unitary cut in the s-oru-channel and also from the reducible part of the “box” diagrams (the contribution without cuts in the complex plane of the variable s). These contributions contain logarithms with the transfer momentum as an infrared cutoff, for example, logarithms that contain ultraviolet cutoff log Λ2/|q2| or the 2 | 2| masses of the interacting particles log mi / q . As shown in [20], the second part of the amplitude can be represented as a matrix element of an effective nonlocal operator of interaction of two energy–momentum tensors (i.e., an effective graviton exchange) in the q2 → 0 limit (but for arbitrary s): √ √ − ˆ 2 2 23 µν 31 2 2 g 23 µν 31 2 L = −gG log q TµνT − T = log q RµνR − R , 0 5 30 (8π)2 5 30 (2) where Tµν,Rµν are the energy–momentum tensor of the matter and the Ricci tensor, respectively. The main feature of the operator is that all descriptions of the interacting particles appear only in the form of a product of the energy–momentum tensors, whereas the “propagator” of the effective graviton is independent of the type of interacting particles.2 In contrast with the operator (2),the irreducible contributions of the diagrams Fig. 5(c), (d) contain nonmultiplicative dependence on momenta and masses of the particles and cannot be represented as operators like (2). On the basis of this operator, power quantum corrections to the Schwarzschild metric were calculated in [20]. Corrections to the interactions dependent on internal angular momentum (spin) of a compound body were also considered. There are three ways to calculate these corrections: first of all, it is possible to calculate quantum corrections to the Kerr metric for rotating compound body and to use covariant spin equation of motion in an external gravitational field [21] as performed in [20]; the second way is to integrate velocity dependent interactions, also calculated in [20], over the volumes of the interacting bodies; the last one is to put spin dependent energy–momentum tensor to the operator (2) explicitly. The purpose of this work is to calculate power quantum corrections to the spin-dependent interactions for particles of spin 1/2. It was shown in an elementary way [20] that spin-averaged contributions containing q2 as an infrared cutoff are exactly the same as (2). The irreducible parts of the diagrams Fig. 5(c), (d) averaged over spins coincide with the scalar ones since the logarithm of the infrared cutoff log λ is canceled by radiation of long-wave gravitons, which is spin-independent. However, it was noticed that the spin effects for rotating compound bodies with scalar constituents can differ from the ones for particles of spin 1/2. The reason is that, apart from the operator (2), the leading contribution to the spin effects can be generated by operators

2 Recall that here we are dealing with scalar particles only. G.G. Kirilin / Nuclear Physics B 728 (2005) 179–191 181 of higher order in q/m. For example, first order in q/m operators have the form: √ ←− −→ ˆ = − 2 G µν α β 1 ¯ − ab + ab O1 g log q R αβ;µt(a)t(b) ψi iDνΣ Σ iDν ψi 16π 4mi i √ ←− −→ 2 G ν ν α β 1 ¯ ab ab = −g log q R ; − R ; t t ψ −iD Σ + Σ iD ψ , 16π β α α β (a) (b) 4m i ν ν i i i (3) √ 2 ←− −→ ˆ 2G ν ν α β 1 ¯ ab ab O = −g log q T ; − T ; t t ψ −iD Σ + Σ iD ψ , 2 2 β α α β (a) (b) 4m i ν ν i i i (4) where left-hand side and right-hand side derivatives are defined as ←− ←− i −→ −→ i D = ∂ − Σabγ t(c), D = ∂ + Σabγ t(c). (5) α α 2 abc α α α 2 abc α ab = i ab Here Σ 2 σ are the Lorentz group generators, γabc are the Ricci rotation coefficients, µ µ = t(a) is a tetrad of basis vectors that are specified by relation t(a)tµ(b) ηab, where ηab is the metric tensor of flat space–time. Taking into account Einstein and Dirac equations in an external gravitational field 1 −→ R = 8πG T − g T , itνγ bD − m ψ = 0, (6) µν µν 2 µν b ν ˆ ˆ note that the difference of the operators O1 and O2 is an operator of second order in q/m √ 2 2 ˆ ˆ 2G O1 − O2 = −g log q T Ti + total derivative 16 m2 i i and makes zero contribution to the spin-dependent interactions. Besides, contributions to the spin–spin interactions can be generated by operators of the second order in q/m: √ ˆ = − 2 µ ¯ 5 a ν ¯ 5 b O3 g log q Dµ t ψ2γ γ ψ2 Dν t(b)ψ1γ γ ψ1 , (7) √ (a) ˆ = − 2 µ ν ¯ 5 a ¯ 5 b O4 g log q D t(a)ψ2γ γ ψ2 Dµ tν(b)ψ1γ γ ψ1 . (8) The plan of this paper is as follows: in Section 2 we describe useful Feynman rules with effective vertices introduced in [20] and also list possible types of spin-dependent amplitudes with arbitrary coefficients (form factors). The explicit expressions for all the form factors in the t → 0 limit are presented in Section 3. Results and conclusions are given in Sections 4 and 5.

2. Effective vertices

The single-graviton vertex for a spinor particle is (see Fig. 1(a))

iκ V (1/2) =− ψ(p¯ ) I P αγ β − η (P/ − 2m) ψ(p), (9) µν 4 µν,αβ µν √ = = + αβ = 1 α β + β α here κ 32πG, P p p , Iµν 2 (δµδν δµδν ) is a sort of a unit operator with the property αβ Iµν tαβ = tµν 182 G.G. Kirilin / Nuclear Physics B 728 (2005) 179–191

Fig. 1. Vertices.

Fig. 2. Compton scattering amplitude. for any symmetric tensor tαβ . The last term in braces, expression (9), proportional to the Minkowski metric tensor ηµν vanishes on-mass-shell. It is convenient to exclude such terms from the diagrams of Compton scattering amplitude containing a pole in the s-oru-channel (see Fig. 2). This term cancels fermion propagator and generates an additional contribution to the contact interaction of a mass-shell fermion with two gravitons (see Fig. 1(b)). On rearrangement, the double-graviton vertex has a form: 3 V (1/2) = i κ2I I δ T αβ , (10) κλ,ρσ 4 κλ,αδ β ,ρσ (1/2) (1/2) where Tαβ is the energy–momentum tensor of the fermion ﬁeld

1 T (1/2) = I P αψ(p¯ )γ β ψ(p). (11) µν 2 µν,αβ For further Feynman rules the reader is referred to the papers [18,20]. The rest of the pole diagrams (Fig. 2) is proportional to the Compton scattering amplitude in quantum electrodynamics. For example, consider diagram shown in Fig. 2(a) 2 iκ ρ φ σ ω Mαβ,γ δ = Iγδ,ρσIαβ,φω(p + l) (p + l) u(p¯ )γ (/l + m)γ u(p) . (12) 16(l2 − m2) It is useful to decompose the amplitude of scattering of photon by an electron into independent spin structures: σω =¯ σ + ω MQED u(p )γ (/l m)γ u(p) 1 i = (P − K)σ δω + (P − K)ωδσ − qσ δω + δσ qω + K ησω J ρ + σωληK S , 2 ρ ρ ρ ρ ρ 2 λ η (13) here P = 2p − q,K = 2k − q, and J µ, Sµ are the vector current and spin four-vector (axial current), respectively: 0 −1 J µ =¯u(p )γ µu(p), Sµ =¯u(p )γ γ µu(p), γ = . (14) 5 5 −10 G.G. Kirilin / Nuclear Physics B 728 (2005) 179–191 183

Diagram shown in Fig. 2(b) can be considered in a similar manner. Vector current J µ can be also decomposed into a convective part and a part containing spin explicitly: 2 µ q µ P i µγ αδ 1 − J = u(p¯ )u(p) + qγ PαSδ. (15) 4m2 2m 4m2 Therefore, all vertices and lines can be decomposed into two structure including bispinor amplitudes of a fermion, namely, u(p¯ )u(p) and Sµ. Thus, matrix element of the fermion–fermion 1,2 scattering can be separated into three groups. The first one does not include Sµ explicitly: 2 2 MN = FN(s)G m1m2(u¯1u1)(u¯2u2) log q , (16) 1,2 the second one contains the terms that are proportional to the first power of Sµ : Sδ Sδ M = F 2 α β γ 2 ¯ + 1 ¯ 2 SO i SO(s)G αβγ δ q P1 P2 (u1u1) (u2u2) log q , (17) m2 m1 and the last part contains terms quadratic in spin: M = 2 F (1) · · + F (2) · 2 SS G SS (s)(q S1)(q S2) SS (s)t(S1 S2) log q . (18) In the first Born approximation corrections to a potential are given by a Fourier transform of nonrelativistic amplitude

3 3 d q − · d q − · M U(r) = A(q)e iq r =− e iq r lim lim √ . (19) 3 3 → + 2 → (2π) (2π) s (m1 m2) t 0 i 2Ei

Since we are interested in amplitudes in the t → 0 limit, the form factors Fi are the functions of the variable s. It is convenient to introduce a new variable s − m2 − m2 γ˜ = 1 2 (20) 2m1m2 that corresponds to the γ -factor of one of the interacting particles in the limit when the mass of the other goes to inﬁnity.

3. Matrix elements

We start the discussion of loops with the graviton–graviton interaction shown in Fig. 3.Itis easy to calculate this contribution using the effective operator derived in [7]

Fig. 3. Graviton–graviton interaction. 184 G.G. Kirilin / Nuclear Physics B 728 (2005) 179–191

Fig. 4. Graviton–fermion interaction. √ − 2 g 21 µν 1 2 L =−log q R Rµν + R . (21) gg (8π)2 15 30 By substituting the Ricci tensor from the Einstein equations (6) one gets the following matrix elements: 1 7 F (γ)˜ =− 42γ˜ 2 + 1 , F (γ)˜ =− γ,˜ (22) N 15 SO 20 7 F (1)(γ)˜ =−F (2)(γ)˜ = 1 − 2γ˜ 2 . (23) SS SS 20 Diagrams containing the graviton–fermion interaction (see Fig. 4) make the following contribution to the matrix elements: γ˜ F (γ)˜ = 2 6γ˜ 2 + 1 , F (γ)˜ = , N SO 2 1 F (1)(γ)˜ =−F (2)(γ)˜ = 1 − 2γ˜ 2 . (24) SS SS 2 These corrections coincide with the corresponding form factor corrections calculated in [19]. By analogy with (21) the contribution of the diagrams including graviton–fermion interaction (Fig. 4) can be represented as an effective operator: √ G L = −g logq2 3RµνT − 2RT − 8Oˆ , (25) gf 4π µν 1 ˆ where the operator O1 is deﬁned by (3). The rest of the diagrams (Fig. 5) corresponds to the fermion–fermion interaction. The sum of the diagrams Fig. 5(a), (b) makes the following contribution to the matrix elements: 15 F (γ)˜ =−3 5γ˜ 2 − 1 , F (γ)˜ =− γ,˜ F (1)(γ)˜ =−F (2)(γ)˜ = 0. (26) N SO 16 SS SS The sum of these diagrams does not contribute to the spin–spin interaction. The corresponding effective operator is √ 15 3 15 L(ab) =− −g logq2G2 T µνT − T 2 + Oˆ , (27) ff 2 µν 2 2 2 ˆ where the operator O2 is deﬁned by (4). There are a number of points to be made before considering the contribution of the diagrams Fig. 5(c), (d). As stated above, the analysis of these diagrams is complicated by the presence of infrared divergencies. Aside from the integrals (A.2)–(A.7) containing q2 as an infrared cutoff, G.G. Kirilin / Nuclear Physics B 728 (2005) 179–191 185

Fig. 5. Fermion–fermion interaction. there are contributions generated by integrals (A.8), (A.10) containing q2 as an ultraviolet cutoff. There is a unitary cut in the s-oru-channel in the last-mentioned integrals, and consequently, there is a nonpolynomial dependence on s. We present the results for the irreducible parts and parts without cuts in the complex plane of the variable s for each type of the matrix elements separately. However, as we shall see later, the reducible part also contains a nonpolynomial dependence on s. The reason is a singularity in the → 0 limit in vector and tensor integrals of (s) the Kµ type (see expression (A.14) in Appendix A) where 1 = (P P )2 − P 2P 2 ,P= p + p ,P = p + p . (28) 4s 1 2 1 2 1 1 1 2 2 2 In the center of mass frame the variable takes the form:

= 4|p|2 cos2 θ/2. (29) Terms proportional to log(sin θ/2)/(cos2 θ/2) are typical for the diagrams of the sort of Fig. 5(c), (d). For example, they occur in the calculation of an polarization vector when an electron is scattered by the Coulomb ﬁeld in the second Born approximation. Terms proportional to 1/ cancel in the spin independent part of the matrix elements (16) in the reducible and irreducible parts independently. However, in the spin dependent terms (17), (18) 1/ cancels only in the sum of the irreducible and reducible parts in the nonrelativistic limit. Taking into account all mentioned above, we write out the matrix elements for the diagrams Fig. 5(c), (d). The reducible parts are 31 1 1 F (γ)˜ =−7 + 15γ˜ 2, F (1)(γ)˜ =−4 1 − 2γ˜ 2 + + , (30) N SS 12 2 γ˜ 2 − 1 47 7 γ˜ 77 1 1 F (γ)˜ = γ˜ + , F (2)(γ)˜ = 4 1 − 2γ˜ 2 − − . (31) SO 16 8 γ˜ 2 − 1 SS 24 2 γ˜ 2 − 1 Since in the expressions (30), (31) we take the limit t → 0, the singularity 1/ turns to the singularity 1/(γ˜ 2 − 1). Terms not proportional to 1/(γ˜ 2 − 1) can be again represented in the form of an effective operator √ 15 7 17 31 77 L(cd) = −g logq2G2 T µνT − T 2 + Oˆ + Oˆ − Oˆ . (32) ff 2 µν 2 2 2 12 3 24 4 The irreducible parts are 2 2 2 2 FN(γ)˜ =−16γ˜ 2γ˜ − 1 WA(γ)˜ + 1 − 2γ˜ WS(γ),˜ (33) 3 ˜ 2 1 7 1 γ 2 F (γ)˜ =− 4γ˜ + + WA(γ)˜ + 2γ˜ − 1 WS(γ),˜ (34) SO 2 8 γ˜ 2 − 1 6 186 G.G. Kirilin / Nuclear Physics B 728 (2005) 179–191 ˜ (1) 3 1 γ 1 2 2 F (γ)˜ = 2γ˜ − 8γ˜ − WA(γ)˜ + 1 − 2γ˜ WS(γ),˜ (35) SS 2 γ˜ 2 − 1 6 ˜ (2) 3 1 γ 1 2 2 1 F (γ)˜ = −2γ˜ + 8γ˜ + WA(γ)˜ + − 2γ˜ − 1 + WS(γ),˜ (36) SS 2 γ˜ 2 − 1 6 12 where functions WS(γ),˜ WA(γ)˜ are deﬁned in Appendix A by Eqs. (A.12), (A.13). As noted above, the terms singular in the γ˜ → 1 limit cancel in the sum of the reducible (30), (31) and irreducible (34)–(36) parts 13 7 9 F (1) =− , F (1)(1) = , F (2)(1) =− . (37) SO 16 SS 12 SS 8

4. Quantum corrections to spin effects

Summing up all the contributions we get the following scattering amplitudes. The terms not 1,2 containing Sµ explicitly are ¯ ¯ 41 2 (u1u1) (u2u2) 2 AN(q) = G m1m2 log q . (38) 5 2m1 2m2 1,2 The terms proportional to the ﬁrst power of Sµ are 8 Sδ (u¯ u ) Sδ (u¯ u ) A (q) = G2 qαP β P γ 2 1 1 + 1 2 2 q2. SO i αβγ δ 1 2 2 2 log (39) 5 2m2 2m1 2m1 2m2 At last, the part containing terms quadratic in spins is 2 G 4 11 2 ASS = (q · S1)(q · S2) + t(S1 · S2) log q . (40) 4m1m2 15 40 Using the nonrelativistic expansions of the bispinor amplitudes of the scattering particles 1 i u(p¯ ∓ q)u(p ) = ± i j k +···, 1,2 1,2 1 2 ij k q p1,2s1,2 (41) 2m1,2 2m1,2 1 δ = 1 ¯ ∓ δ ¯ ≈ S1,2 u(p1,2 q)γ5γ u(p1,2) (0, 2s1,2), (42) 2m1,2 2m1,2 i m qαP β P γ Sδ =± n k m ∓ 2,1 n k m , 2 αβγ δ 1 2 1,2 8i nkmq p2,1s1,2 8i nkmq p1,2s1,2 (43) 2m1,2 m1,2 yields the amplitudes in the momentum space:

41 2 2 AN(q) = G m1m2 log q , (44) 5 A = 87 2 − m2 i j k + m1 i j k 2 SO(q) G i ij k q p1s1 i ij k q p2s2 log q 10 m1 m2 + 64 2 n k m − n k m 2 G inkmq p2s1 inkmq p1s2 log q , (45) 5 16 11 A (q) = G2 (q · s )(q · s ) + q2(s · s ) log q2. (46) SS 15 1 2 10 1 2 G.G. Kirilin / Nuclear Physics B 728 (2005) 179–191 187

With the expressions (A.1) from Appendix A we obtain the corrections in the coordinate space. First of all, according to [18,20], the power quantum correction to the Newton law is the same as for scalar particles (1)

41 hG¯ 2m m U =− 1 2 . (47) N 10 c3πr3 The quantum correction to the gravitational spin–orbit interaction is 261 hG¯ 2 m m U = 2 + 1 · , SO 5 5 s1 s2 l (48) 20 c πr m1 m2 i ij k l = (r1 − r2)j pk, (49) where p = p1 =−p2 is the relative momentum. The quantum correction to the Lense–Thirring effect is

96 hG¯ 2 U = (s + s ) · l. (50) LT 5 c5πr5 1 2

The quantum correction to the interaction of the two internal angular momenta si , i.e., to the gravitational spin–spin interaction is hG¯ 2 5 U = 8(n · s )(n · s ) + (s · s ) . (51) SS c5πr5 1 2 2 1 2

5. Discussion and conclusions

Hence, as expected, the quantum corrections to the classical Hamiltonian (48), (50), (51) for a spinor particle differ from the full sum of the similar corrections obtained in [20] for a rotating compound body with scalar constituents. This difference is associated with the irreducible part as well as with the reducible one that can be represented in the form of an effective operator (see (2), (21), (25), (27), (32)) 31 77 L = L − 8Oˆ + 16Oˆ + Oˆ − Oˆ , (52) 0 1 2 12 3 24 4 ˆ where the new operators Oi take part. Following are several points concerning the operator (52). As demonstrated in [20],after averaging over the spins, the Compton amplitude of fermion–graviton scattering and the scalar– graviton one coincide with the required accuracy. In particular, it means that the effects linear in spin (see expressions (48), (50)) for fermion–fermion and fermion–scalar scattering are the same. However, we see that the analysis is complicated by the singularity 1/. Therefore, it is not unreasonable to make sure that the separation into the reducible and the irreducible parts is thesameasinEqs.(27), (30), (32), (33) for the case of fermion scattering by a scalar particle (evidently, there is only a contribution of one of the particles under the summation sign in the operators (32), (33) in this case). We use the effective vertices derived in [20] for scalar particles. The diagrams Fig. 5(a), (b) make the following contribution to the spin effects: 3 F (γ)˜ = 5 − 16γ˜ 2, F (γ)˜ =− γ.˜ (53) N SO 4 188 G.G. Kirilin / Nuclear Physics B 728 (2005) 179–191

The reducible part of the diagrams Fig. 5(c), (d) yields 11 7 γ˜ F (γ)˜ =−9 + 16γ˜ 2, F (γ)˜ = γ˜ + . (54) N SO 4 8 γ˜ 2 − 1 Note that the sum of (53) and (54) agree with the sum of corresponding expressions (26) and (30). The irreducible part is also equal to (34). Therefore the singularities 1/(γ˜ 2 − 1) cancel in the same way as in the case of fermion–fermion scattering. Thus, the effective operator (52) is universal for scattering of scalar particles and particles of spin 1/2. Being interested in the quantum corrections to spin effects at the leading order in 1/c (where c is the speed of light), we considered only the matrix elements like (16)–(18). Formally, it is possible to consider one more matrix element at the same order in q/mi (P S ) (P S ) M(s, t) = F(s, t)t 1 2 2 1 , (55) m2 m1 that however does not contribute to the spin effects at the leading order in 1/c. Nevertheless, it is possible to make sure that the contribution of the diagrams Figs. 3, 4, 5(a), (b) and the reducible part of the diagram Fig. 5(c), (d) to the form factor F(s, t) can also be described by the operator (52). This fact is trivial for the diagrams presented in Fig. 3 since their contribution originally contains the energy–momentum tensors of interacting particles. Contribution of the diagrams Fig. 4 is γ˜ F(γ)˜ =− , (56) 4 that corresponds to the contribution of the operator (25). The diagrams shown in Fig. 5(a), (b) yield F(γ)˜ = 0, (57) that is also equal to the contribution of the corresponding operator (27). The reducible contribution of the diagrams Fig. 5(c), (d) is 33γ˜ γ˜ F (γ)˜ = 2γ˜ − + . (58) red 16(1 −˜γ 2) 4(1 −˜γ 2)2 Note that the terms not containing 1/(1 −˜γ 2) also correspond to the contribution of the operator (32). Hence, the operator (52) describes all the contributions to the matrix element (55) correctly. Terms in the expression (58) singular in the γ˜ → 1 limit cancel with the irreducible part of the diagrams Fig. 5(c), (d) in the same manner as for the matrix elements (17), (18): ˜ 3 ˜ γ γ 2 9 1 F (γ)˜ = + WS + −2 1 +˜γ + − WA, irred 6 48(1 −˜γ 2) 4(1 −˜γ 2) 4(1 −˜γ 2)2 (59) 19 lim Fred(γ)˜ + Firred(γ)˜ =− . (60) γ˜→1 40

Acknowledgements

I would like to thank I.B. Khriplovich and A.A. Pomeransky for their helpful comments and discussions. The investigation was supported by the Russian Foundation for Basic Research G.G. Kirilin / Nuclear Physics B 728 (2005) 179–191 189 through Grant No. 05-02-16627-a and by Institute für Theoretische Physik-II Ruhr-Universität through Graduiertenkolleg “Physik der Elementarteilchen an Beschleunigern und im Univer- sum”.

Appendix A

To calculate the correction to spin-dependent interactions, it is needed to consider the following Fourier transforms:

3 3 2 −iq·r d q 1 2 −iq·r d q 3 ri log q e =− , −iqi log q e = , (2π)3 2πr3 (2π)3 2π r5

3 3 2 2 −iq·r d q 3 2 −iq·r d q 3 q log q e = , qiqj log q e = (5ninj − δij ). (2π)3 πr5 (2π)3 2πr5 (A.1)

According to the choice 1(p1 − q),2(p2 + q)|1(p1), 2(p2) of kinematics and the sign in the exponent of the Fourier transforms, the vector r is directed from the second body to the ﬁrst one, i.e., r = r1 − r2. Logarithmic parts of the integrals occurring are given below, using the notation L =− i × (4π)2 log |q2|:

dDk 1 I = = L +···, (A.2) (2π)D k2(k − q)2

dDk k q I = µ = µ L +···, (A.3) µ (2π)D k2(k − q)2 2 dDk k k 1 q2 I = µ ν = q q − δ L +···, (A.4) µν (2π)D k2(k − q)2 3 µ ν 4 µν

dDk 1 L J = = +···, (A.5) (2π)D k2(k − q)2((p − k)2 − m2) 2m2

D d k kµ pµ − qµ Jµ = =− L +···, (A.6) (2π)D k2(k − q)2((p − k)2 − m2) 2m2

dDk k k J = µ ν µν (2π)D k2(k − q)2((p − k)2 − m2) L q2 q2 = pµpν − (pµqν + pνqµ) + 2qµqν − gµν +···. (A.7) 4m2 m2 2 For the diagrams Fig. 5(c), (d) to be calculated, it is necessary to consider integrals with four Feynman propagators with the infrared divergency regularized by the “mass” of graviton λ.The imaginary part of the diagram Fig. 5(c) will be omitted because it does not contribute to the potential of the two-body interaction. It corresponds to the second Born approximation. The real part of the integral for the diagram Fig. 5(c) is

dDk 1 1 K(s) = D 2 + − 2 + − 2 − 2 + + 2 − 2 + (2π) (k i0)((q k) i0) ((p1 k) m1 i0)((p2 k) m2 i0) 190 G.G. Kirilin / Nuclear Physics B 728 (2005) 179–191 2 2 s − m− − s − m+ =− i 1 t 2m1m2 +··· 2 log 2 log (4π) m1m2t λ 2 2 2 2 (s − m+)(s − m−) s − m− + s − m+ i 1 t =− K (s) +···, 2 log 2 s (A.8) (4π) m1m2t λ where notations m+ = m1 +m2, m− = m1 −m2 were introduced, and dependence on the variable s denoted as the function Ks(s). To simplify expressions it is convenient to introduce a new ˜ = − 2 − 2 variable γ (s m1 m2)/(2m1m2). 1 2 Ks(s) = log γ˜ − γ˜ − 1 . (A.9) γ˜ 2 − 1 In the diagram Fig. 5(d) the same integral can be derived by analytic continuation of all the = 2 = 2 + 2 − − expressions from the s-totheu-channel (t q , u 2m1 2m2 s t): 2 2 m+ − u + m− − u K =−K 2 + 2 − − = 2m1m2 u(u) s 2m1 2m2 s t log . 2 2 2 2 (m+ − u)(m− − u) m+ − u − m− − u (A.10) µ In the limit when the momentum transfer q goes to zero the function Ku(u) can be expressed as a function of Ks(s) up to terms linear in t: t γ˜K (s) + 1 K (u) ≈−K (s) + s . u s 2 (A.11) 2m1m2 γ˜ − 1 In the course of the calculation, it is convenient to express amplitudes as a half-sum and half- difference of the functions Ks(s) and Ku(u): 6m1m2 1 γ˜Ks(s) + 1 WS(γ)˜ =−lim Ks(s) + Ku(u) =−3 , (A.12) t→0 t 2 γ˜ 2 − 1 1 WA(γ)˜ = lim Ku(u) − Ks(s) =−Ks(s), (A.13) t→0 2 the functions WS(γ),˜ WS(γ)˜ are normalized to unity in the nonrelativistic limit, i.e., WS(1) = WA(1) = 1. Vector and tensor integrals with four propagators are considered with respect to the relations already obtained (A.2)–(A.8), (A.10). As an example, consider an integral (P1 = 2p1 − q,P2 = 2p2 + q):

dDk k K(s) = µ µ D 2 2 2 2 2 2 (2π) (k − λ )((k − q) − λ )(k − 2p1k)(k + 2p2k) = AP1µ + BP2µ + Cqµ. (A.14) Using the following relations for the logarithmic parts, we obtain the coefﬁcients A, B, C:

q2 qµK(s) = K(s), (A.15) µ 2 q2 P µK(s) =−J − K(s), (A.16) 1 µ 2 2 G.G. Kirilin / Nuclear Physics B 728 (2005) 179–191 191

q2 P µK(s) = J + K(s), (A.17) 2 µ 1 2 here J1 and J2 are the integrals (A.5) with the propagators of the ﬁrst and the second particles, respectively, 4s q2 q2 A = J + K(s) (P P ) + J + K(s) P 2 , (A.18) 1 2 1 2 2 2 2 4s q2 q2 B =− J + K(s) (P P ) + J + K(s) P 2 , (A.19) 2 2 1 2 1 2 1 1 C = K(s), (A.20) 2 where we introduce the following notation 1 = (P P )2 − P 2P 2 ,s= (p + p )2. (A.21) 4s 1 2 1 2 1 2 Tensor integrals can be considered in the same way.

References

[1] A.F. Radkowski, Ann. Phys. 56 (1970) 319. [2] D.M. Capper, M.J. Duff, L. Halpern, Phys. Rev. D 10 (1974) 461. [3] D.M. Capper, M.J. Duff, Nucl. Phys. B 44 (1974) 146. [4] M.J. Duff, J.T. Liu, Phys. Rev. Lett. 85 (2000) 2052, hep-th/0003237. [5] D.M. Capper, G. Leibbrandt, M. Ramón Medrano, Phys. Rev. D 8 (1973) 4320. [6] M.J. Duff, Phys. Rev. D 9 (1973) 1837. [7] G. ’t Hooft, M. Veltman, Ann. Inst. H. Poincaré A 20 (1974) 69. [8] D.G. Boulware, S. Deser, Ann. Phys. 89 (1975) 193. [9] J.F. Donoghue, Phys. Rev. Lett. 72 (1994) 2996, gr-qc/9310024. [10] J.F. Donoghue, Phys. Rev. D 50 (1994) 3874, gr-qc/9405057. [11] J.F. Donoghue, Introduction to the effective field theory description of gravity, in: F. Cornet, M.J. Herrero (Eds.), Advanced School on Effective Theories, World Scientific, Singapore, 1996, gr-qc/9512024. [12] J.F. Donoghue, Perturbative dynamics of quantum general relativity, in: T. Piran, R. Ruffini (Eds.), Proceedings of the Eighth Marcel Grossmann Meeting on General Relativity, World Scientific, Singapore, 1999, gr-qc/9712070. [13] I.J. Muzinich, S. Vokos, Phys. Rev. D 52 (1995) 3472, hep-th/9501083. [14] A. Akhundov, S. Belucci, A. Shiekh, Phys. Lett. B 395 (1998) 19, gr-qc/9611018. [15] H. Hamber, S. Liu, Phys. Lett. B 357 (1995) 51, hep-th/9505182. [16] D. Dalvit, F.D. Mazzitelli, Phys. Rev. D 56 (1997) 7779, hep-th/9708102. [17] I.B. Khriplovich, G.G. Kirilin, Zh. Eksp. Teor. Fiz. 95 (2002) 1139, Sov. Phys. JETP 95 (2002) 981, gr-qc/0207118. [18] N.E.J. Bjerrum-Bohr, J.F. Donoghue, B.R. Holstein, Phys. Rev. D 67 (2003) 084033, hep-th/0211072. [19] N.E.J. Bjerrum-Bohr, J.F. Donoghue, B.R. Holstein, Phys. Rev. D 68 (2003) 084005, hep-th/0211071. [20] I.B. Khriplovich, G.G. Kirilin, Zh. Eksp. Teor. Fiz. 125 (2004) 1219, Sov. Phys. JETP 98 (2004) 1063, gr- qc/0402018. [21] I.B. Khriplovich, A.A. Pomeransky, Zh. Eksp. Teor. Fiz. 113 (1998) 1537, Sov. Phys. JETP 86 (1998) 839, gr- qc/9710098. Nuclear Physics B 728 (2005) 192–206

Anomaly mediation, Fayet–Iliopoulos D-terms and precision sparticle spectra

R. Hodgson a,I.Jacka, D.R.T. Jones a,G.G.Rossb

a Departament of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK b The Rudolf Peierls Centre for Theoretical Physics, Oxford University, 1 Keble Road, Oxford OX1 3NP, UK Received 17 August 2005; accepted 8 September 2005 Available online 22 September 2005

Abstract We consider the sparticle spectra that arise when anomaly mediation is the source of supersymmetry- breaking and the tachyonic slepton problem is solved by a Fayet–Iliopoulos D-term. We also show how this can lead to a minimal viable extension of anomaly mediation, in which the gauge symmetry associated with this D-term is broken at very high energies, leaving as its footprint in the low energy theory only the required D-terms and seesaw neutrino masses.  2005 Elsevier B.V. All rights reserved.

1. Introduction

Increasing precision of sparticle spectrum calculations is an important part of theoretical preparation for the LHC and the ILC. Much of this work has concentrated on the MSUGRA scenario, where it is assumed that the uniﬁcation of gauge couplings at high energies is accom- panied by a corresponding uniﬁcation in both the soft supersymmetry-breaking scalar masses and the gaugino masses; and also that the cubic scalar interactions are of the same form as the Yukawa couplings and related to them by a common constant of proportionality, the A-parameter. This paradigm is not, however, founded on a compelling underlying theory and therefore it is worth- while exploring other possibilities. In this paper we focus on anomaly mediation (AM) [1–22]. This is a framework in which a single mass parameter determines the φ∗φ, φ3 and λλ supersymmetry-breaking terms in terms of calculable and moreover renormalisation group (RG) invariant functions of the dimension-

E-mail address: [email protected] (D.R.T. Jones).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.09.013 R. Hodgson et al. / Nuclear Physics B 728 (2005) 192–206 193 less couplings, in an elegant and predictive way; too predictive, in fact, in that the theory in its simplest form leads to tachyonic sleptons and fails to accommodate the usual electroweak vacuum state. There is a natural solution to this, however, which restores the correct vacuum while retaining the RG invariance (and hence the ultra-violet insensitivity) of the predictions. This is achieved simply (and without introducing another source of explicit supersymmetry-breaking) by the introduction of a Fayet–Iliopoulos (FI) D-term or terms. This possibility was first explored in detail in Ref. [17], and subsequently by a number of authors [18–22]. The main purpose of this paper is to present the most precise spectrum calculations to date in the AMSB scenario. We also show how the low energy theory employed can arise in a natural way from a theory with an additional anomaly-free U1 broken at a high scale. We examine the decoupling in this case and show how only the soft mass contributions from the D-terms remain, which can naturally eliminate the tachyonic slepton problem. This provides a minimal extension of anomaly mediation. In the original scenario of Ref. [17], FI terms corresponding to two distinct U1 groups were introduced, one being the standard model U1, and the other the second mixed-anomaly-free (or completely anomaly-free if right-handed neutrinos are included) U1 admitted by the MSSM. This U1 may be chosen to be B − L [18], or some linear combination of it and the MSSM U1 [17]. Or, as emphasised in Ref. [20], a single new U1 may be employed if the charges are chosen appropriately. If these FI terms are added to the masses with constant coefficients (as in Eq. (4)) rather than as genuine gauge linear D-terms, then as discussed in Ref. [17] and at more length in Ref. [20], the choices made in Refs. [17–20] are simply reparametrisations of each other. As indicated above, we will see how this scenario can emerge naturally at low energies in a specific theory with an additional gauged anomaly-free U1.

2. The minimal supersymmetric standard model

The MSSM is deﬁned by the superpotential: c c c W = H2QYt t + H1QYbb + H1LYτ τ + µH1H2 (1) with soft breaking terms: 3 = 2 ∗ + 2 + 1 + LSOFT mφφ φ m3H1H2 Miλiλi h.c. = 2 φ i 1 c c c + H2Qht t + H1Qhbb + H1Lhτ τ + h.c. , (2) where in general Yt,b,τ and ht,b,τ are 3 × 3 matrices. We work throughout in the approximation that the Yukawa matrices are diagonal, and neglect the Yukawa couplings of the ﬁrst two generations.

3. The AMSB solution

Remarkably the following results are RG invariant: = Mi m0βgi /gi, =− ht,b,τ m0βYt,b,τ , 1 d m2 i = m2µ γ i , j 2 0 dµ j 194 R. Hodgson et al. / Nuclear Physics B 728 (2005) 192–206

2 = − m3 κm0µ m0βµ. (3)

Here βgi are the gauge β-functions, γ the chiral supermultiplet anomalous dimension, and βYt,b,τ 2 i are the Yukawa β-functions. Moreover, the RG invariance is preserved if we replace (m ) j in Eq. (3) by 1 d m¯ 2 i = m2µ γ i + kY δi , (4) j 2 0 dµ j i j where k is a constant and Yi are charges corresponding to a U1 symmetry of the theory with no mixed anomalies with the gauge group. Of course the kY term corresponds in form to a FI D-term. The expressions for M, h and m2 given in Eq. (3) are obtained if the only source of breaking is a vev in the supergravity multiplet itself: the AMSB scenario (m0 is then the gravitino 2 mass). Note the parameter κ in the solution for m3; some treatments in the literature omit this term (based on top–down considerations). However, Eq. (3) is RG invariant for arbitrary κ and 2 so we retain it. This means that m3 will be determined in the usual way by the electroweak minimisation. In the following two sections we will show how Eq. (4) can arise via spontaneous breaking of a U1 symmetry.

4. Anomaly-free U1 symmetries

The MSSM (including right-handed neutrinos) admits two independent generation-blind anomaly-free U1 symmetries. The possible charge assignments are shown in Table 1. Note that an anomaly-free, flavour-blind U1 necessarily corresponds to equal and opposite charges for H1,2 and hence an allowed Higgs µ-term; for an attempt at a Froggatt–Nielsen style origin for the Higgs µ-term using a flavour dependent U1, see for example Ref. [23]. One of the attractive features of anomaly mediation is that squark/slepton mediated flavour changing neutral currents are naturally small; this feature is preserved by a generation-blind U1 but not by a flavour dependent U1, so we stick to the former here. SM = =− The SM gauged U1 is L 1, e 2; this U1 is of course anomaly free even in the absence c B−L =− = c 3 of ν . U1 is L e 1; in the absence of ν this would have U1 and U1-gravitational anomalies, but no mixed anomalies with the SM gauge group, which would suffice to maintain SM B−L the RG invariance of the AMSB solutions. We can introduce FI terms for both U1 and U1 , SM or for U1 and a linear combination of them [17], or indeed simply have a single U1 with the same sign for L and e [20]. We will follow Ref. [20] here; however in the decoupling scenario described in the next section, the low energy theories corresponding to the single U1 case and the double U1 case of Ref. [17] are simply reparametrisations of each other.

5. Spontaneously broken U1

With the MSSM augmented by an additional U1, it is natural to ask at what scale this U1 is broken. It is possible that this scale is at around 1 TeV [24]; here, however, we concentrate on

Table 1 Anomaly free U1 symmetry for arbitrary lepton doublet and singlet charges L and e respectively c c c Qu d H1 H2 ν − 1 − − 2 + 4 − − + − − 3 L e 3 Le3 L e LeL 2L e R. Hodgson et al. / Nuclear Physics B 728 (2005) 192–206 195 the idea that it is broken at very high energies and that the only low energy remnant of it is the set of FI-type terms that we require. It would be natural to think that if a U1 is broken at some high scale M then, by the decoupling theorem, all effects of the U1 would be suppressed at energies E M by powers of 1/M.We shall see that with a FI term this is not the case and it is quite natural for there to be O(MSUSY) scalar mass contributions arising from the presence of the FI term. It is straightforward to construct a model with an additional gauged U1 in such a way that the only effect on the low-energy theory is the appearance of the FI terms we require. ¯ =± + We introduce a pair of MSSM singlet fields φ,φ with U1 charges qφ,φ¯ (4L 2e) and a gauge singlet s, with a superpotential 1 W = λ φφs¯ + λ νcνcφ. (5) 1 2 2 The choice of charges is essentially determined by the requirement that the φ,φ¯ fields decouple from the MSSM while generating a large mass for νc. The scalar potential takes the form 2 2 ∗ 2 ¯∗ ¯ 1 ∗ ¯∗ ¯ ∗ V = m φ φ + m ¯ φ φ +···+ ξ − q φ φ − φ φ − e χ χ +···, (6) φ φ 2 φ i i i matter where χi stands for all the MSSM scalars, and ei their U1 charges, and we have introduced a 2 FI term for U1. We will take ξ>0, qφ > 0 and ξ m0, and assume that the scalar masses in Eq. (6) (apart from the Higgs µ-term) and other supersymmetry-breaking terms in the theory are the anomaly-mediation contributions. We now√ proceed to minimise the scalar potential. As we shall see, this will result in a vev for φ of order ξ√; this means that the appropriate scale at which we should minimise the potential is also around ξ. As a consequence, we of course include at this stage the U1 contributions in the anomalous dimensions of the fields. It therefore follows 2 that as long as λ1,2 are somewhat smaller than the U1 charges then we will have mφ < 0, which we will assume in the following analysis. If we look for an extremum with only φ nonzero we find − 2 ∗ 1 qφξ m φ φ ≡ v2 = φ φ 2 (7) 2 qφ √ = ≈ 2 so φ O( ξ)for large ξ and V mφξ/qφ. Note that since as indicated above we have chosen 2 parameters so that mφ <√0wehaveV<0 at the minimum. Expanding about the minimum, i.e., = + with φ (vφ H (x))/ 2, where H is the (real) physical U1 Higgs, we find 2 4 m ξ m 1 ∗ e ∗ V = φ − φ + m2 + m2 + v2 λ2 φ¯ φ¯ − i m2 χ χ 2 φ φ¯ φ 1 φ i i qφ 2qφ 2 qφ 1 ∗ 1 ∗ 1 ∗ ∗ + v2 λ2s s + v2 λ2 νc νc + v q H − q φ¯ φ¯ + e χ χ 2 +···. (8) 2 φ 1 2 φ 2 2 φ φ φ i i i For large ξ (i.e., large vφ) all trace of the U1 in the effective low energy Lagrangian disappears, except for contributions to the masses of the matter fields which are naturally of the same order as the AMSB ones. We can see this either by treating the heavy H -field as nonpropagating and eliminating it via its equation of motion, or by noting that the quartic (χ ∗χ)2 D-term still present 196 R. Hodgson et al. / Nuclear Physics B 728 (2005) 192–206 in Eq. (8) is cancelled (at low energies) by the H -exchange graph using two Hχ∗χ vertices. In the large ξ limit the breaking of U1 preserves supersymmetry; thus the U1 gauge boson, its gaugino, ψH and H form a massive supermultiplet which decouples from the theory. The√ fact that supersymmetry is good at large ξ protects the light χ fields from obtaining masses of O( ξ) from loop corrections. Moreover vφ via the superpotential gives large supersymmetric masses to φ¯, s and also νc, thus naturally implementing the see-saw mechanism. The low energy theory contains just the MSSM fields with the only modification being the FI-type mass contributions 2 proportional to mφ. This is simply another manifestation of the nondecoupling of soft mass corrections from D-terms [25]. We now need only choose the charges L,e for the lepton doublet and singlet so that the 2 contributions to their slepton masses are positive; that is, we choose L,e > 0 since mφ < 0. It is easy to show that (modulo electroweak breaking) this represents the absolute minimum = of the potential√ (note that λ1 plays a crucial role here in that for λ1 0theD-flat direction φ=φ¯ ξ would lead to an potential unbounded from below). The model constructed here is similar in spirit to those of Harnik et al. [19], in that the U1 breaking is at a high scale so that only the D-term contributions survive in the low energy theory. Like [19] we assume that the anomaly mediated contribution to SUSY breaking is dominant, something that can be justified in the conformal sequestered scheme of Luty et al. [16].The main difference is that we use a FI term to trigger the U1 breaking rather than an F -term. Here we have just assumed the existence of the FI term as one of the terms allowed by the symmetries of the theory. We will return elsewhere to a discussion of how such a term may be generated in an underlying theory. In the next section we will explore the region of the (e, L) parameter space such that electroweak-breaking via the higgses is obtained as usual.

6. The sparticle spectrum

We turn now to the effective low energy theory. Evidently in the scenario described in Sec- tion 5, we have decoupling of the U1 at low energies so that the anomalous dimensions of the fields are as in the MSSM; thus for the higgses and 3rd generation matter fields we have (at one loop): 3 3 16π 2γ = 3λ2 + λ2 − g2 − g2, H1 b τ 2 2 10 1 3 3 16π 2γ = 3λ2 − g2 − g2, H2 t 2 2 10 1 3 3 16π 2γ = λ2 − g2 − g2, L τ 2 2 10 1 8 3 1 16π 2γ = λ2 + λ2 − g2 − g2 − g2, Q b t 3 3 2 2 30 1 2 2 8 2 8 2 16π γ c = 2λ − g − g , t t 3 3 15 1 2 2 8 2 2 2 16π γ c = 2λ − g − g , b b 3 3 15 1 2 2 6 2 16π γ c = 2λ − g , (9) τ τ 5 1 R. Hodgson et al. / Nuclear Physics B 728 (2005) 192–206 197 where λt,b,τ are the third generation Yukawa couplings. For the first two generations we use the same expressions but without the Yukawa contributions. The soft scalar masses are given by

¯ 2 = 2 − 1 ¯ 2 = 2 − 2 + mQ mQ Lξ , mtc mtc L e ξ , 3 3 2 2 4 2 2 m¯ c = m c + L + e ξ , m¯ = m + Lξ , b b 3 L L 2 2 2 2 m¯ c = m c + eξ , m¯ = m ∓ (e + L)ξ , (10) τ τ H1,2 H1,2 where 2 = 1 2 d = 1 2 ∂ mQ m0µ γQ m0βi γQ (11) 2 dµ 2 ∂λi (where λi includes all gauge and Yukawa couplings) and so on, and we have written the effective FI parameter as 2 m ξ =− φ . (12) qφ The 3rd generation A-parameters are given by

=− + c + At m0(γQ γt γH2 ),

=− + c + Ab m0(γQ γb γH1 ),

=− + c + Aτ m0(γL γτ γH1 ) (13) and we set the corresponding ﬁrst and second generation quantities to zero. The gaugino masses are given by

βgi Mi = m0 . (14) gi The scale of the FI contributions is set by the AMSB contribution to the φ-mass, and hence is naturally expected to be the same order as the other AMSB contributions. In the examples considered below this is indeed the case. Clearly these FI contributions depend on two parameters, Lξ and eξ. For notational simplicity we will set ξ = 1 (TeV)2 from now on. We begin by choosing input values for m0,tanβ, L, e and signµ, and then we calculate the appropriate dimensionless coupling input values at the scale MZ by an iterative procedure involving the sparticle spectrum, and the loop corrections to α1...3, mt , mb and mτ , as described in Ref. [27]. We then determine a given sparticle pole mass by running the dimensionless couplings up to a certain scale chosen (by iteration) to be equal to the pole mass itself, and then using Eqs. (11), (13), (14) and including full one-loop corrections from Ref. [27], and two-loop corrections to the top quark mass [26].AsinRef.[31], we have compared the effect of using one, two and three-loop anomalous dimensions and β-functions in the calculations. Note that when doing the three-loop calculation, we use in Eq. (11), for example, the three loop approximation for both βi and γQ, thus including some higher order effects. We will present results for µ>0 and m0 = 40 TeV, for which value the gluino mass is around 900 GeV. The allowed region in (e, L) space corresponding to an acceptable vacuum is shown in Fig. 1.

To deﬁne the allowed region, we have imposed mτ˜ > 82 GeV, mν˜τ > 49 GeV and mA > 90 GeV. 198 R. Hodgson et al. / Nuclear Physics B 728 (2005) 192–206

Fig. 1. The region of (e, L) space corresponding to an acceptable electroweak vacuum, for m0 = 40 TeV and tan β = 10.

The region is to a good approximation triangular, with one side of the triangle corresponding to mA becoming too light (and quickly imaginary just beyond the boundary, with breakdown of the electroweak vacuum) and the other two sides to one of the sleptons (usually a stau) becoming too light. (Note that Ref. [20] sets e = 1 rather than ξ = 1, which is why the allowed region in their Fig. 1 has a different shape; the ﬁgures are in fact (roughly) equivalent.) Certain features of the spectrum are apparent from Eq. (10). Since to avoid tachyonic sleptons we must choose L,e > 0 we can see that the heaviest squarks (especially at low tan β) are likely to be the mainly right handed sdown and sbottom. As an example of an acceptable spectrum, we give the results for m0 = 40 TeV, tan β = 10, L = 1/25, e = 1/10, sign µ =+as derived using the one, two and three loop approximations for the anomalous dimensions and β-functions. In Table 2 we have used mt = 178 GeV, while in Table 3 we have used [28] mt = 172.7 GeV. The spectrum is not very much affected by this choice, the most noticeable alteration being in the mass of the light top squark. The rest of the results we present will be for mt = 178 GeV. This point in (e, L) space is near the boundary of the allowed region (see Fig. 1) and is characterised by a light stau. The notation in Table 2,etc., ˜ ˜ is fairly standard, see for example Ref. [27]; note in particular that in all our examples t1, b1, τ˜1 refer to the mainly left-handed particles. The results exhibit the same feature as found for the Snowmass Benchmark (SPS) points in Ref. [31]; that is, the effect of using 3-loop β-functions has a surprisingly large effect on the squark spectrum. This effect was most marked in the SPS cases when the gluino mass was signiﬁcantly larger than the squark masses, which is not the case here; nevertheless, for the light top squark, for example, it is still noticeable. A characteristic feature of AMSB distinguishing it from MSUGRA is that M2

Table 2 Mass spectrum for mt = 178 GeV, m0 = 40 TeV, tan β = 10, L = 1/25, e = 1/10 Mass (GeV) 1 loop 2 loops 3 loops g˜ 914 891 888 ˜ t1 770 761 751 ˜ t2 543 540 529 u˜L 834 820 809 u˜R 767 756 744 ˜ b1 738 728 718 ˜ b2 928 920 910 ˜ dL 838 824 813 ˜ dR 937 929 919 τ˜1 124 109 110 τ˜2 284 284 284 e˜L 132 118 119 e˜R 285 285 285 ν˜e 104 86 86 ν˜τ 99 79 80 χ1 105 129 129 χ2 354 362 361 χ3 540 563 555 χ4 552 575 566 ± χ1 106 129 129 ± χ2 549 572 563 h 117 117 117 H 315 351 336 A 314 351 336 ± H 324 360 345 ± − χ1 χ1 (MeV) 230 240 240

LSP) is predominantly the neutral wino and the lighter chargino (often the NLSP) is almost degenerate with it. In Table 2 we have given all results for masses to the nearest GeV; however we have calculated the χ±, χ masses using the full one-loop results and expect our results for ± − χ1 χ1 to be good to 10 MeV as quoted. A clear account of the dominant contribution to wino mass splitting and the associated phenomenology appears in Ref. [4]. Our splitting of around 240 MeV is consistent with their results. Note that in Table 2 the τ -sneutrino is the LSP; it is interesting that the argument of Ref. [29], which excluded the possibility of a sneutrino LSP in the MSUGRA scenario, does not apply here. The claim was [29] that with MSUGRA boundary conditions, a sneutrino LSP would necessarily have a mass less than half the Z-mass and so contribute to the invisible Z width. Evidently that is not the case here. For a recent discussion of sneutrinos as dark matter see [30]. An interesting feature of the results is that for low tan β we ﬁnd that the light CP-even Higgs mass, mh, is less than the experimental (standard model) lower bound of 114 GeV. Although the generally quoted supersymmetric bound is signiﬁcantly lower, we must take seriously the SM 200 R. Hodgson et al. / Nuclear Physics B 728 (2005) 192–206

Table 3 Mass spectrum for mt = 172.7GeV,m0 = 40 TeV, tan β = 10, L = 1/25, e = 1/10 Mass (GeV) 1 loop 2 loops 3 loops g˜ 914 890 888 ˜ t1 767 758 748 ˜ t2 519 516 505 u˜L 835 820 809 u˜R 767 756 744 ˜ b1 732 723 713 ˜ b2 928 920 910 ˜ dL 838 824 813 ˜ dR 937 929 919 τ˜1 123 108 109 τ˜2 284 284 284 e˜L 132 118 119 e˜R 285 285 285 ν˜e 104 85 86 ν˜τ 99 79 80 χ1 106 129 130 χ2 355 362 362 χ3 563 584 576 χ4 574 595 587 ± χ1 106 130 130 ± χ2 571 592 584 h 116 115 115 H 354 385 372 A 353 384 372 ± H 362 393 381 ± − χ1 χ1 (MeV) 220 230 230 bound here, since we ﬁnd generally that for us sin(β − α) ∼ 1, so that h couples to the Z-boson like the SM Higgs. However as tan β is increased, mh increases above this bound. The allowed range of tan β depends on the choice of L,e.InFigs. 2, 3 we plot mh and the CP-odd Higgs mass mA against tan β for L = 1/25, e = 1/10. The electroweak vacuum fails for tan β>25 in this case. We also plot the lighter stau mass (Fig. 4) and the tau sneutrino mass (Fig. 5)against tan β. We see that acceptable values of mh are obtained for 7 < tan β<25, and of the stau mass for tan β<19. As can be seen from Table 2, mh is actually essentially unchanged by whether we use two or three-loop β-functions; in fact we have used the two-loop β-functions to generate Figs. 2, 3. In Table 4, we give the results for another point in (e, L)-space, chosen to be in the centre of the allowed region, where this time the lightest neutralino is the LSP. Finally in Table 5 we give results for (e, L) = (0.05, 0.05), a point again near the boundary in (e, L) space, with light sleptons and heavy squarks, and also a large charged Higgs mass of over 400 GeV. This point is interesting because of the fact that previous authors have noted that the R. Hodgson et al. / Nuclear Physics B 728 (2005) 192–206 201

Table 4 Mass spectrum for m0 = 40 TeV, tan β = 10, L = 0.08, e = 0.07 Mass (GeV) 1 loop 2 loops 3 loops g˜ 914 890 888 ˜ t1 762 753 743 ˜ t2 554 541 530 u˜L 826 812 801 u˜R 769 758 746 ˜ b1 728 719 709 ˜ b2 940 932 923 ˜ dL 830 816 805 ˜ dR 949 941 932 τ˜1 212 208 208 τ˜2 250 247 247 e˜L 228 228 228 e˜R 241 234 235 ν˜e 227 220 220 ν˜τ 225 218 218 χ1 106 130 130 χ2 353 361 361 χ3 530 554 545 χ4 543 566 557 ± χ1 106 130 130 ± χ2 539 562 553 h 117 117 117 H 277 319 303 A 277 319 302 ± H 288 329 313 ± − χ1 χ1 (MeV) 240 250 250

fact that M3 and M2 have opposite signs disfavours at ﬁrst sight a supersymmetric explanation of the well-known discrepancy between theory and experiment for the anomalous magnetic moment SUSY of the muon, aµ. This is because if sign (µM2) is chosen so as to create a positive aµ then sign (µM3) leads to constructive interference between various supersymmetric contributions to B(b → sγ), and consequent restrictions on the allowed parameter space. However, with light sleptons (to generate a contribution to aµ) and heavy squarks and charged Higgs (to suppress contributions to B(b → sγ)) this conclusion can be evaded (as was already argued in Ref. [32]). There has been a considerable amount of work in recent years on two loop corrections to mh [33–36]. For some regions of the MSSM parameter space these can be substantial; therefore since we have presented predictions of around 115–118 GeV we have to worry about them since they generally reduce mh. Using the useful web resource from Ref. [33], we obtain, for the input parameters of Table 5,theresultmh = 116.2 ± 1.4 GeV, in excellent agreement with our results, which suggests that the two-loop corrections are in fact not very large in our scenario. Other points in (e, L) space give similar results. Thus for m0 = 40 TeV we predict that mh is less than 202 R. Hodgson et al. / Nuclear Physics B 728 (2005) 192–206

Table 5 Mass spectrum for m0 = 40 TeV, tan β = 10, L = 0.05, e = 0.05 Mass (GeV) 1 loop 2 loops 3 loops g˜ 914 890 888 ˜ t1 772 763 753 ˜ t2 579 576 566 u˜L 832 818 807 u˜R 795 785 773 ˜ b1 737 727 718 ˜ b2 909 900 890 ˜ dL 836 822 811 ˜ dR 917 909 899 τ˜1 140 130 131 τ˜2 194 191 191 e˜L 168 158 158 e˜R 178 177 177 ν˜e 148 137 137 ν˜τ 144 133 133 χ1 106 130 130 χ2 355 362 362 χ3 577 599 590 χ4 587 609 601 ± χ1 106 130 130 ± χ2 585 606 598 h 117 117 117 H 429 455 444 A 428 454 443 ± H 436 462 451 ± − χ1 χ1 (MeV) 220 230 230

about 118.4 GeV (see Fig. 2). If we increase m0 then this bound does increase somewhat (to around 125 GeV at m0 = 100 TeV, for example) but at the price of considerable electroweak ﬁne-tuning.

7. Mass sum rules

By taking appropriate linear combinations of masses it is straightforward to derive a set of interesting sum rules [17]. In the following equations, if we substitute the tree values for the various masses on the left- hand side, the (e, L) dependent terms and the electroweak breaking contributions to the masses cancel. We have calculated the numerical coefﬁcients on the right-hand side from Table 2,using the two-loop sparticle mass predictions; it is easy to then check that to the indicated accuracy the same equations hold for the results in Tables 4, 5. Thus these sum rules are to an excellent approximation independent of (e, L), and also in fact of m0; the numerical coefﬁcients are slowly R. Hodgson et al. / Nuclear Physics B 728 (2005) 192–206 203

Fig. 2. The light CP-even Higgs mass mh as a function of tan β,forL = 1/25,e= 1/10. The dotted line is the SM lower limit (114 GeV).

Fig. 3. The CP-odd Higgs mass mA as a function of tan β,forL = 1/25,e= 1/10. 204 R. Hodgson et al. / Nuclear Physics B 728 (2005) 192–206

Fig. 4. The light stau mass as a function of tan β,forL = 1/25, e = 1/10. The dotted line is the lower limit (82 GeV).

Fig. 5. The τ -sneutrino mass as a function of tan β,forL = 1/25, e = 1/10. R. Hodgson et al. / Nuclear Physics B 728 (2005) 192–206 205 varying functions of tan β and the input top pole mass. 2 + 2 + 2 + 2 − 2 = 2 m˜ m˜ m ˜ m ˜ 2mt 2.76(mg˜ ) , t1 t2 b1 b2 2 2 2 2 2 2 m ˜ + m ˜ + m˜ + m˜ − 2m = 1.14(mg˜ ) . (15) τ1 τ2 t1 t2 t

2 2 2 2 m˜ + 2m ˜ + m ˜ = 2.60(mg˜ ) , eL uL dL 2 2 2 2 2 m ˜ + m ˜ + m ˜ + m ˜ = 3.51(mg˜ ) , uR dR uL dL 2 2 2 2 2 m ˜ + m ˜ − m ˜ − m˜ = 0.88(mg˜ ) . (16) uL dL uR eR 2 2 2 2 m − 2sec2β m˜ + m˜ = 0.40(mg˜ ) , A eL eR 2 2 2 2 2 m − 2sec2β m + m − 2m = 0.39(m ˜ ) . (17) A τ˜1 τ˜2 τ g The existence of these sum rules will be a useful distinguishing feature of the AMSB scenario.

8. Conclusions

Despite remarkable advances in the understanding of string theory, a coherent high energy theory spawning the MSSM as an effective low energy theory remains elusive. This has led to exploration of such outré possibilities as little Higgs models and split supersymmetry. Remaining within the conservative world of low energy supersymmetry, the AMSB scenario is an attractive alternative to (and easily distinguished from) MSUGRA. We have shown how a U1 gauge symmetry broken at high energies can lead in a natural way to the FI-solution to the tachyonic slepton problem in the context of anomaly mediation. The result is a sparticle spectrum described by the parameter set m0,e,L,tan β,sign(µ); and it is only for a comparatively restricted set of (e, L) that an acceptable spectrum is obtained. Moreover we have presented a set of sum rules which are independent of m0, L and e. At the very least, the scenario we describe has the merit of being immediately testable should sparticles be discovered in experiments at the LHC.

Acknowledgements

D.R.T.J. was supported by a PPARC Senior Fellowship, and a CERN Research Associateship, and both he and G.G.R. were visiting CERN while part of this work was done. D.R.T.J. thanks Howie Haber for correspondence, Sabine Kraml for conversations and Sven Heinemeyer for a patient introduction to the simplicities of FeynHiggs. This work was partially supported by the EC 6th Framework Programme MRTN-CT-2004-503369.

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On open membranes, cosmic strings and moduli stabilization

Evgeny I. Buchbinder

School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA Received 9 August 2005; accepted 6 September 2005 Available online 15 September 2005

Abstract We discuss how cosmic strings can be created in heterotic M-theory compactifications with stable moduli. We conclude that the only appropriate candidates seem to be fundamental open membranes with a small length. In four dimensions they will appear as strings with a small tension. We make an observation that, in the presence of the vector bundle moduli, it might be possible to stabilize a five-brane very close to the visible sector so that a macroscopic open membrane connecting this five-brane and the visible brane will have a sufficiently small length. We also discuss how to embed such cosmic strings in heterotic models with stable moduli and whether they can be created after inflation.  2005 Elsevier B.V. All rights reserved.

1. Introduction

String theory seems to have appropriate candidates for cosmic strings. The possibility of view- ing superstrings as cosmic strings was ﬁrst discussed by Witten in [1]. He pointed out some serious problems in this direction. For example, one of the common problems is that the tension of fundamental superstrings is much bigger than the observational bound on the cosmic string tension. Another very common problem is that cosmic superstrings appear as boundaries of axion domain walls. This will force the strings to collapse after the axion gets a mass [2]. Cosmic strings in the context of string theory were recently reconsidered in [3–6]. It was argued that the previous problems might be avoided thanks to recent developments in string theory, including D-, NS- and M-branes, compactiﬁcations with large warp factors and models with large extra dimensions. In particular, small tension of cosmic superstrings can be explained by a large

E-mail address: [email protected] (E.I. Buchbinder).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.09.002 208 E.I. Buchbinder / Nuclear Physics B 728 (2005) 207–219 warp factor. A systematic analysis of fundamental and Dirichlet cosmic strings was performed by Copeland et al. in [6]. Results obtained in [6] suggest that in type IIB compactifications (at least in models with many Klebanov–Strassler throats [7]) it is possible to have brane/anti-brane inflation [8] followed by production of metastable F- and D-strings and de Sitter (dS) vacua [9]. Furthermore, it was proposed in [10–12] that cosmic D-strings can be identified with solitons in supergravity with Fayet–Iliopoulos (FI) terms. A holographic dual of D-strings in the throat was studied in [13]. The most phenomenologically attractive four-dimensional vacua arise in string/M-theory from compactifications of strongly coupled heterotic string [14–16]. Recently, it was demonstrated [17,18] that heterotic string theory on smooth non-simply connected Calabi–Yau manifold with Z3 × Z3 homotopy group can give rise to a four-dimensional theory whose spectrum differs form the spectrum of the Standard Model only by one extra Higgs multiplet. A progress has also been made in heterotic models towards moduli stabilization [19–23] and inflation [24,25]. In this paper, we ask a question whether it is possible to find cosmic strings in heterotic compactifications with stable moduli. In Section 2, we discuss several ways how cosmic strings could be created and come to conclusion that the only candidates seem to be fundamental open membranes with a small length. In four dimensions, an open membrane appears as a string. Such membranes could arise if one of five-branes is stabilized close to one of the orbifold fixed planes. In Section 3, we show that it might be possible to stabilize a five-brane close to the visible brane due to its non-perturbative interaction with the vector bundle moduli. In Section 4, we discuss stabilization of the remaining moduli. We slightly generalize the earlier results [20,23] and point out that it should be possible to stabilize the interval in the phenomenological range in the presence of many five-branes in the bulk without any extra assumptions. Cosmic strings are unstable in the presence of axion domain walls. In [6], it was shown that axion domain walls will not be formed if the axion is charged under a U(1) gauge group. In heterotic string theory this happens if in four dimensions there is an anomalous U(1) whose anomaly is canceled by the Green– Schwarz mechanism [26]. However, to keep this argument valid, it is important that the axion does not receive any potential. Since in heterotic M-theory the axion is the imaginary part of the volume multiplet [27,28], no superpotential for the volume should be present. This makes it difficult to stabilize the volume. Fortunately, the presence of the anomalous U(1) implies the presence of volume-dependent FI-terms. We show that together with the F-terms, the FI-terms naturally provide stabilization of all moduli in a non-supersymmetric anti-de Sitter (AdS) vacuum. In Section 4.2, we speculate how cosmic strings can be embedded into compactifications with dS vacua. We show that it might be possible in non-supersymmetric compactifications. In Section 5, we discuss how the cosmic strings, studied in this paper, can be created after inflation.

2. Cosmic strings in heterotic M-theory

In this section, we will point out that it is harder to find string-like objects in Calabi–Yau compactifications of heterotic M-theory comparing to type II compactifications. Let us start with discussing whether cosmic strings can arise after inflation considered in [24]. The inflationary phase in [24] is represented by a five-brane, wrapped on an non-isolated genus zero or a higher genus holomorphic curve in a Calabi–Yau threefold, approaching the visible brane.1

1 Throughout the paper, we will refer to one of the orbifold ﬁxed planes as to the visible brane, or the visible sector, and to the other one as to the hidden brane, or the hidden sector. E.I. Buchbinder / Nuclear Physics B 728 (2005) 207–219 209

It was argued in [24] that as the five-brane comes too close the transition vector bundle moduli [29,30] associated with this holomorphic curve become tachyonic terminating inflation. In the post-inflationary phase, the five-brane disappears (becomes massive) through the small instanton transition [31–33], whereas the transition vector bundle moduli can be stabilized by non-perturbative effects. Thus, the post-inflationary physics does not contain any rolling moduli. All the remaining moduli are stabilized both during and after inflation. The cosmological constant, generically, changes after the small instanton transition. This leads to the possibility of obtaining dS vacua with a small cosmological constant in the post-inflationary phase. A five-brane wrapped on a Riemann surface of genus one or higher carries one or more U(1) gauge fields on its world-volume [34]. For simplicity, consider a five-brane wrapping a torus so that there is only one U(1) factor. During the small instanton transition, this U(1) gets broken. Therefore, it seems conceivable that cosmic strings can be created by the Kibble mechanism [35]. However, it is not the case. First, problem is that the small instanton transition may not be de- scribable by field theory, whereas the Kibble argument is based on the Lagrangian description. Second, even if there is a field theory description of the small instanton transition (one might expect it if a Calabi–Yau threefold is not simply connected), one should recall that the theory we 1 deal with is the S /Z2 orbifold. Therefore, when a five-brane coincides with one of the orbifold fixed plains, it also coincides with its image. In this case, one should expect that the gauge group which gets restored is SU(2). As a result, after the small instanton transition the group which is broken is SU(2) and no strings are formed. This set-up has one more candidate for cosmic string in the vector-bundle branch. When a five-brane coincides with an orbifold fixed plane, a tensionless string arises [36]. Can it get a tension in the vector bundle branch, with its tension being set by the vector-bundle moduli? The answer is again negative. This tensionless string should be understood as the analog of a monopole becoming light. This analogy suggests that this string will get a tension in the five-brane branch. In the vector bundle branch, it will not exist for the same reason why monopoles do not exist in the Higgs branch. This discussion demonstrates that no cosmic strings are formed after inflation presented in [24]. Below, in the paper, we will modify the set-up of [24] to allow cosmic strings to be formed after inflation. Let us now discuss whether it is possible to create cosmic strings after a hypothetical (not yet constructed) five-brane/anti-five-brane inflation. After five-brane/anti-five-brane annihilation, a membrane bound state can be formed [37]. This membrane will be parallel to the orbifold fixed planes. If our Calabi–Yau threefold is non-simply connected, one can wrap such a membrane on a one-cycle with non-trivial π1 and form a string. However, this string will be stable in M-theory but not in heterotic M-theory. A membrane parallel to the orbifold fixed planes is not supersymmetric [38,39]. Therefore, one should expect that it will quickly annihilate with its image. Our discussion shows that it is hard to find cosmic strings in heterotic M-theory. However, we have not yet considered the most natural candidate, a fundamental open membrane. From the four-dimensional viewpoint such a membrane looks like a string. The tension of such a string behaves as ∼ 3 µ M11y, (2.1) where M11 is the eleven-dimensional Planck scale and y is the length of the membrane in the eleventh dimension. It is easy to estimate that in order to fulfill the observational bound Gµ < 10−7–10−6 (see, for example, [40] and references therein), the ratio of y to the size of the interval πρ should be of order y − − ∼ 10 4–10 3. (2.2) πρ 210 E.I. Buchbinder / Nuclear Physics B 728 (2005) 207–219

To obtain (2.2), we set the volume scale vCY and the interval scale πρ to be in the phenom- 1/6 ∼ 16 −1 ∼ 15 −1 enological range, vCY (10 GeV) and πρ (10 GeV) and used the fact that the eleven-dimensional and four-dimensional Planck scales, M11 and MPl, are related as follows 2 = 9 MPl M11vCYπρ. (2.3) Production of strings with such a tension implies some constraints on the five-brane stabilization. One of the five-branes in the bulk has to be stabilized very close to either one of the orbifold fixed planes or another five-brane. In the next section, we will argue that it is possible to stabilize a five-brane close to the visible brane. The key role in this process is played by the vector bundle moduli and their interaction. In later sections, we will extend results of Section 3 to include the remaining moduli and the inflaton.

3. Five-brane stabilization and open membranes with a small length

We consider a Calabi–Yau compactiﬁcation of heterotic M-theory with the following complex moduli

I S, T ,Y,Zα; Y0,φi. (3.1) Here S = V +iσ, where V is the volume modulus and a is the axion, T I ’s are the Kähler moduli. They are constructed as follows [27,28]

T I = RbI + ipI ,I= 1,...,h1,1, (3.2) where R is the interval modulus, bI are the (1, 1) moduli of the Calabi–Yau threefold and pI ’s come from the components of the M-theory three-form C along the interval and the Calabi– Yau manifold. The moduli Y and Y0 correspond to five-branes wrapped on an isolated genus zero holomorphic curve in the Calabi–Yau threefold. The real part of each of the two five-brane multiplets is the position of the corresponding five-brane in the interval. We denote them by y and y0, respectively. The imaginary part is related to the axions on the five-brane worldvolume. See [41] for details. One five-brane is needed to stabilize the interval in a phenomenological range as in [20,23]. This five-brane has to be located relatively close to the hidden brane. The other five-brane will be the one that we will attempt to stabilize in this section close to visible brane so that Eq. (2.2) is satisfied. Zα are the complex structure moduli, whose actual number is irrelevant. We will assume that the (3, 0, 1) flux G is turned on. Such a flux produces the superpotential for Zα of the form [42,43] 2 MPl 11 Wf = dx G ∧ Ω. (3.3) vCYπρ CY This superpotential is expected, generically, to stabilize all the complex structure moduli. We will assume that it is the case. We will also assume that the complex structure moduli are stabilized at slightly higher scale that all the other moduli so that Wf can be considered as constant in the low-energy field theory. At last, φi are the moduli of the vector bundle on the visible brane. For simplicity, we will assume that the bundle on the hidden brane either is trivial or does not have any moduli. The moduli V and R are assumed to be dimensionless and normalized with 16 −1 15 −1 respect to the reference scales vCY ∼ (10 GeV) and R ∼ (10 GeV) . The moduli y and y0 are also dimensionless and less that R. To obtain the four-dimensional coupling constants in E.I. Buchbinder / Nuclear Physics B 728 (2005) 207–219 211 a phenomenological range, V and R have to be stabilized at (or be slowly rolling near) a value of order one. In this section, we will concentrate on the moduli Y0 and φ assuming that all the other moduli are stable. They will be considered in the next section. The Kähler potential of the system (in the four-dimensional Planck units) is given by ¯ 2 K(Y0,φ)= K0 + K1(Y0 + Y0) + kK(φ). (3.4)

Here K0 and K1 are functions of the remaining moduli independent of Y0 whose precise form is irrelevant for us. K(φ) is the Kähler potential for the vector bundle moduli φi and k is a dimensionless parameter of order 10−5 [20]. It is much less than one because the gauge sector in Horava–Witten theory [14,15] arises at the sub-leading order in the gravitational coupling constant comparing to the gravity sector. The superpotential (in the four-dimensional Planck units) is given by

−τY0 W = W0 + α Pfaff(D−)e . (3.5) Here the second term is the non-perturbative superpotential due to a membrane instanton [38,39, 44–48], with α being a dimensionless parameter, and W0 is the superpotential depending on the remaining moduli. For our purposes it is a constant. The parameter τ is given by 1 π 1/3 τ = (πρ)v , (3.6) 2 2κ11 where v is the area of the holomorphic curve on which the membrane is wrapped. Generically, it is of order 100. Pfaff(D−) is the Pfaffian of the Dirac operator depending on the vector bundle moduli. It is very difficult to compute it explicitly because no gauge connections on Calabi–Yau threefolds are known. However, in [47,48], this factor was computed in a number of examples for the case the Calabi–Yau threefold elliptically fibered over the Hirzebruch surface using algebraic geometry techniques. It was found to be a high degree homogeneous polynomial in the vector bundle moduli. We will denote it as P(φ). If the five-brane is close to the visible sector, all other φi and Y0 contributions to the non-perturbative superpotential are negligibly small. If we ignore all other moduli (they will be discussed later), the minimum of the potential energy for the Y0,φi system is given by = = Dφi W 0,DY0 W 0, (3.7) = + W where DW is the Kähler covariant derivative, DW ∂W 2 ∂K.FromEqs.(3.4) and (3.5) MPl we have

∂P(φ) − ∂K(φ) = τY0 + = Dφi W e k W 0 (3.8) ∂φi ∂φi and − =− P τY0 + = DY0 W τ (φ)e 4K1y0W 0. (3.9) It is easy to realize that since τ is much bigger than one and k is much less than one, in the second termsofEqs.(3.8) and (3.9), we can replace W with W0. If the degree of the polynomial P(φ) is sufficiently high, the first terms is Eqs. (3.8) and (3.9) are practically the same. The second term in Eq. (3.8) is proportional to k whereas the second term in Eq. (3.9) is proportional to y0. These two equations are consistent only if y0 ∝ k, that is y0 is much less than one. Whether or not the 212 E.I. Buchbinder / Nuclear Physics B 728 (2005) 207–219 solution for y0 is consistent with (2.2) depends on details of the compactification, in particular, on the Kähler potential K(φ) which is unknown how to evaluate explicitly on a Calabi–Yau threefold. Nevertheless, the form of Eqs. (3.8) and (3.9) suggests that in some models it might be possible to find a solution for y0 consistent with (2.2). To get a better understanding how a solution for y0 might look like, let us make a reasonable model for the Kähler potential K(φ). Let us choose, for simplicity, K(φ) to be flat 2 K(φ) = |φi| . (3.10) i=1 It corresponds to the Kähler potential for the size and orientation moduli of centered Yang–Mills 4 instantons on R if the number of moduli φi is even [49–51]. The Pfaffian will be assumed to be a generic homogeneous polynomial of degree d. It is obvious that for a generic Pfaffian it is possible to find a solution for the phases of φ’s and the imaginary part of Y0.Letri be the absolute value of φi . Consider Eq. (3.8) for ri . Since we are interested in solutions with τy0 1, − the exponential factor e τy0 in Eqs. (3.8), (3.9) can be set to unity. Then using Eqs. (3.10), (3.8) for ri can be written as follows k|W | F (r ) = 0 , (3.11) i j αd where each Fi is a homogeneous function of rj of degree d − 2. The factor d in the denominator in the left-hand side comes form differentiating the Pfaffian.2 From Eq. (3.11) it follows that every ri is of the form − k|W | 1/(d 2) r ∝ 0 , (3.12) i αd where the proportionality coefficient is a number depending on details of the Pfaffian. Then from Eq. (3.9) it follows that d/(d−2) ατ k|W0| y0 ∼ . (3.13) 4K1|W0| αd If d 1, we have τk y0 ∼ . (3.14) 4K1d −5 Taking, for example, k ∼ 10 , τ ∼ 100, d ∼ 10, 4K1 ∼ 1, we find that −4 y0 ∼ 10 (3.15) which is consistent with condition (2.2). Note that y0 is inverse proportional to d. A large value of d, which is consistent with results of [47,48], might also be a reason for a small value of y0. In this section, we demonstrated that it is possible to stabilize a five-brane, wrapped on an isolated genus zero holomorphic curve, close to the visible brane. From the four-dimensional viewpoint, a macroscopic open membrane stretched between this five-brane and the visible brane will look like a cosmic string with a relatively small tension which might be consistent with the observational bound.

2 d Without loss of generality, we can assume that the Pfafﬁan contains terms rj . E.I. Buchbinder / Nuclear Physics B 728 (2005) 207–219 213

4. Stabilization of the remaining moduli

4.1. AdS vacua

In this section, we will discuss how cosmic strings from Section 3 can be embedded into compactifications with stable moduli. For this we need to stabilize the remaining moduli in (3.1), that is S,T I and Y . The moduli T I and Y can be stabilized by the same methods as in [20,23]. However, stabilization of S represents a problem. The imaginary part of S is the axion. It has been known that cosmic strings of the type constructed in the previous section are boundaries of axion domain walls [1]. If the axion gets a mass the strings become unstable [2]. This problem can be avoided if the axion is charged under some U(1) [6]. In heterotic M-theory such a situation occurs if there is an anomalous U(1) in four dimensions. The anomaly of such a U(1) is canceled by the four-dimensional version of the Green–Schwarz mechanism. The axion, in this case, transforms under U(1) as δσ = λ. (4.1) Now the axion becomes a pure gauge degree of freedom and can be gauged away. As a result, no axion domain walls are formed. The above arguments will become invalid if the axion re- ceives a potential. This, in particular, indicates that stability of cosmic strings requires that no superpotentials for the S modulus should be present. An anomalous U(1) whose presence, as we just explained, is important for stability of cosmic strings, gives rise to the moduli dependent FI-terms [26]. In heterotic M-theory they are of the form [23] U g2 FI ∼ . 4 2 (4.2) MPl V Here g is the gauge coupling constant which is itself moduli dependent g2 g2 = 0 , (4.3) Re(S +···) and the ellipsis stands for the T I - and Y -dependent corrections. The precise form of the corrections depends whether an anomalous U(1) is in the visible or in the hidden sector. They can be found, for example, in [34]. To simplify equations below, we will assume that they are small and can be ignored. Their presence can be shown not to lead to any conceptually new results below. The order of magnitude of UFI was estimated in [23] and was found to be, generically of order 2 2 Wf /MPl. As we said, the moduli T I and Y can be stabilized by methods presented in [20,23].Let us briefly discuss them with a slight generalization. For simplicity, we will consider the case h1,1 = 1. Generalizations for h1,1 > 1 is straightforward but technically more complicated. In this case, there is only one T -modulus whose real part is the length of the interval. If the five- brane whose modulus has been denoted by Y is located close to the hidden sector, the leading contribution to the superpotential W0 is of the form −τ(T−Y) W0 = Wf + βe . (4.4) The scale of the coefficient β will be assumed to be set by the eleven-dimensional Planck scale. The equation of motion for R, schematically, is [20,23]

− − |Wf | e τ(R y) ∼ . (4.5) βτ 214 E.I. Buchbinder / Nuclear Physics B 728 (2005) 207–219

In order this equation to have a solution, the right-hand side has to be less than one. First, Wf is 2/3 quantized in units of (κ11/4π) . Therefore, in the limit of large volume and large interval, the ratio |Wf |/β is less than one. Second, the order of magnitude of Wf might be reduced by Chern– Simons invariants [21].Third,τ is much greater than one. This guarantees that, generically, the right-hand side in Eq. (4.5) is less than one. However, to stabilize R in the phenomenological regime, R ∼ 1, the five-brane Y has to be stabilized close to the hidden sector so that R − y is sufficiently small. Whether or not Y can be stabilized close enough depends on details of the compactification. However, there is one obvious possibility of making R − y small, which has not been considered before. Let us assume that in the bulk there are N five-branes wrapped on the same holomorphic curve. Open membrane instantions will generate contributions to the superpotential inverse proportional to the exponent of the length between the five-branes. Then the equation of motion for the ith five-brane will, schematically, be as follows | | −τ(y −y − ) −τ(y + −y ) Wf −β e i i 1 + β + e i 1 i ∼ . (4.6) i i 1 τ Writing similar equations for all N five-branes and the interval, it is possible to show that the five-branes and the interval will be stabilized in such a way that the distance between any two adjacent objects will approximately be the same and, therefore, of order R/N. Taking N big enough, it is always possible to find a solution

R ∼ 1 (4.7) from Eq. (4.5). Let us point out that our analysis in Section 3 and in the present section shows that a five-brain in the bulk might have several different minima. In particular, there is a minimum close to the visible sector whose existence is due to the vector bundle moduli. On the other hand, there is a minimum at a generic point in the interval due to the non-perturbative exponential interaction with the adjacent branes (five-branes or the end-of the-world branes) on the left and on the right. Putting N five-branes will make the distance between any two adjacent branes of order R/N. However, stabilization of the volume becomes a problem. In [20,21,23], the volume was stabilized by balancing the gaugino condensation superpotential against the fluxes. As we pointed out, any volume dependent superpotential is also a potential for the axion which will destabilize cosmic strings. Thus, in this paper, we will assume that no superpotential for the S-modulus is generated. In this subsection, we will consider the simplest potential for the volume generated by the Kähler potential and by the FI-terms. Note that existence of the FI-terms is not an extra assumption. It already follows from existence of anomalous U(1) which is necessary to gauge away the axion. The Kähler potential for S is as follows [27,28,41] c K(S) =−ln(S + S)¯ + , (4.8) S + S¯ where c depends on T I - and Y -moduli which are assumed to be stable in this paper. To keep themetricintheS,T I ,Y moduli space positive definite, at least in the large volume and the large interval limit, c has to be sufficiently small so that the second term is sub-leading. Then by studying equations of motion for V , it is possible to realize that the second term in (4.8) does not play an important role in existence of extrema in the V -direction. Thus, for simplicity, we can assume that c is sufficiently small and can be ignored to the leading order. It is also possible to show that the Kähler covariant derivatives for Kähler and five-brane moduli will be shifted from zero by terms proportional to 1/τ which is much less than one. Thus, approximately, dynamics E.I. Buchbinder / Nuclear Physics B 728 (2005) 207–219 215

Fig. 1. Potential U(V) (appropriately normalized) for A = 3, B = 1. There is an AdS minimum. of V is governed by the potential3 K(S) −1 ¯ ¯ A B U(V)= e G D WD¯ W − 3WW + U ≈− + . (4.9) SS¯ S S FI V V 3 Here A and B are constants (depending on vevs. of other moduli). The scale of A is set by the fluxes and the scale of B is set by the FI-terms. Generically, both A and B are of the same order of magnitude [23]. This potential has a non-supersymmetric AdS minimum. See Fig. 1.Our discussion in this section shows that it is conceivable to find stable cosmic strings in heterotic compactifications with a stable non-supersymmetric AdS vacuum. To conclude this subsection, let us make sure that addition of new moduli does not destabilize the moduli φ and y0 from Section 3. It is easy to show that up to terms linear in y0 the equation of motion for φi is still = Dφi W 0. (4.10)

Similarly, taking into account the precise dependence of the FI-terms on y0 [23,34], one can = show that equation DY0 W 0 is modiﬁed by terms linear in y0. This means that the form of Eqs. (3.8) and (3.9) will not change and a solution y0 1 will still exist.

4.2. dS vacua?

Let us now discuss how these results can be extended to compactiﬁcations with dS vacua. The most common strategy to obtain dS vacua [9,23,52] is to use the F-terms to obtain a supersymmetric AdS vacuum and then raise with extra terms (breaking supersymmetry) in the potential energy. However, as we discussed in the previous subsection, in the presence of cosmic strings it is problematic to obtain a supersymmetric AdS vacuum because we cannot use a superpotential for the volume multiplet. As a result, we had to use FI-terms to stabilize the volume rather than to raise an AdS vacuum. One way to create dS vacua can be to stabilize the volume by higher-order corrections to the Kähler potential. This was studied recently in type IIB compact- iﬁcations in [53–55]. However, it is not known how to extend these results to M-theory. One

3 The FI terms will modify equations of motion for the interval and the ﬁve-branes. However, since the order of magnitude of FI terms is the same as that of the ﬂuxes, a solution for these moduli will still exist. 216 E.I. Buchbinder / Nuclear Physics B 728 (2005) 207–219

Fig. 2. Potential U(V)+ U(V ) (appropriately normalized) for A = 0.25, B = 1.2, C = 1. There is a dS minimum. can also worry that, generically, this approach will stabilize the volume away from a phenomenologically interesting range. Another approach could be to search for extra corrections to the potential, in addition to the F-terms and the FI-terms. Such corrections arise, for example, in non-supersymmetric compactifications, which, in general, represent an approach for obtaining dS vacua [23,56]. Below, we will show that under some conditions a dS vacuum can be found in non- ¯ supersymmetric E8 × E8 compactifications. In such compactifications, there is an extra volume- dependent contribution to the potential of the form [23] a U(V ) = . (4.11) V The coefficient a depends on the h1,1 moduli and on the interval. Generically, its order of magnitude is bigger than that of the fluxes and the FI-terms. However, it depends on the h1,1 moduli. Here, we will assume that the h1,1 moduli are stabilized in such a way that U(V ) is compara- ble with U(V). It would be interesting to understand under what circumstances it is possible to stabilize a two-cycle at a very small size. We will not discuss it in this paper. Let us consider the potential U(V)+ U(V ) including linear terms in c/V . It looks as follows A C B U(V)+ U(V ) = − + . (4.12) V V 2 V 3 Here A is a−A. We will assume that it is positive. C is linear in c. It is always positive and comes from the factor eK(S) in Eq. (4.9). It is obvious that under some conditions on the parameters A,B and C, this potential can have a metastable dS vacuum. See Fig. 2.

5. Inﬂation

In this section, we will briefly discuss whether cosmic strings studied in this paper can be formed after inflation. For concreteness, we will consider the inflationary model of [24].One should also be able to extend this discussion to another heterotic M-theory inflation studied in [25]. To the system of moduli in Eq. (3.1), we add another five-brane multiplet corresponding to a five-brane wrapped on a non-isolated genus zero or higher genus holomorphic curve. Moduli E.I. Buchbinder / Nuclear Physics B 728 (2005) 207–219 217 of such a five-brane will not have any non-perturbative superpotential. Instead, the translational modulus has a potential with inflationary properties.4 The process of inflation is represented by such a five-brane approaching the visible brane. It was argued in [24] that when the five-brane comes close to the visible sector, the transition vector bundle moduli [29,30] become tachyonic and terminate inflation. Eventually, the five-brane disappears through the small instanton transition which modifies the vector bundle on the visible brane, in particular, new vector bundle moduli are created which can be stabilized by non-perturbative effects. Thus, formation of macroscopic open membranes with a small length should be related to this change of the vector bundle. Since after the small instanton transition, the vector bundle, the number of the vector bundle moduli and the Pfaffian change, the position of the five-brane discussed in Section 3 will also change. This could explain why it is stabilized close to the visible sector only after inflation. For example, let the bundle before inflation not have any moduli. Then from discussion in Section 3 it follows that no minimum for the five-brane close to the visible sector will exist. Therefore, in this case, the five-brane will be stabilized at a generic point along the interval. After inflation, vector bundle moduli will be created and, according to our discussion in Section 3, they might force the five-brane to move closer to the visible sector so that Eq. (2.2) is satisfied. The presence of vector bundle moduli might also stabilize a five-brane close to the visible brane after inflation in the model proposed in [25]. Thus, it might be possible to create cosmic strings in the inflationary scenario in [25]. It would be interesting to understand it in detail.

Acknowledgements

The author is very grateful to Juan Maldacena for lots of helpful discussions, explanations and comments on the preliminary version of the paper. The author would also like to thank Willy Fischler for discussions. The work is supported by NSF grant PHY-0070928.

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Determination of neutrino mass texture for maximal CP violation

Ichiro Aizawa, Teruyuki Kitabayashi, Masaki Yasuè

Department of Physics, Tokai University, 1117 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan Received 25 July 2005; received in revised form 29 August 2005; accepted 14 September 2005 Available online 22 September 2005

Abstract

We show a general form of neutrino mass matrix (M), whose matrix elements are denoted by Mij (i, j = e,µ,τ) as flavor neutrino masses, that induces maximal CP violation as well as maximal atmospheric neutrino mixing. The masses of Mµµ, Mττ and Mµτ +σMee (σ =±1) turn out to be completely determined by Meµ and Meτ for given mixing angles. The appearance of CP violation is found to originate from the interference between the µ–τ symmetric part of M and its breaking part. If |Meµ|=|Meτ |, =− iθ =− iθ ∗ giving either Meµ σe Meτ or Meµ σe Meτ with a phase parameter θ, is further imposed, we find that |Mµµ|=|Mττ| is also satisfied. These two constraints can be regarded as an extended version of the constraints in the µ–τ symmetric texture given by Meµ =−σMeτ and Mµµ = Mττ. Majorana CP iθ violation becomes active if arg(Mµτ ) = arg(Meµ) + θ/2forMeµ =−σe Meτ and if arg(Mµτ ) = θ/2 =− iθ ∗ for Meµ σe Meτ .  2005 Elsevier B.V. All rights reserved.

PACS: 14.60.Lm; 14.60.Pq

1. Introduction

To understand property of neutrinos is one of the key issues in the current quark-lepton physics that demands a new leptonic sector beyond the standard model. Oscillating neutrinos, which have been first confirmed to exist in Nature by the Super-Kamiokande Collaborations in 1998 [1], require the presence of their mass terms in the leptonic sector [2]. Since their masses if they exist should be extremely small, there should be a specific mechanism that keeps neutrinos extremely

E-mail addresses: [email protected] (I. Aizawa), [email protected] (T. Kitabayashi), [email protected] (M. Yasuè).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.09.016 I. Aizawa et al. / Nuclear Physics B 728 (2005) 220–232 221 light, which is known to be provided by the seesaw mechanism [3,4] and the radiative mechanism [5,6]. Neutrino oscillations arise as a result of mixings among three flavor neutrinos of νe,µ,τ , whose masses are supplied by these mechanisms. This lepton mixing can be described by the PMNS mixing matrix [2] in as much the same way as the quark mixing is described by the CKM mixing matrix [7]. The flavor neutrinos are converted into three massive neutrinos of ν1,2,3, whose masses are to be denoted by m1,2,3, and they are related to each other as ν = Uiνi for = e,µ,τ and i = 1, 2, 3, where U represents the PMNS mixing matrix. The presence of the leptonic mixing analogous to the quark mixing has opened the possibility that CP violation also arises in the lepton sector since CP violation exists in the quark mixing. Furthermore, CP violation is an indispensable ingredient generating the excess of the baryon number of universe [8].Ifthe leptonic CP violation exists in an appropriate form, the baryon number can be generated by leptogenesis [9]. The leptonic CP violation has two sources: one from a CP violating Dirac phase and the other from two CP violating Majorana phases [10] if neutrinos are Majorana particles. It has been, then, argued that leptonic CP violating effects show up in various neutrino-induced processes [11,12]. These phases are contained in U parameterized by three mixing angles θij , one Dirac phase δ and three Majorana phase β1,2,3, where the CP violating Majorana phases are given by two combinations of β1,2,3 such as βi − β3 (i = 1, 2, 3). We adopt the parameterization given by PDG [16]: U = UνK with   −iδ c12c13 s12c13 s13e  iδ iδ  Uν = −c23s12 − s23c12s13e c23c12 − s23s12s13e s23c13 , iδ iδ s23s12 − c23c12s13e −s23c12 − c23s12s13e c23c13 K = diag eiβ1 ,eiβ2 ,eiβ3 , (1) where cij = cos θij and sij = sin θij (i, j = 1, 2, 3). The observed mass squared differences 2 2 (m, matm) and mixing angles (θ, θatm, θCHOOZ) can be, respectively, identified with 2 − 2 | 2 − 2| (m2 m1, m3 m2 ) and (θ12, θ23, θ13). These parameters are experimentally constrained to be [17,18]:

−5 2 2 −5 2 5.4 × 10 eV <m < 9.5 × 10 eV , × −3 2 2 × −3 2 1.2 10 eV <matm < 4.8 10 eV , (2) 2 2 2 0.70 < sin 2θ < 0.95, 0.92 < sin 2θatm, sin θCHOOZ < 0.05. (3) If the CP phases are included in neutrino oscillations, it is not obvious that what types of ﬂavor neutrino masses are appropriate to explain the observed properties of neutrino oscillations although several speciﬁc textures are proposed [13–15]. We have argued that the Dirac CP violating phase can be determined by [19]

−iδ s23Meµ + c23Meτ =|s23Meµ + c23Meτ |e , (4) where M is a Hermitian matrix deﬁned by M = M†M with M being a ﬂavor neutrino mass matrix parameterized by   Mee Meµ Meτ   M = Meµ Mµµ Mµτ , (5) Meτ Mµτ Mττ 222 I. Aizawa et al. / Nuclear Physics B 728 (2005) 220–232

T 1 and U MU = diag(m1,m2,m3). TherelationinEq.(4) reﬂects the general property that all phases in Mij (i, j = e,µ,τ) arise from δ because the Majorana phases in K are cancelled in † 2 2 3 + U MU giving diag(m1,m2,m2). The same quantity of s23Meµ c23Meτ is also linked to θ13 determined by

|s M + c M | tan 2θ = 2 23 eµ 23 eτ . (6) 13 2 + 2 + − s23Mµµ c23Mττ 2s23c23 Re(Mµτ ) Mee

In the limit of sin θ13 = 0, we have |s23Meµ + c23Meτ |=0, which indicates that the CP violating phase δ deﬁned in Eq. (4) is irrelevant as expected. Furthermore, the atmospheric neutrino mixing θ23 is calculated from

Im(Meµ) tan θ23 = . (7) Im(Meτ )

This relation is of great signiﬁcance because it requires the presence of imaginary parts of Meµ 2 and Meτ to consistently deﬁne θ23. Because the presence of the non-vanishing imaginary parts also demands the presence of a non-vanishing sin θ13 in Eq. (6), it is physically meaningful to argue how we obtain the magnitude 2 of sin θ13 as small as possible in order to get the smaller magnitude to favor sin θ13 < 0.05. Since |s23Meµ +c23Meτ | |s23 Im(Meµ)+c23 Im(Meτ )| in Eq. (6), the minimal magnitude of tan 2θ13 is achieved at s23 Re(Meµ) + c23 Re(Meτ ) = 0, leading to

Re(Meτ ) tan θ23 =− , (8) Re(Meµ) and s23Meµ + c23Meτ =±i|s23Meµ + c23Meτ |. The Dirac CP violation becomes maximal at Eq. (8) because e−iδ =±i is required by Eq. (4). Therefore, one can minimize the effect in sin θ13 by adjusting Meµ,eτ if CP violation is maximal. To reach the minimized magnitude of 2 3 sin θ13 for given sets of ﬂavor neutrino masses prefers the observed suppression of sin θ13. By combining Eqs. (7) and (8), we obtain that

=− ∗ + sin 2θ23Meτ Meµ cos 2θ23Meµ, (9) which is the condition to provide maximal CP violation. In this article, we would like to present a general form of ﬂavor neutrino mass matrix that ensures the appearance of maximal CP violation satisfying Eq. (9). To make our argument simpler, we only consider the case of the maximal atmospheric neutrino mixing, which is compatible with the observation. The relation between Meµ,eτ of Eq. (9) becomes simpler and is given by

=− ∗ Meτ σ Meµ, (10)

1 It is understood that the charged leptons and neutrinos are rotated, if necessary, to give diagonal charged-current interactions and to define the flavor neutrinos of νe, νµ and ντ . 2 In the case of Im(Meµ) = Im(Meτ ) = 0, see the discussion in Ref. [21]. 3 Of course, the strength of Dirac CP violation proportional to sin θ13 sin δ gets maximized for the maximally allowed 2 magnitude of sin θ13 for δ =±π/2. Our statement is purely theoretical one to have the smaller magnitude of sin θ13 to 2 approach the suppressed magnitude of sin θ13. I. Aizawa et al. / Nuclear Physics B 728 (2005) 220–232 223 √ 4 where c23 = σs23 = 1/ 2 (σ =±1) for the maximal atmospheric neutrino mixing. We have already performed the similar analyses and have constructed three specific textures by guesswork so as to satisfy Eq. (10) [20,21]. However, we have not yet clarified their common structure by solving Eq. (10) in terms of M itself, which supplies necessary and sufficient condition on M for the appearance of maximal CP violation. In the next section, Section 2, we derive conditions on flavor neutrino masses that provide maximal CP violation and estimate neutrino masses, from which we show how the mass hier- 2 2 archy of matm m is realized. In Section 3, to see characteristic property in the texture obtained in Section 2, we rely upon the µ–τ symmetry [14,22–25], where the presence of Meτ =−σMeµ and Mττ = Mµµ is required, to divide M into a µ–τ symmetric part and its breaking part. The CP violation is found to arise from the interference of these two parts, which is examined by applying Eq. (4) to the texture. To see the relation of our texture to the µ–τ symmetric texture, the condition of |Meτ |=|Meµ| is imposed [14] and its consequence is discussed. The final section is devoted to summary.

2. Neutrinos in maximal CP violation

Let us begin with presenting a set of equations that determines masses and mixing angles in terms of M [20]:

sin 2θ12(λ1 − λ2) + 2 cos 2θ12X = 0, (11) − (M − M ) sin 2θ − 2M cos 2θ = 2s e iδX, (12) ττ µµ 23 µτ 23 13 −iδ iδ sin 2θ13 Meee − λ3e + 2 cos 2θ13Y = 0, (13) and 1 + cos 2θ 1 − cos 2θ −2iβ1 12 −2iβ2 12 m1e = λ2 − X, m2e = λ2 + X, sin 2θ12 sin 2θ12 2 2 −2iδ − c λ3 − s e Mee m e 2iβ3 = 13 13 . (14) 3 2 − 2 c13 s13

The mass parameters of λ1,2,3, X and Y are given by = 2 − iδ + 2 2iδ = 2 + 2 − λ1 c13Mee 2c13s13e Y s13e λ3,λ2 c23Mµµ s23Mττ 2s23c23Mµτ , = 2 + 2 + λ3 s23Mµµ c23Mττ 2s23c23Mµτ , (15) c23Meµ − s23Meτ X = ,Y= s23Meµ + c23Meτ , (16) c13 √ iδ and c23 = σs23 = 1/ 2 and e =±i will be chosen for our calculation. There are six constraints to have a diagonalized U T MU. Three of them supply three masses given by Eq. (14) and the other three are just Eqs. (11)–(13). Since M has six complex parameters, using three equations of Eqs. (11)–(13) for given mixing angles, we can ﬁx three complex parameters, which are taken to be Mµµ,ττ and Mµτ + σMee.

4 The sign of s23 denoted by σ is retained because, for a too large magnitude of sin θ13, the smaller magnitude of sin θ13 can be obtained if to reverse the sign results in the cancellation of s23Meµ + c23Meτ ≈ 0 in the numerator of Eq. (6). 224 I. Aizawa et al. / Nuclear Physics B 728 (2005) 220–232 √ iδ For c23 = σs23 = 1/ 2 and e =±i, we can ﬁnd a solution to Eqs. (11)–(13) consisting of √ −iδ 2M = P(M − σM ) + Q+e (σ M + M ), (17) √ µµ eµ eτ eµ eτ ∗ −iδ 2Mττ = P (Meµ − σMeτ ) + Q+e (σ Meµ + Meτ ), (18) √ + ∗ P P −iδ 2σ(M + σM ) =− (M − σM ) + Q−e (σ M + M ), (19) µτ ee 2 eµ eτ eµ eτ where

1 −iδ 1 t13 P = − σt13e ,Q± = ± . (20) c13 tan 2θ12 tan 2θ13 2

Therefore, Mµµ,ττ and Mµτ ±Mee depending on the sign of σ are completely determined by two parameters Meµ,eτ . Hereafter, we choose Meµ,eτ,µτ as free parameters. The solution is indeed a solution to Eq. (10), which becomes ∗ + ∗ + + ∗ + ∗ + ∗ ≡ = Meτ Mττ Meµ(Mµτ σMee) σ MµµMeµ Mµτ σMee Meτ ( J) 0. (21) By inserting Eqs. (17)–(19) into Eq. (21), we obtain = iδ − ∗ + ∗ − −iδ ∗ − ∗ + J e σ(Meµ σMeτ ) σMeµ Meτ e σ Meµ σMeτ (σ Meµ Meτ ) + −iδ ∗ − − iδ ∗ − ∗ t√13 2e σMeτ (Meµ σMeτ ) 2e Meµ Meµ σMeτ , (22) 2 2 which turns out to identically vanish because of e−iδ =−eiδ. The mass matrix is expressed in terms of Meµ,eτ,µτ :   − √1 (P + P ∗) 1 −σ  2 √  Meµ − σMeτ M =  1 2P 0  √ 2 −σ 0 2P ∗  √  −iδ 2Q−e √ 1 σ +  −iδ  Meµ σMeτ + 1 2Q+e √ 0 − 2 σ 0 2Q+e iδ   −σ 00   + 001Mµτ , (23) 010 which correctly describes the maximal atmospheric neutrino mixing and maximal CP violation. The neutrino masses can be calculated from Eq. (14) and are found to be: − − − 2iβ1 = − 2iβ2 = + 2iβ3 =− + m1e A B, m2e A B, m3e A C, (24) 2 ∗ 2 2 2 ∗ 1 2 m = 4Re B A (> 0), m = |B| −|C| + 2Re C A + m, (25) atm 2 where 2 + c13 −iδ σMeµ Meτ A = A − σMµτ ,A= √ e , 2 sin 2θ13 M − σM 2 B = √ eµ eτ ,C= B θ + A . cos 2 12 2 (26) 2c13 sin 2θ12 c13 I. Aizawa et al. / Nuclear Physics B 728 (2005) 220–232 225

The three parameters A, B and C, respectively, represent freedoms associated with Mµτ , − + 2 2 Meµ σMeτ and Meµ σMeτ . The observed mass hierarchy of matm m can be roughly realized by the following requirement:

−2iβ −2iβ −2iβ (M1) A ≈ B with |C||B| leading to m1e 1 ≈ 0, m2e 2 ≈ 2B and m3e 3 ≈ C with 2 ≈ | |2 2 ≈| |2 m 4 B and matm C known as the normal mass hierarchy, −2iβ −2iβ −2iβ 2 (M2) B ≈ 0 and A ≈ C leading to m1e 1 ≈ m2e 2 ≈ A and m3e 3 ≈ 0 with m ≈ ∗ 2 ≈| |2 4Re(B A) and matm C known as the inverted mass hierarchy, −2iβ −2iβ −2iβ 2 (M3) A ≈ C ≈ 0 leading to m1e 1 ≈−m2e 2 ≈−B and m3e 3 ≈ 0 with m ≈ ∗ 2 ≈| |2 4Re(B A) and matm B known as the inverted mass hierarchy, −2iβ −2iβ (M4) |B|∼|C||A| with C = kB for a real k (k = O(1)) leading to −m1e 1 ≈ m2e 2 ≈ − ∗ 2iβ3 ≈ 2 ≈ 2 ≈| − 2 | m3e B with m 4Re(B A) and matm (1 k )B known as the case of degenerate neutrinos, −2iβ −2iβ −2iβ 2 (M5) |A||C||B| leading to m1e 1 ≈ m2e 2 ≈−m3e 3 ≈ A with m ≈ ∗ 2 ≈ | ∗ | 4Re(B A) and matm 2 Re(C A) known as the case of degenerate neutrinos.

−2iβ Some of the cases differ with each other in the relative sign of mae a (a = 1, 2, 3). If no cancellation occurs between Meµ and Meτ in C, |C||B| is satisﬁed because of the smallness of | sin θ13|.

3. Usefulness of the µ–τ symmetry

Although we have succeeded in obtaining general flavor neutrino masses yielding maximal CP violation, it is not obvious that what kind of characteristic property is present in our texture. To see this, we use the µ–τ symmetry in neutrino mixings, which is defined by the invariance of the flavor neutrino mass terms in the Lagrangian under the interchange of νµ ↔ ντ or νµ ↔−ντ .As a result, we obtain Meτ = Meµ and Mµµ = Mττ for νµ ↔ ντ or Meτ =−Meµ and Mµµ = Mττ for νµ ↔−ντ . We use the sign factor σ heretohaveMeτ =−σMeµ for the µ–τ symmetric part under the interchange of νµ ↔−σντ . For any textures that do not respect the µ–τ symmetry, (±) (±) we can define Meµ = (Meµ ± (−σMeτ ))/2 and Mµµ = (Mµµ ± Mττ)/2 and divide M into a µ–τ symmetric part (Msym) and its breaking part (Mbreaking) [21]:   + + M M( ) −σM( )  ee eµ eµ  = (+) (+) Msym  Meµ Mµµ Mµτ  , + + −σM( ) M M( )  eµ µτ µµ  − − 0 M( ) σM( )  eµ eµ  = (−) (−) Mbreaking  Meµ Mµµ 0  , (27) (−) (−) σMeµ 0 −Mµµ (+) (−) (+) (−) (+) where obvious relations of Meµ = Meµ + Meµ , Meτ =−σ(Meµ − Meµ ), Mµµ = Mµµ + (−) (+) (−) 5 Mµµ and Mττ = Mµµ − Mµµ are used. The µ–τ symmetric part is known to have θ13 = 0.

√ 5 From M , it is easy to see that the eigenvector corresponding to |ν is given by (0,σ,1)/ 2 with the eigenvalue sym √ 3 (+) Mµµ + σMµτ , leading to s13 = 0andc23 = σs23 = 1/ 2inUν of Eq. (1). This result of θ23 is consistent with our assignment of θ23. 226 I. Aizawa et al. / Nuclear Physics B 728 (2005) 220–232

In fact, tan 2θ13 is proportional to Y as in Eq. (13), which vanishes for Msym if Eq. (13) is appropriately applied to Msym, and θ13 is equal to 0. The vanishing θ13 in turn yields cos 2θ23 = 0 from Eq. (12) applied to Msym, indicating the presence of the maximal atmospheric neutrino mixing. To see what this separation physically means, we compute Eq. (4), i.e., σ Meµ + Meτ = −iδ |σ Meµ + Meτ |e , for each parts and observe that σ Meµ + Meτ identically vanishes. Namely, δ itself is absent in Msym and Mbreaking. The non-vanishing δ is produced by the interference † † between Msym and Mbreaking given by MsymMbreaking and MbreakingMsym in M, which certainly gives pure imaginary value if Eqs. (17)–(19) are inserted into Eq. (4), leading to maximal CP violation. This observation shows that the separation of M into Msym and Mbreaking is physically relevant and that Msym and Mbreaking act as if they were the real and imaginary part as far as their contribution to δ is concerned. It should be noted that the separation shown in Eq. (27) is a general procedure applied to any textures giving the maximal atmospheric neutrino mixing. What is speciﬁc to our present texture is to use the solution of Eqs. (17)–(19) for the presence of maximal CP violation. Therefore, it would be possible to argue CP property of any textures by focusing on the behavior of the simpler texture of Mbreaking. As another characteristic property, we discuss that if we require [14]

|Meµ|=|Meτ |, (28) then, we ﬁnd another relation

|Mµµ|=|Mττ|, (29) which is necessarily satisﬁed. Therefore, a symmetry that requires these two relations can be called “complex” µ–τ symmetry. From Eq. (28), we can use

=− =− ∗ Meτ σzMeµ, or Meτ σzMeµ, (30) where z = eiθ with θ being a phase parameter and the factor −σ is a matter of convention implied√ = | |2 −| |2 by Eq. (10). Because J of Eq. (21) is calculated√ to be J iσ Im(P )( Meµ Meτ )/ 2 2 2 for Msym and J =−iσ Im(P )(|Meµ| −|Meτ | )/ 2forMbreaking, the requirement of Eq. (28) gives J = 0 for both Msym and Mbreaking. Namely, the necessary condition for the presence of maximal CP violation is individually satisfied by Msym and Mbreaking although the real story is that their interference is the source of the presence of δ. For later convenience, we introduce ∗ ω± = (ω ± zω )/2 for a complex number ω. For Meτ =−σzMeµ, it is useful to consider Mµµ/Meµ, Mττ/Meµ and (Mµτ + σMee)/Meµ as well as 1 ± z =±z(1 ± z∗) and e−iδ =−eiδ in Eqs. (17)–(19). We, then, find that M∗ + M∗ + σM∗ Mττ = µµ Mµτ σMee = µτ ee z ∗ , z ∗ , (31) Meµ Meµ Meµ Meµ leading to ∗ ∗ ∗ = 2i arg(Meµ) + = 2i arg(Meµ) + Mττ ze Mµµ,Mµτ σMee ze Mµτ σMee , (32) which indicate that Mµµ + Mττ and Mµτ + σMee have a definite phase arg(Meµ) + θ/2 and that |Mµµ|=|Mττ| is indeed satisfied. Equivalently, Mµµ,ττ is expressed by Meµ as ∗ Mµµ = κMeµ,Mττ = κ zMeµ, (33) I. Aizawa et al. / Nuclear Physics B 728 (2005) 220–232 227 where 1 1 −iδ 1 t13 −iδ κ = √ − σt13e (1 + z) + + e σ(1 − z) . (34) 2 c13 tan 2θ12 tan 2θ13 2 The texture takes the form of   Mee Meµ −σzMeµ (A) =   Mν Meµ κMeµ Mµτ . (35) ∗ −σzMeµ Mµτ zκ Meµ (A) For a given Mν , κ+, κ− and Mee are determined by Eqs. (11)–(13) so that free parameters are =− ∗ given by Meµ, Mµτ and θ as expected. Similarly, for Meτ σzMeµ, we find that = ∗ + = ∗ + ∗ Mττ zMµµ,Mµτ σMee z Mµτ σMee , (36) ± ∗ =± ± ∗ ∗ where we have used the property that Meµ zMeµ z(Meµ zMeµ) . The relation of |Mµµ|=|Mττ| is satisfied. The flavor neutrino masses of Mµµ + Mττ and Mµτ + σMee have a (B) definite phase θ/2. The texture can be expressed as Mν :   − ∗ Mee Meµ σzMeµ (B) =   Mν Meµ Mµµ Mµτ . (37) − ∗ ∗ σzMeµ Mµτ zMµµ (B) For a given Mν , Mµµ+, Mµµ− and Mee are determined by Eqs. (11)–(13) so that free parameters are also given by Meµ, Mµτ and θ. In both cases, we observe that the assumed relation of |Meµ|=|Meτ | demands the presence of another relation of |Mµµ|=|Mττ|.Ourthreetex- (A,B) tures obtained in Ref. [20] correspond to the specific choices of Mµτ =−σMee in Mν and = iθ ∗ (B) Mee,µτ e Mee,µτ in Mν . Contributions to the Majorana phases can be estimated from Eq. (24) by considering the property that Mµµ + Mττ and Mµτ + σMee have a definite phase, which is transmitted to A , B and C. The neutrino masses are then given by −2iβ1 i(arg(Meµ)+θ/2) m1e = e |A |−|B| − σMµτ , −2iβ2 i(arg(Meµ)+θ/2) m2e = e |A |+|B| − σMµτ , −2iβ3 i(arg(Meµ)+θ/2) m3e = e −|A |+|C| + σMµτ , (38) (A) for Mν and −2iβ1 iθ/2 m1e = e |A |−|B| − σMµτ , −2iβ2 iθ/2 m2e = e |A |+|B| − σMµτ , −2iβ3 iθ/2 m3e = e −|A |+|C| + σMµτ , (39) (B) for Mν , where the possible ± sign in front of |A |, |B| and |C| is suppressed. The Majorana iθ CP violation can generally arise if arg(Mµτ ) = arg(Meµ) + θ/2forMeµ =−σe Meτ and if = =− iθ ∗ arg(Mµτ ) θ/2forMeµ σe Meτ . Finally, let us comment on neutrino masses and mixings in the simplest case of θ = 0for (B) (B) Mν , where Mν is reduced to the similar texture to the one discussed in Refs. [15,19,21] that, 228 I. Aizawa et al. / Nuclear Physics B 728 (2005) 220–232

(B) however, turns out to be over-constrained. This case gives Mν in the form of   − ∗ Mee Meµ σMeµ (B) =   Mν θ=0 Meµ Mµµ Mµτ . (40) − ∗ ∗ σMeµ Mµτ Mµµ

It is required that Mee + σMµτ be real from Eq. (19). Since both Mee and Mµτ are set to be real in Refs. [15,19], it is over-constrained for the presence of maximal CP violation. The mixing angles are calculated to be: √ = 2 2Re(Meµ) tan 2θ12 2 , s13 c13 Re(Mµµ) − (Mee + σMµτ ) − (Re(Mµµ) + Mee + σMµτ ) cos 2θ13 √ σ Im(M ) 2 2σ Im(M ) θ =−√ µµ ieiδ, θ = eµ ieiδ, sin 13 tan 2 13 + + (41) 2Re(Meµ) Re(Mµµ) σ(Mee σMµτ ) and masses are calculated from √ σ Im(M ) − 2Re(M ) A = √ eµ ie iδ − σM ,B= eµ , µτ c sin 2θ 2t13 √ 13 12 2 2σ Im(Meµ) −iδ C = B cos 2θ12 + ie , (42) sin 2θ13 where the Majorana CP violation arises if Mµτ is a complex number. Since there are sufﬁcient 2 2 freedoms, the hierarchy of matm m is readily realized. The closer relation to the µ–τ (B) symmetry in Eq. (40) is more transparent if we notice that Re(Mν |θ=0) is µ–τ symmetric (B) while Im(Mν |θ=0) gives its breaking term. For comparison, we also show results for M(A) with θ = 0 given by   −(σ Mµτ + Re(κ)Meµ)Meµ −σMeµ (A) =   Mν = Meµ κMeµ Mµτ . (43) θ 0 ∗ −σMeµ Mµτ κ Meµ The mixing angles are calculated to be: √ 2 σ Im(κ) iδ tan 2θ12 = , sin θ13 =− √ ie , (44) c13 Re(κ) 2 where Eq. (13) for tan 2θ13 is identically satisﬁed, and masses are calculated from √ 2Meµ A =−σMµτ ,B= ,C= B cos 2θ12, (45) c13 sin 2θ12 where the Majorana CP violation arises if arg(Mµτ ) = arg(Meµ) as can be read off from Eq. (38). (A) The requirement of Im(κ) = 0inMν |θ=0 provides the µ–τ symmetric texture. The hierarchy of 2 2 | |∼| || | matm m can be realized by the presence of degenerate neutrinos with B C A as ∼ 2 2 = ∗ 2 = 2 | |2 + in the 4th type (M4), giving m1,2,3 matm, m 4Re(B A) and matm sin 2θ12 B 2 2 ≈ 2 | |2 c12m sin 2θ12 B . We can estimate the effective neutrino mass mββ used in the detection of the absolute neutrino mass [26], which is given by |Mee|. In our prediction, mββ is computed to − + | |∼| || | be (σ Mµτ Re(κ)Meµ), which is roughly equal to Re(κ)Meµ fromB C A . Because | |≈ 2 | |≈ 2 of Eq. (44) for Re(κ) and sin 2θ12 B matm, we obtain mββ matm/ tan 2θ12. I. Aizawa et al. / Nuclear Physics B 728 (2005) 220–232 229

4. Summary and discussions

Summarizing our discussions, we have found constraints on flavor neutrino masses that provide the maximal atmospheric neutrino mixing and maximal CP violation. Our texture Eq. (23) has three parameters given by Meµ,eτ,µτ and other flavor neutrino masses are determined by Eqs. (17)–(19) as the solution to Eq. (10) for the given mixing angles. We have also clarified that the separation of neutrino mass matrix into the sum of the µ–τ symmetric part and its breaking part is very useful to understand how CP violating phase arises. With the aid of Eq. (4),itis shown that each part has no phase δ but the interference between these two parts provides CP violation and that our solution indeed gives maximal CP violation. Furthermore, the appearance 2 of maximal CP violation favors the smallness of sin θ13 because maximal CP violation gives the minimum value for the denominator of tan 2θ13 in Eq. (6). To obtain the observed mass hi- 2 2 erarchy of matm m, we have calculated the required correlation for the flavor neutrino masses giving five types of mass ordering (M1)–(M5) for m1,2,3 depending on their plus and minus signs. To see the close connection with the µ–τ symmetric texture, we have imposed |Meµ|=|Meτ | and have found that the consistency demands the relation of |Mµµ|=|Mττ|, which is satisfied = = ∗ iθ =− iθ = iθ ∗ by Mµµ κMeµ and Mττ κ e Meµ with Meτ σe Meµ and by Mττ e Mµµ with =− iθ ∗ Meτ σe Meµ. These relations can be regarded as an extended version of the µ–τ symmetric texture with Meµ =−σMeτ and Mµµ = Mττ [14] and a symmetry that requires the relations of |Meµ|=|Meτ | and |Mµµ|=|Mττ| can be called “complex” µ–τ symmetry. The resulting textures are given by   iθ Mee Meµ −σe Meµ (A) =   Mν Meµ κMeµ Mµτ , (46) iθ iθ ∗ −σe Meµ Mµτ e κ Meµ and   − iθ ∗ Mee Meµ σe Meµ (B) =   Mν Meµ Mµµ Mµτ , (47) − iθ ∗ iθ ∗ σe Meµ Mµτ e Mµµ (A) as in Eqs. (35) and (37). The texture Mν is characterized by the fact that Mµµ + Mττ ∗ iθ (= κMeµ + κ e Meµ) and Mµτ + σMee have the common phase arg(Meµ) + θ/2 and the Ma- (B) jorana CP violation exists if arg(Mµτ ) = arg(Meµ) + θ/2. On the other hand, the texture Mν + = + iθ ∗ + exhibits the property that Mµµ Mττ ( Mµµ e Mµµ) and Mµτ σMee have the common phase θ/2 and that the Majorana CP violation exists if arg(Mµτ ) = θ/2. (A) In the simplest cases with θ = 0, the µ–τ symmetry is recovered by Im(κ) = 0forMν while (B) (B) (B) Re(Mν ) and Im(Mν ), respectively, provide the µ–τ symmetric and breaking part in Mν . In other words, we observe that sin θ13 is determined by imaginary parts of the textures with = θ 0, which are given by either Im(Meµ) and Im(Mµµ) or Im(κ), because to preserve the µ–τ θ = M(B) m ∼ m2 symmetry is linked to sin 13 0. For ν , we predict ββ atm because of the presence ∼ 2 of the degenerate neutrinos with m1,2,3 matm. It should be noted that there is no freedom in the choice of the PMNS mixing matrix because it is completely determined once a specific neutrino√ mass texture is given. Our texture Eq. (23) is diagonalized by Eq. (1) with c23 = σs23 = 1/ 2 and δ =±π/2. To see if a given mass matrix 230 I. Aizawa et al. / Nuclear Physics B 728 (2005) 220–232 can be diagonalized by U of Eq. (1), we have to check the consistency among constraints imposed = † on M. Note that the advantage to use M( Mν Mν) is lies in the fact that fictitious phases as well as Majorana phases do not appear in M because all phases except for phases related to δ are cancelled as stated in the Introduction. The additional constraints [19] are given by6 sin 2θ cos 2θ Re(M ) = 23 (M − M ) − t X cos δ, (48) 23 µτ 2 ττ µµ 13 Im(M ) t = µτ , (49) 13 X sin δ where X = Re(c23Meµ − s23Meτ ). In the present case, we have cos δ = 0 and sin δ =±1. The constraint Eq. (48) should be consistent with the prediction of Eq. (7), which is cos 2θ23 = 0 and sin 2θ23 = σ for the maximal atmospheric neutrino mixing. Similarly,√ the prediction of Eq. (49) for θ13 should be compatible with Eq. (6) for c23 = σs23 = 1/ 2. Our solution Eq. (19) certainly satisfies these constraints. Furthermore, unphysical diagonal phase matrix (to be denoted by P ) may generally arise in the left of Uν and νf = PUνmass is satisfied. This phase P can = −1 = be absorbed by the redefinition of flavor neutrino fields νf P νf Uνmass; therefore, νf is transformed into νmass just by U. We believe that we exhaust all possible textures with the maximal atmospheric neutrino mixing and maximal CP violation, if they are at all diagonalized by U of Eq. (1), by choosing a variety of Meµ, Meτ and Mµτ .

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6 These constraints are most easily seen by expressing M in terms of the neutrino masses, the neutrino mixing angles and the Dirac phase. I. Aizawa et al. / Nuclear Physics B 728 (2005) 220–232 231

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CUMULATIVE AUTHOR INDEX B721–B728

Aarts, G. B726 (2005) 93 Björnsson, J. B727 (2005) 77 Abada, A. B728 (2005) 55 Bobeth, C. B726 (2005) 252 Ablikim, M. B727 (2005) 395 Bona, M. B726 (2005) 252 Aguado, M. B723 (2005) 234 Bonciani, R. B723 (2005) 91 Aissaoui, H. B728 (2005) 55 Borhade, P. B727 (2005) 471 Aizawa, I. B728 (2005) 220 Borsányi, S. B727 (2005) 244 Albino, S. B725 (2005) 181 Botella, F.J. B725 (2005) 155 Ananth, S. B722 (2005) 166 Boucaud, P. B721 (2005) 175 Anastasiou, C. B724 (2005) 197 Boughezal, R. B725 (2005) 3 Angelantonj, C. B725 (2005) 115 Boulanger, N. B722 (2005) 225 Ansari, M.H. B726 (2005) 494 Boulton, L. B724 (2005) 380 Antonov, E.N. B721 (2005) 111 Bourrely, C. B722 (2005) 149 Aoki, M. B727 (2005) 163 Bozi, D. B725 (2005) 421 Ashoorioon, A. B727 (2005) 63 Branco, G.C. B725 (2005) 155 Azcoiti, V. B723 (2005) 77 Brandhuber, A. B721 (2005) 98 Brink, L. B722 (2005) 166 Bai, J.Z. B727 (2005) 395 Britto, R. B725 (2005) 275 Bakulev, A.P. B721 (2005) 50 Buchbinder, E.I. B728 (2005) 207 Bal, S. B727 (2005) 196 Buchbinder, I.L. B727 (2005) 537 Balog, J. B725 (2005) 531 Buras, A.J. B726 (2005) 252 Ban, Y. B727 (2005) 395 Burguet-Castell, J. B725 (2005) 306 Barger, V. B726 (2005) 149 Bartels, J. B726 (2005) 53 Cachazo, F. B725 (2005) 275 Batrachenko, A. B726 (2005) 275 Cai, X. B727 (2005) 395 Bedford, J. B721 (2005) 98 Calloni, E. B726 (2005) 441 Behrndt, K. B721 (2005) 287 Campbell, J. B726 (2005) 109 Beisert, N. B727 (2005) 1 Caprini, I. B722 (2005) 149 Bekaert, X. B722 (2005) 225 Cardella, M. B725 (2005) 115 Belhaj, A. B727 (2005) 499 Carmelo, J.M.P. B725 (2005) 421 Belitsky, A.V. B722 (2005) 191 Casper, D. B725 (2005) 306 Bellorín, J. B726 (2005) 171 Chacko, Z. B725 (2005) 207 Bellucci, S. B722 (2005) 297 Chan, C.-T. B725 (2005) 352 Bellucci, S. B726 (2005) 233 Chen, H.F. B727 (2005) 395 Berges, J. B727 (2005) 244 Chen, H.S. B727 (2005) 395 Berman, D.S. B723 (2005) 117 Chen, H.X. B727 (2005) 395 Bernreuther, W. B723 (2005) 91 Chen, J. B727 (2005) 395 BES Collaboration B727 (2005) 395 Chen, J.C. B727 (2005) 395 Bian, J.G. B727 (2005) 395 Chen, S.-L. B724 (2005) 423 Bimonte, G. B726 (2005) 441 Chen, Y.B. B727 (2005) 395

0550-3213/2005 Published by Elsevier B.V. doi:10.1016/S0550-3213(05)00838-2 234 Nuclear Physics B 728 (2005) 233–238

Cherednikov, I.O. B721 (2005) 111 Gambino, P. B726 (2005) 35 Chi, S.P. B727 (2005) 395 Gao, C.S. B727 (2005) 395 Chu, C.-S. B725 (2005) 327 Gao, P. B721 (2005) 287 Chu, Y.P. B727 (2005) 395 Gao, Y.N. B727 (2005) 395 Cirelli, M. B727 (2005) 99 Gasser, J. B728 (2005) 31 Cirigliano, V. B728 (2005) 121 Gayral, V. B727 (2005) 513 Colangelo, G. B721 (2005) 136 Gehrmann, T. B723 (2005) 91 Copland, N.B. B723 (2005) 117 Genovese, L. B721 (2005) 212 Costa, M.S. B728 (2005) 148 Germani, C. B725 (2005) 15 Couce, E. B725 (2005) 306 Giannakis, I. B723 (2005) 255 Cui, X.Z. B727 (2005) 395 Giedt, J. B726 (2005) 210 Cvetic,ˇ M. B721 (2005) 287 Giménez, V. B721 (2005) 175 Giovannangeli, P. B721 (2005) 1 Dai, Y.S. B727 (2005) 395 Giovannangeli, P. B721 (2005) 25 Da Rold, L. B721 (2005) 79 Giudice, G.F. B726 (2005) 35 Degrassi, G. B724 (2005) 183 Gómez-Cadenas, J.J. B725 (2005) 306 Deng, Z.Y. B727 (2005) 395 Gracia-Bondía, J.M. B727 (2005) 513 Denner, A. B724 (2005) 247 Gribanov, V.V. B727 (2005) 564 Derkachov, S.É. B722 (2005) 191 Grinstein, B. B728 (2005) 121 de Vega, H.J. B726 (2005) 464 Gu, S.D. B727 (2005) 395 D’Hoker, E. B722 (2005) 81 Gu, Y.T. B727 (2005) 395 Di Bari, P. B727 (2005) 318 Günaydin, M. B727 (2005) 421 Di Carlo, G. B723 (2005) 77 Guo, Y.N. B727 (2005) 395 Dittmaier, S. B724 (2005) 247 Guo, Y.Q. B727 (2005) 395 Doikou, A. B725 (2005) 493 Gürsoy, U. B725 (2005) 45 Dong, L.Y. B727 (2005) 395 Gutperle, M. B722 (2005) 81 Dong, Q.F. B727 (2005) 395 Hadasz, L. B724 (2005) 529 Dorsner, I. B723 (2005) 53 Haefeli, C. B721 (2005) 136 Du, S.X. B727 (2005) 395 Hagiwara, K. B727 (2005) 163 Du, Z.Z. B727 (2005) 395 Hallowell, K. B724 (2005) 453 Dudas, E. B721 (2005) 309 Hanada, M. B727 (2005) 196 Dudas, E. B727 (2005) 139 Hatanaka, T. B726 (2005) 481 Duff, M.J. B726 (2005) 275 He, K.L. B727 (2005) 395 Dürr, S. B721 (2005) 136 He, M. B727 (2005) 395 Heged˝us, Á. B725 (2005) 531 Eden, B. B722 (2005) 119 Heinesch, R. B723 (2005) 91 Ejiri, H. B727 (2005) 406 Heng, Y.K. B727 (2005) 395 Engels, J. B724 (2005) 357 Henkel, M. B723 (2005) 205 Esposito, G. B726 (2005) 441 Herdeiro, C.A.R. B728 (2005) 148 Ewerth, T. B726 (2005) 252 Hernández, P. B725 (2005) 306 Ho, P.-M. B725 (2005) 352 Fang, J. B727 (2005) 395 Hodgson, R. B728 (2005) 192 Fang, S.S. B727 (2005) 395 Holtmann, S. B724 (2005) 357 Faraggi, A.E. B728 (2005) 83 Hovdebo, J.L. B727 (2005) 63 Feng, B. B725 (2005) 275 Hu, H.M. B727 (2005) 395 Fernando, S. B727 (2005) 421 Hu, T. B727 (2005) 395 Fileviez Pérez, P. B723 (2005) 53 Huang, X.P. B727 (2005) 395 Fornengo, N. B727 (2005) 99 Huang, X.T. B727 (2005) 395 Frigerio, M. B724 (2005) 423 Hwang, S. B727 (2005) 77 Fu, C.D. B727 (2005) 395 Hyun, S. B727 (2005) 421 Fuchs, J. B724 (2005) 503 Fujiwara, K. B723 (2005) 33 Inami, T. B725 (2005) 327 Fukuma, M. B728 (2005) 67 Irges, N. B725 (2005) 115 Irie, H. B728 (2005) 67 Gade, R.M. B726 (2005) 363 Isidori, G. B728 (2005) 121 Galante, A. B723 (2005) 77 Israël, D. B722 (2005) 3 Nuclear Physics B 728 (2005) 233–238 235

Itoyama, H. B723 (2005) 33 Leineweber, T. B723 (2005) 91 Ivanov, E. B722 (2005) 297 Li, G. B727 (2005) 301 Ivanov, E.A. B726 (2005) 131 Li, G. B727 (2005) 395 Ivanov, M.A. B728 (2005) 31 Li, H.B. B727 (2005) 395 Li, H.H. B727 (2005) 395 Jack, I. B728 (2005) 192 Li, J. B727 (2005) 395 Jarczak, C. B722 (2005) 119 Li, R.Y. B727 (2005) 395 Jaskólski, Z. B724 (2005) 529 Li, S.M. B727 (2005) 395 Jellal, A. B725 (2005) 554 Li, T. B726 (2005) 149 Jentschura, U.D. B725 (2005) 467 Li, T. B727 (2005) 301 Ji, X.B. B727 (2005) 395 Li, W.D. B727 (2005) 395 Jiang, J. B726 (2005) 149 Li, W.G. B727 (2005) 395 Jiang, X.S. B727 (2005) 395 Li, X.-Q. B727 (2005) 301 Jiao, J.B. B727 (2005) 395 Li, X.L. B727 (2005) 395 Jin, D.P. B727 (2005) 395 Li, X.Q. B727 (2005) 395 Jin, S. B727 (2005) 395 Li, Y.L. B727 (2005) 395 Jin, Y. B727 (2005) 395 Liang, Y.F. B727 (2005) 395 Jing, J. B728 (2005) 109 Liao, H.B. B727 (2005) 395 Jones, D.R.T. B728 (2005) 192 Lin, C.-J.D. B721 (2005) 175 Lipatov, L.N. B721 (2005) 111 Kadyshevsky, V.G. B727 (2005) 564 Lipatov, L.N. B726 (2005) 53 Kaneko, H. B725 (2005) 93 Liu, C.X. B727 (2005) 395 Karabali, D. B726 (2005) 407 Liu, F. B727 (2005) 395 Kashani-Poor, A.-K. B728 (2005) 135 Liu, F. B727 (2005) 395 Liu, H.H. B727 (2005) 395 Kato, J. B721 (2005) 229 Liu, H.M. B727 (2005) 395 Kawai, H. B727 (2005) 196 Liu, J. B727 (2005) 395 Kawamoto, N. B721 (2005) 229 Liu, J.B. B727 (2005) 395 Kehagias, A. B725 (2005) 15 Liu, J.P. B727 (2005) 395 Ketov, S.V. B726 (2005) 481 Liu, J.T. B726 (2005) 275 Kim, C.H. B727 (2005) 218 Liu, R.G. B727 (2005) 395 Kim, S.-S. B722 (2005) 166 Liu, Z.A. B727 (2005) 395 Kirilin, G.G. B728 (2005) 179 Losada, M. B728 (2005) 55 Kitabayashi, T. B728 (2005) 220 Lu, F. B727 (2005) 395 Kitazawa, Y. B725 (2005) 93 Lu, G.R. B727 (2005) 395 Knechtli, F. B726 (2005) 421 Lu, H.J. B727 (2005) 395 Kniehl, B.A. B725 (2005) 181 Lu, J.G. B727 (2005) 395 Kobayashi, Y. B726 (2005) 481 Lubicz, V. B721 (2005) 175 Korchemsky, G.P. B722 (2005) 191 Lucietti, J. B724 (2005) 343 Korthals Altes, C.P. B721 (2005) 1 Luo, C.L. B727 (2005) 395 Korthals Altes, C.P. B721 (2005) 25 Lüst, D. B727 (2005) 264 Kramer, G. B725 (2005) 181 Krykhtin, V.A. B727 (2005) 537 Ma, E. B724 (2005) 423 Kubo, F. B727 (2005) 196 Ma, F.C. B727 (2005) 395 Kühn, J.H. B727 (2005) 368 Ma, H.L. B727 (2005) 395 Kulesza, A. B727 (2005) 368 Ma, L.L. B727 (2005) 395 Kunduri, H.K. B724 (2005) 343 Ma, Q.M. B727 (2005) 395 Kuraev, E.A. B721 (2005) 111 Ma, W.-G. B727 (2005) 301 Ma, X.B. B727 (2005) 395 Lai, Y.F. B727 (2005) 395 Maltoni, F. B724 (2005) 183 Laliena, V. B723 (2005) 77 Manashov, A.N. B722 (2005) 191 Langacker, P. B726 (2005) 149 Mann, R.B. B727 (2005) 63 Lange, B.O. B723 (2005) 201 Mao, Z.P. B727 (2005) 395 Leder, B. B726 (2005) 421 Markopoulou, F. B726 (2005) 494 Lee, H.K. B728 (2005) 1 Martinelli, G. B721 (2005) 175 Lee, J.-C. B725 (2005) 352 Martínez Resco, J.M. B726 (2005) 93 Lee, K.S.M. B721 (2005) 325 Mawatari, K. B727 (2005) 163 236 Nuclear Physics B 728 (2005) 233–238

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