Gauge threshold corrections in warped geometry

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New Journal of Physics

Gauge threshold corrections in warped geometry

Kiwoon Choi1,3, Ian-Woo Kim2 and Chang Sub Shin1 1_{Department of Physics, KAIST, Daejeon 305-701, Korea}

2_{Department of Physics, University of Wisconsin, Madison, WI 53706, USA} E-mail:kchoi@kaist.ac.kr,ikim@physics.wisc.eduand

csshin@muon.kaist.ac.kr

New Journal of Physics12 (2010) 075014 (32pp)

Received 31 December 2009 Published 16 July 2010 Online athttp://www.njp.org/

doi:10.1088/1367-2630/12/7/075014

Abstract. In this paper, we discuss the Kaluza–Klein threshold correction to low-energy gauge couplings in theories with warped extra dimension, which might be crucial for the gauge coupling unification when the warping is sizable. Explicit expressions of one-loop thresholds are derived for generic five-dimensional (5D) gauge theory on a slice of 5D anti-de Sitter space (AdS5), where some of the bulk gauge symmetries are broken by orbifold boundary conditions and/or by bulk Higgs vacuum values. The effects of the mass mixing between the bulk fields with different orbifold parities are included, as such a mixing is required in some classes of realistic warped unification models.

Contents

1. Introduction 2

2. Some generic features of KK threshold corrections 3

3. Warped gauge thresholds 11

4. Conclusion 21

Acknowledgments 24

Appendix A. The N-function 24

Appendix B. KK thresholds with boundary matter fields 29

References 30

3_{Author to whom any correspondence should be addressed.}

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1. Introduction

In theories with unified gauge symmetry at high energy scale, threshold corrections due to heavy particles often affect the predicted low-energy gauge couplings significantly [1]. Since symmetry breaking leads to a mass splitting between particles in the same representation of unified gauge symmetry, the couplings in low-energy effective theory acquire generically non-universal quantum corrections when heavy particles are integrated out. In four-dimensional (4D) theories, the resulting differences between low-energy gauge couplings are proportional to the logarithms of the mass ratios. Therefore, the threshold effects can be particularly important when mass splitting occurs over a wide range of energy scales and/or for many degrees of freedom.

Such a situation can be realized in higher dimensional gauge theories (including string theories), in which there exist generically an infinite tower of gauge-charged Kaluza–Klein (KK) states. Higher dimensional gauge theories can employ novel classes of symmetry breaking mechanisms such as the one by boundary condition [2] or by the vacuum expectation value (VEV) of the extra-dimensional component of the gauge field [3]. Such mechanisms might successfully address various naturalness problems of grand unified theories (GUTs) [4] and/or explain the origin of the Higgs field [5]. In higher dimensional theories with broken gauge symmetry, the whole KK tower of higher dimensional fields are split. This splitting can yield a large threshold correction because of the infinite number of KK modes and also a large-scale difference between the lowest KK mass and the cutoff scale of the theory [6].

On the other hand, the calculation of the KK thresholds in higher dimensional gauge theories requires a careful treatment of the UV divergences associated with an infinite tower of massive KK modes. Summing up the logarithmic contribution from each KK mode, it is expected that power-law-divergent contributions will appear [7]. All the UV divergences must be absorbed into local counterterms that are consistent with the defining symmetry of the theory. In models with unified gauge symmetry in bulk spacetime, these power-law divergences are universal and can be absorbed into a renormalization of the unified higher dimensional gauge coupling at the cutoff scale 3. However, if the unified gauge symmetry is broken by a boundary condition at the orbifold fixed point, there can be non-universal logarithmically divergent counterterms localized at the fixed point. Those logarithmic divergences are associated with the renormalization group (RG) runnings of the fixed-point gauge coupling constants [8], which lead to a controllable consequence in the predicted low-energy gauge couplings as in the case of conventional 4D GUTs [6]. After identifying the UV-divergent pieces of the KK threshold corrections, the finite calculable parts are unambiguously defined4_.

In general, these finite corrections heavily depend on the parameters of the model, including the symmetry breaking VEVs and the masses of higher dimensional fields, as well as on the structure of the background spacetime geometry.

It has been of particular interest to study quantum corrections in warped geometry. Warped extra dimension might be responsible for the weak scale to the Planck scale hierarchy [11] or the supersymmetry breaking scale to the Planck scale hierarchy [12, 13] or even the Yukawa coupling hierarchies [14]. There have also been studies on higher dimensional GUTs in

4 _{In string theory, the full threshold corrections including stringy thresholds are finite with the cutoff scale} ₃

replaced by the string scale. For an early discussion of threshold corrections in compactified string theory, see for instance [9,10].

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warped geometry, showing quite distinct features arising from the warping [15]–[19]. In warped models, gauge threshold corrections might be of crucial importance for a successful unification when the lowest KK scale mKK is hierarchically lower than the conventional unification scale

M_GUT_{∼ 2 × 10}16GeV. A series of studies on quantum corrections in anti-de Sitter space (AdS) show that KK threshold corrections in warped gauged theory are enhanced by the large logarithmic factor ln(e) [15], [20]–[23], where e is an exponentially small warp factor. Explanation of this logarithmic factor has been attempted in various contexts, including those based on the AdS/CFT correspondence that states that a 5D theory on a slice of AdS5 can be regarded as a 4D conformal field theory (CFT) with conformal symmetry spontaneously broken at mKK[24,25].

In [23], analytic expressions of the KK thresholds in 5D gauge theory on a slice of AdS5 have been derived for the case that bulk gauge symmetries are broken by orbifold boundary conditions and there is no mixing between bulk fields with different orbifold parities. In the present paper, we wish to extend the analysis of [23] to the more general cases that involve symmetry breaking by bulk scalar VEVs and also nonzero mass mixings among bulk fields with different orbifold parities. Our results then cover most of the warped GUT models discussed so far in the literature.

The organization of this paper is as follows. In section2, we first discuss some features of KK thresholds that are relevant for our later discussion and then examine a simple example of the 5D scalar threshold to illustrate our computation method. In section 3, we consider generic 5D gauge theory defined on a slice of AdS5, and derive an analytic expression of one-loop KK thresholds induced by 5D gauge and matter fields when some of the bulk gauge symmetries are broken by orbifold boundary conditions and/or by bulk Higgs vacuum values. To be general, we also include the effects of mass mixing between the bulk fields with different orbifold parities. In section 4, we give the conclusion. We provide, in appendix A, a detailed discussion of the

N-function whose zeros correspond to the KK spectrum; a discussion of boundary matter fields in appendixB.

2. Some generic features of KK threshold corrections

The 5D gauge theory under consideration here can be defined as a Wilsonian effective field theory with the action

S_W_{= −} Z d5x√_−G 1 4 1 g2_5a + κa 4π2 δ(y) √ G55 + κ 0 a 4π2 δ(y − π) √ G55 Fa M NF_{M N}a

+S_{gauge−fixing}+ Sghost+ Smatter, (1)

where Sgauge−fixing and Sghost are the gauge-fixing term and the associated ghost action, respectively, Smatter is the model-dependent action of 5D scalar and fermion matter fields, and the 5D spacetime metric GM N is assumed to take a generic 4D Poincaré-invariant form:

ds2= GM NdxMdxN = e2(y)ηµνdxµdxν+ R2dy2 (0 6 y 6 π),

whereπ R is the proper distance of the interval, ηµν corresponds to the 4D graviton zero mode that is used by the low-energy observer to measure the external 4D momentum pµas well as the KK mass spectrum, and we are using the warp factor convention: e(y=0)= 1 and e(y=π)6 1. Here, we do not include any boundary matter field separately as it can be considered as the

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localized limit of the bulk matter field, which is achieved by taking some mass parameters to the cutoff scale. (For a discussion of this point, see appendixB.) Note that the range of the 5th dimension is taken as 0 6 y 6π with the convention: R₀πdyδ(y) = R₀πdyδ(y − π) = 1/2.

In order for the theory to be well defined, one also needs to specify the UV cutoff scheme along with the Wilsonian action. Then all the Wilsonian couplings in SW depend implicitly on the associated cutoff scheme 3, and this 3-dependence of Wilsonian couplings should cancel the3-dependence of regulated quantum corrections, rendering all the observable quantities to be independent of3.

The quantity of our concern is the low energy one-particle-irreducible (1PI) gauge couplings of 4D gauge boson zero modes. It can be obtained by evaluating

ei0[8cl]₌

Z

D8qu eiSW[8cl+8qu], ₍₂₎

where 8cl denotes background field configuration, which includes the 4D gauge boson zero modes Aa(0)

µ as well as the vacuum values of scalar fields, and 8qu stands for quantum fluctuations of the 5D gauge, matter and ghost fields in the model. The resulting 1PI gauge coupling ga(p) of A2 a(0)

µ (p) carrying an external 4D momentum pµis given by (−p2ηµν_{+ p}µ_pν) g2 a(p) ≡ δ 20 δ Aa(0) µ (p)δ Aaν(0)(−p) Aa_µ(0)=0 . (3)

As the gauge boson zero modes have a constant wavefunction over the 5th dimension, the 4D gauge couplings at tree level are simply given by

1 g2 a tree = π R g2 5a + 1 8π2 κa+κ 0 a . (4)

To compute quantum corrections, one needs to introduce a suitable regularization scheme that might involve a set of regulator masses collectively denoted by 35_{. One also needs to deal} with a summation over the KK modes whose mass eigenvalues {mn} depend on various model parameters that will be collectively denoted by λ, for instance the bulk or boundary masses of the matter and gauge fields as well as the AdS vacuum energy density that would determine the warp factor. Note that the 4D momentum pµ of the gauge boson zero modes and the KK mass eigenvalues {mn} are defined in the 4D metric frame of the graviton zero mode ηµν, whereas3, λ and 1/R are the 5D mass parameters invariant under the 5D general coordinate transformation.

Schematically, one-loop correction to the 4D 1PI gauge coupling is given by 1 8π21a(p, 3, R, λ) = X 80,8n Z _d4_l (2π)4 fa(p, l, mn(R, λ)), (5)

where 80 denotes the light zero modes with a mass m0 p, while 8n stands for the massive KK modes with mn _{p. In the limit p mKK} and3 λ, where mKKis the lowest KK mass,

5 _{At this stage, we assume a mass-dependent cutoff scheme introducing an appropriate set of Pauli–Villars}

regulating fields and/or higher derivative regulating terms, although eventually we will use a mass-independent dimensional regularization that is particularly convenient for the computation of gauge boson loops.

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the above one-loop correction takes the form [6,21] 1 8π21a= γa 24π33π R + 1 8π2 h ˜baln(3π R) − baln(pπ R) + ˜1a(R, λ)i +O _p2 m2_{K K} + O λ 3, 1 3R . (6)

Then the low-energy 1PI couplings are given by 1 g2 a(p) = 1 g2 a tree + 1 8π21a = π R ˆg5a2 + 1 8π2ˆκa(ln p, λ, R) + O _p2 m2_{K K} + O λ 3, 1 3R , (7) where 1 ˆg5a2 = 1 g_5a2 (3)+ γa 24π33

ˆκa= κa(3) + κa(3) + ˜ba0 ln(3π R) − baln(pπ R) + ˜1a(R, λ). (8) The above expression of 4D 1PI coupling is valid only for p< mKK. However, it still provides a well-defined matching between the observable low-energy gauge couplings and the fundamental parameters in the 5D action defined at the cut-off scale 3 mKK. Note that the Wilsonian couplings g2

5a, κa andκa0 depend on3 in such a way as to make ˆg5a2 and ˆκa independent of3. The linearly divergent piece in (8) originates from the KK modes around the cutoff scale3, and therefore its coefficientγa severely depends on the employed cutoff scheme. For instance, in a mass-dependent cutoff scheme introducing an appropriate set of Pauli–Villars (PV) regulating fields and/or higher derivative regulating terms, each γa has a nonzero value depending on the detailed structure of the regulator masses and the regulator coefficients, while it vanishes in a mass-independent cutoff scheme such as dimensional regularization [26]6. Note that this does not affect the calculable prediction of the theory, which is determined by the scheme-independent combination 1/ˆg_5a2 . Unlike the coefficient of power-law divergence, the coefficients of ln p and ln3 are unambiguously determined by the physics below 3 [6,21]. As ln p originates from the light zero modes with m0 p, one immediately finds

ba= 1 6 X ϕ(0) Tr(T_a2(ϕ(0))) +2 3 X ψ(0) Tr(T_a2(ψ(0))) −11 3 X A(0)_µ Tr(T_a2(A(0)_µ )), (9)

whereϕ(0),ψ(0)and A(0)_µ denote the 4D real scalar, 4D chiral fermion and 4D real vector boson zero modes that originate from 5D matter and gauge fields, and Ta(8) is the generator of the unbroken gauge transformation of 8. Note that ϕ(0) can originate from a 5D vector field. The logarithmic divergence appears because of the orbifold fixed points. This implies that ˜ba are determined just by the orbifold boundary condition of 5D fields if there is no 4D matter field confined at the fixed point. The logarithmic divergence generically takes the form

− Z d5x√_−G ln3 16π2 λa0δ(y) √ G55 +λaπ√δ(y − π) G55 F_µνa Faµν, (10)

6 _{A novel extension of dimensional regularization for higher dimensional gauge theory has been suggested}

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and the coefficients λa0 and λa_π are independent of the smooth geometry of the underlying spacetime. It is then straightforward to determine λa0 and λa_π in the flat orbifold limit, which yields [6,21] λa0 = X zz0 z 24 Tr(T_a2(φzz0)) − 23Tr(T_a2(AM_zz₀)), (11) λaπ = X zz0 z0 24 Tr(T_a2(φzz0)) − 23 Tr(T_a2(AM_zz0)), and thus ˜ba= λa0+λaπ = X zz0 (z + z0) 24 Tr(T_a2(φzz0)) − 23 Tr(T_a2(AM_zz0)), (12)

where φzz0 and AM_zz₀ (z, z0= ±1) denote 5D real scalar and vector fields with the orbifold

boundary condition:

φzz0(−y) = zφzz0(y), φzz0(−y + π) = z0φzz0(y + π), A_zzM0(−y) = zMA M zz0(y), A M zz0(−y + π) = z0MA M zz0(y + π) (13) whereµ= 1 and 5= −1.

The last part of the one-loop correction, i.e. ˜1a(R, λ), is highly model dependent as it generically depends on various parameters of the underlying 5D theory, e.g. the curvature of background geometry, matter and gauge field masses in bulk and at the fixed points, and also on the orbifold boundary conditions of 5D fields. Note that all of these features affect the KK mass spectrum, and thus the KK thresholds. In many cases, it can be an important part of quantum correction, even a dominant part in the warped case. The aim of this paper is to provide an explicit expression of ˜1a as a function of the fundamental parameters in 5D theory in a context as general as possible. Let us now consider a specific example of the 5D scalar threshold to see some of the features discussed above. We start with the case of a massless 5D complex scalar fieldφzz0 in the flat spacetime background:

Smatter= − Z d5x√_−GX z,z0 GM NDMφ†_zz0DNφzz0, (14) where ds2= GM NdxMdxN = ηµνdxµdxν+ R2dy2.

In this case, one can easily find an explicit form of the KK spectra:

mn(φzz0) =                  n R for φ++ 2n + 1 2R for φ+− and φ−+, n+ 1 R for φ−−. (15)

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where n is a non-negative integer. The corresponding one-loop correction can be obtained using a simple momentum cutoff:

1 8π2(p 2_ηµν − pµpν)1a(φzz0) = X z,z0 3R X n=0 Tr(T_a2(φzz0)) Z _d4_l (2π)4 f µν_, ₍₁₆₎ where fµν₌ 2η µν_{((p + l)}2_{+ m}2_n(φzz 0)) − (p + 2l)µ(p + 2l)ν i((p + l)2_{+ m}2n(φzz₀))(l2_{+ m}2 n(φzz0)) , which gives (in the limit p mKK= 1/R)

1a= 1 3Tr(T 2 a(φ++)) ln 3 p +1 3 h Tr(T_a2(φ++)) + Tr(T_a2(φ−−))i 3R X n=1 ln 3R n +1 3 h Tr(T_a2(φ+−)) + Tr(T_a2(φ−+))i 3R X n=1 ln 23R 2n − 1 + O(1) = 1 3 h Tr(T_a2(φ++)) + Tr(T_a2(φ−−)) + Tr(T_a2(φ+−)) + Tr(T_a2(φ−+))i3R +1 6 h Tr(T_a2(φ++)) − Tr(T_a2(φ−−))iln(3π R) − 1 3Tr(T 2 a(φ++)) ln(pπ R) + O(1). (17) Obviously, in the case with a unified gauge symmetry in bulk spacetime, the coefficients of linear divergence, i.e.P

z,z0Tr(Ta2(φzz0)), are universal. Also, the above result gives

˜ba₌ 1 6 h

Tr(T2

a(φ++)) − Tr(Ta2(φ−−))i,

which confirms the result of (12). Note thatφzz0 here are complex scalar fields, whileφzz0 in (12)

are real scalar fields. One can generalize the above result by introducing a nonzero bulk mass. To see the effect of bulk mass, let us considerφ++ with a 5D mass MS p in the flat spacetime background. It is still straightforward to find the explicit form of the KK spectrum:

mn= r

M_S2+ n 2

R2. (18)

In this case, there is no light mode since MS p, and therefore ba= 0. Again the one-loop threshold can be computed with a simple momentum cutoff:

1a(φ++) =1 6Tr(T 2 a(φ++)) 3R X n=0 ln 32_R2 M2 SR2+ n2 + O(1) =1 6Tr(T 2 a(φ++)) 23R + ln 3 MS − ln sinh(MSπ R) + O(1) .

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For the warped spacetime background, the KK spectrum takes a more complicated form, and its explicit form is usually not available. Furthermore, as the 4D loop momentum lµ and the KK spectrum {mn} are defined in the metric frame of 4D graviton zero mode, the cutoff scales for lµ and {mn} depend on the position in warped extra-dimension. One can avoid these difficulties using the Pole function method with dimensional regularization [23, 26,28], which will be described below. As the one-loop correction takes the form

1 8π21a= ∞ X n=0 Z d4l (2π)4 fa(p, l, mn), (19)

where fa→ 1/(l2+ m2_n)2 in the limit l2∼ m2n→ ∞, one can introduce a meromorphic pole function: P(q) =1 2 ∞ X n=0 1 q − mn + 1 q+ mn (20) with which 1 8π21a= Z dq 2πi d4_l (2π)4 P(q) fa(p, l, q), (21)

where the integration contour is illustrated as C1 in figure 1. This pole function has the following asymptotic behavior at |q| → ∞:

P(q) → A

q + B(Im(q)) + O(q

−2_), ₍₂₂₎

where (x) = x/|x|, and A and iB are real constants. With simple dimensional analysis, one easily finds that A and B are associated with logarithmic divergence and linear divergence, respectively. In particular, iB corresponds to the spectral density of the KK spectrum in the UV limit mn→ ∞, which is common to the generic 5D field 8(x, y) with a definite 4D spin and 4D chirality, i.e.8 = φ(x, y) or Aµ(x, y) or ψL,R(x, y) with γ5ψL_,R(x, y) = ±ψL_,R(x, y). As

Ais associated with the coefficient (12) of logarithmic divergence, we have A ∝ (z + z0), where

z, z0_{= ±1 are the orbifold parities of the associated 5D field at y = 0, π.}

One may regulate the 5D momentum integral (21) by introducing an appropriate set of 5D PV regulator fields and/or higher derivative regulating terms in the 5D action. However, as we eventually need to include the gauge boson loops, it is more convenient to use a dimensional regularization scheme in which

1 8π21a= Z _dD5_q 2πi dD4_l (2π)4 P(q) fa(p, l, q) = ca 8π2 A (4 − D4)+ 1 8π21 finite a , (23)

where ca is some group theory coefficient, and1_afiniteis finite in the limit D5→ 1 and D4→ 4. In this regularization scheme, the irrelevant linear divergence is simply thrown away, while the logarithmic divergence appears through 1/(D4_{− 4).}

After the integration over lµ, the remaining integration over q can be done by deforming the integration contour appropriately. For the one-loop corrections (16) induced by 5D scalar

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Im q Re q C1 Im q Re q C1 C2

Figure 1. Integration contours on the q-plane. Crosses along the real axis represent the KK masses {±mn} that correspond to the poles of P(q). The branch cut along the imaginary axis arises from Ga(p, q), and the contour C1 can be deformed to the contour C2for the integration involving Pfinite(q).

fields, we find 1 8π21a= Z C1 dD5_q 2πi P(q)Ga(p, q), (24) where Ga(p, q) = Tr(Ta2(φ)) 16π2 Z 1 0 dx(1 − 2x)2 2 4 − D4− ln(q 2_{+ x}_{(1 − x)p}2₎ . Since it depends on q2 logarithmically, Ga contains a branch cut in the complex plane of q, and we can take a branch cut line along the imaginary axis with q2_{+ x}_{(1 − x)p}2_{< 0. It is then} convenient to divide the Pole function into three pieces:

P(q) = A

q + B(Im q) + Pfinite(q), (25)

where

Pfinite(q) → O(q−2) for |q| → ∞.

One can then use the original contour C1 for the integration involving B(Im(q)), an infinitesimal circle around q = 0 for the integration involving A/q, and finally the contour deformed as C2 in figure 1for the integration involving Pfinite. Applying this procedure to the integral of the form

0X = Z C1 dD5_q 2πi  A q + B(Im(q)) + Pfinite X₀_{− ln(q}2+ X2₁), one obtains 0X|D5→1 = A X0− ln X 2 1 − 2iB|X1| + Z C2 dq 2πiPfinite(q) X0− ln(q2+ X12) = AX0− ln N (q)|_q=i|X₁_|, (26)

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where the N -function is defined as P(q) =1 2 d dq ln N(q), 1

2ln N(i|q|) → A ln |q| + iB|q| for |q| → ∞. (27)

We then find that the one-loop correction due to a complex 5D scalar field is given by 1 8π21a= Tr(T2 a(φ)) 16π2 2 A 3(4 − D4)− Z 1 0 dx(1 − 2x)2ln N(ipx(1 − x)p2) , showing that the model-parameter dependence of low energy couplings at p2

 m2KK is determined essentially by the behavior of N(q) in the limit q → 0. For a given 5D gauge or matter field, the corresponding N(q) can be uniquely determined, as will be discussed in appendixA. To complete the computation in dimensional regularization, one needs to subtract the 1/(4 − D4) pole to define the renormalized coupling. The subtraction procedure should take into account that dimensional regularization has been applied for the momentum integral defined in the 4D metric frame of ηµν, while the correct renormalized coupling should be defined in generic 5D metric frame as a quantity invariant under the 5D general coordinate transformation. The 1/(4 − D4) pole is associated with the renormalization of the fixed point gauge couplings, κa andκ_a0, in the action (1). For warped spacetime with

ds2= e2(y)ηµνdxµdxν+ R2dy2 (e(0)= 1),

the logarithmic divergence structure of (10) indicates that the correct procedure is to subtract 1/(4 − D4) with the counter term λa0ln(3) + λa_πln(3e(y=π)), which would yield

1a = (λa0+λa_π) ln 3 + λa_πln e(y=π) +1finite_a . (28) One can now apply the above prescription to the one-loop correction due to a 5D complex scalar fieldφ++ on a slice of AdS5:

ds2= e−2k|y|ηµνdxµdxν+ R2dy2.

Forφ++ with a 5D mass MS p, there is no zero mode, and we find

1a(φ++) = 1 6Tr(T 2 a(φ++)) ln 3 MS −1 2ln α2_{− 4} α2 − ln sinh(απk R) , where α = q

4 + M2S/k2_{. In fact, one can obtain the same result using the PV regularization} scheme in which 1a(φ++)|PV =1 3Tr(T 2 a(φ++)) ∞ X n=0 ln mn(8 PV ++) mn(φ++) , (29)

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where mn(8PV

++) is the KK spectrum of the PV regulator field 8PV++, which has a bulk mass3. In the limit n → ∞, mn(φ++) and mn(8PV

++) have the same asymptotic form mn → nπk/(eπk R− 1). We then have 1a(φ++)|PV =1 3Tr(T 2 a(φ++)) ∞ X n=0 ln 30 mn(φ++) − ln 30 mn(8PV₊₊) = 1a(φ++)|D R− 1a(8PV ++) D R =1 6Tr(T 2 a(φ++)) ln 3 MS −1 2ln α2 − 4 α2 − ln sinh(απk R) +3π R + O(1) ,

where 30 is an arbitrary mass parameter, the subscript DR means dimensional regularization, and the PV regulator mass is taken as 3 k, 1/R. As we have noted, the linearly divergent part of1a depends on the employed regularization scheme, and such a scheme dependence can be absorbed into the renormalization of the Wilsonian 5D gauge couplings. A constant piece of order unity in1ais also scheme dependent, and can be absorbed into the renormalization of the fixed-point gauge couplings. On the other hand, the terms depending on the model parameters

MS, k and R correspond to the calculable part of 1a, which should be scheme independent. The

above result confirms that the two regularization schemes, DR and PV, indeed give the same calculable part of1a.

3. Warped gauge thresholds

In this section, we discuss the one-loop gauge thresholds in generic 5D gauge theory on a slice of AdS5, where some of the bulk gauge symmetries are broken by orbifold boundary conditions and/or by bulk Higgs vacuum values. The effective action of the 4D gauge boson zero modes

Aa_µ(0)can be obtained by evaluating ei0[8cl]

= Z

D8qu eiSW[8cl+8qu], ₍₃₀₎

where 8cl denotes a background field configuration that includes Aa(0)

µ as well as the Higgs vacuum values, and 8qu stands for the quantum fluctuations of all gauge, matter and ghost fields in the model. To compute the one-loop effective action, we need the quadratic action of those quantum fluctuations. To derive the quadratic action of8qu, let us start with the Wilsonian action given by

SW = Sgauge+ Smatter+ Sgauge−fixing+ Sghost, (31)

where S_gauge_{= −} Z d5x√_−G 1 4 1 g_{5 A}2 + κA 4π2 δ(y) √ G₅₅+ κ0 A 4π2 δ(y − π) √ G₅₅ FAM NF_{M N}A , S_matter_{= −} Z d5x√_−G 1 2D M_φI_D_MφI + V(φ) + i ¯ψp(δpqγMDM+ MF pq(φ))ψq + _√δ(y) G₅₅ V0(φ) + 2iµpq(φ) ¯ψ p_ψq −δ(y − π)√ G₅₅ Vπ(φ) + 2i ˜µpq(φ) ¯ψ p_ψq

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for the 5D gauge fields AA

M, Dirac fermionsψ

p_{and real scalar fields}φI_{. Here S}

gauge−fixingis the gauge-fixing term and Sghostis the associated ghost action. We fix the background spacetime to be a slice of AdS5:

ds2= GM NdxMdxN = e−2k R|y|ηµνdxµdxν+ R2dy2 (0 6 y 6 π), and impose the Z2× Z20 orbifold boundary conditions:

AA_M(−y) = zAMAAM(y), AAM(−y + π) = z0AMAA_M(y + π),

φI_{(−y) = z}

IφI(y), φI(−y + π) = z0Iφ

I_{(y + π),}

(32) ψp_{(−y) = zp}_γ5ψp_{(y), ψ}p_{(−y + π) = z}0_pγ5ψp_{(y + π),}

where zA,I,p, z0A,I,p= ±1, µ= 1 and 5= −1. Here we ignore the boundary kinetic terms of matter fields since they are not relevant for the discussion of one-loop gauge couplings. As for the boundary scalar potentials V0 and Vπ, we assume for simplicity that they share (approximately) a common minimum with the bulk scalar potential V , and as a result the scalar field vacuum values are (approximately) constant along the 5th dimension:

hφIi = vI. (33)

Then there can be two independent sources of gauge symmetry breaking, one is the bulk Higgs vacuum valuesvI and the other is the orbifold boundary conditions imposed on the gauge fields. Let us now set up the notations. In the following, Aσ_M denotes the 5D gauge fields not receiving a mass from the Higgs vacuum values vI_{, B}α

M are the other gauge fields that obtain a nonzero 5D mass, πα are the associated Goldstone bosons and, finally,ϕi _{are the real-valued} physical scalar field fluctuations in the non-Goldstone direction. These gauge and scalar field fluctuations have the following form of the kinetic and mass terms:

1 g2 5 A FAM NF_{M N}A ₌ 1 g2 5σ Fσ M NF_{M N}σ + 1 g2 5α BαM NB_{M N}α , DMφIDMφI = ∂Mϕi∂Mϕi+∂Mπα∂Mπα+λ2_αBMαBαM+ · · · , (34) V(φ) = hV i +1₂M2 Si jϕ iϕj + · · · , V0(φ) = hV0i + mi jϕiϕj_{+ · · · ,} V_π(φ) = hV_π_{i + ˜}mi jϕiϕj_{+ · · · ,}

where F_{M N}σ and B_{M N}α are the field strength tensors of Aσ_M and Bα_M, respectively. Here, each 5D field can have arbitrary orbifold parities, and then Z2× Z20 symmetry implies that the mass matrices take the form

M_{Si j}2 _{= M}_{Si j}2 z_{i j}z_{i j}0 (y), MF pq = MF pq¯zpq¯z0pq(y),

mi j= mi jδzizj, m˜i j = ˜mi jδz0iz0j, (35)

µpq = µpqδ¯zpzq, µpq˜ = ˜µpqδ¯z0pz0q,

where M2

Si j and MF pq are constants, zi j= zizj, z0i j= z0iz0j, etc, ¯z = −z, ¯z0= −z0, and the kink functionzz0(y) is defined as

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Note that MSi j, MF pq, mi j and ˜mi j have the mass dimension one, while µpq and ˜µpq are dimensionless parameters. For generic forms of mass matrices, there can be nonzero mass mixing between matter fields with different orbifold parities. Our aim is to compute the one-loop gauge couplings as a function of the mass parameters and of the orbifold parities defined above.

As we are going to compute the low-energy effective action of Aa_µ(0)(x), we regard all 5D gauge fields as quantum fluctuations around a background configuration of Aa(0)

µ (x), which originate from Aσ_M|σ=a with the orbifold parity z = z0_{= 1. To proceed, we choose the following} form of the gauge fixing terms:

S_{gauge−fixing}_{= −} Z d5x√_−G " 1 2g2_5σ e2k R|y|ηµνD_µAσ_ν+ 1 R2e

2k R|y|_∂y_(e−2k R|y|_Aσ 5) 2 + 1 2g₅2_α e2k R|y|ηµνD_µB_να+ 1 R2e 2k R|y|_∂ y(e−2k R|y|B₅α) − g₅2_αλαπα 2# , (36)

where D_µ= ∂µ− iAaµ(0)Ta is the covariant derivative involving the background gauge boson zero modes. The corresponding ghost action is given by

Sghost= Z d5x√_−G e2k R|y|¯cσAD 2_cσ A+ e2k R|y| R2 ¯c

σ_A(∂y_e−2k R|y|_∂y_cσ_A)

+ e2k R|y|¯cαBD 2_cα B+ e2k R|y| R2 ¯c α

B(∂ye−2k R|y|∂ycαB) − g 2

5αλ2α¯cαBcαB+ · · ·

, (37)

where cσ_A and cα_B are the ghost fields for Aσ_M and Bi

M, respectively, and D2= ηµνDµDν.

In the model under consideration, there are three classes of field fluctuations, each of which can have arbitrary orbifold parities: (i) 5D gauge fields Aσ_Mthat do not get a mass from the Higgs vacuum valuesvI_{, and the associated ghost fields c}σ

A, (ii) 5D gauge fields BMα that get a nonzero 5D mass MV_α= g5αλα fromvI_{, and the associated Goldstone bosons and ghost fields,}πα _and

cα_B, and (iii) 5D Dirac fermions ψp _{and the physical scalar fields} ϕi_{. After the following field} redefinition: Aσ_M _→_√1 R g₅_σAσ_M, Bα_M_→ _√1 R g₅_αBαM, πα_→_√1 Rπ α_, (38) ϕi → √1 Rϕ i_{, ψ}p → √1 Rψ p_{, c}σ A→ 1 √ R cσA, cα_B _→_√1 R cαB,

we find that each class of field fluctuations has the quadratic action:

S₂₌ Z d4xdy(LA+ LB+ LM) , (39) where LA = − 1 2η µν_Aσ µ1Aσν− e−2k R|y| 2R2 (∂yA σ µ)2− e−2k R|y| 2 A σ 51Aσ5 − e−4k R|y| 2R2 (∂yA σ 5) 2 −1 2 e−4k R|y| R −4k 2 + 4k(δ(y) − δ(y − π) (Aσ₅)2 −e−2k R|y|¯cσA1cσA+

e−2k R|y|

R2 ¯c

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LB _{= −}1 2η µν_Bα µ1Bνα− e−2k R|y| 2R2 (∂yB α µ)2− e−2k R|y| 2 g 2 5αλ 2 α(Bµα)2 −1 2e −2k R|y| _πα_1πα_{+ B}α 51B5α − e−4k R|y| 2R2 (∂yπ α₎2₊_(∂y_Bα 5) 2 −1 2e −4k R|y| _πα _Bα 5 g 2 5αλ2α −2kg5αλα−−(y) −2kg5αλα−−(y) g52αλ2α− 4k2 ! πα B₅α −e −4k R|y| R 2k(δ(y) − δ(y − π))(B α 5) 2 −e−2k R|y|¯cαB1cαB+ e−2k R|y| R2 ¯c α

B(∂ye−2k R|y|∂ycαB) − e−4k R|y|g 2 5αλ2α¯cαBcαB, LM _{= −}1 2e −2k R|y|_ϕi_1ϕi −e −4k R|y| 2R2 (∂yϕ i₎2 −e −4k R|y| 2 M 2 Si jϕ i_ϕj −e −4k R|y| R mi jδ(y) − ˜mi jδ(y − π) ϕ i_ϕj − ie−3k R|y|ψ¯pγµDµψp −ie −2k R|y| R ψ¯ p_γ5(∂ ye−2k R|y|ψp) − ie−4k R|y|MF pqψ¯ pψq −ie −4k R|y| R 2µpqδ(y) − 2 ˜µpqδ(y − π) ¯ψ p_ψq_.

Here the gauge-covariant operator1 is defined as 18 = −ηµν_D

µDν+ Fµν(0)Jjµν

8, (40)

where F_µν(0)_{= F}a(0)

µν Ta is the field strength of the gauge boson zero modes Aaµ(0), and J_jµν is the 4D Lorentz generator for a field with 4D spin j , which is normalized as tr(J_jµνJ_jρλ) = C( j)(ηµρηνλ_{− η}µληνρ), where C( j) = (0, 1/2, 2) for j = (0, 1/2, 1). Here and in the following,8(x, y) stands for a 5D field that has a definite value of 4D spin j and also of 4D chirality, e.g.8 = A_µ(x, y) with j = 1, 8 = ψL_,R(x, y) with j = 1/2 and γ5ψL_,R = ±ψL,R, 8 = A5(x, y) or ϕ(x, y) or cA(x, y) with j = 0. Note that the AdS curvature k generates a mixing between B₅α andπα in the quadratic action. Since the Goldstone bosonπαhas the same orbifold parity as B_µα, this is a mixing between 4D scalar fields with opposite orbifold parities.

With the quadratic action (39), the one-loop effective action of the gauge boson zero modes is given by 0[A(0) µ ] = −_4gπ R2 5a Z d4x F_µνa(0)Fa(0)µν+ i 2 TrA_µln ˜1A_µ+ TrA5ln ˜1A5− 2 TrcAln ˜1cA +i 2 TrB_µln ˜1B_µ+ TrB5,πln ˜1B5,π− 2 TrcB ln ˜1cB +i 2 Tr_ϕln ˜1ϕ− TrψL ln ˜1ψL− TrψR ln ˜1ψR , (41) where Tr₈ln ˜1₈= X n Tr₈_n ln 1 + m2_n(8) = Z dD5_q 2πi P8(q)Tr ln 1 + q 2 . (42)

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Here8n denotes the nth KK modes with the mass eigenvalue mn(8): 8(x, y) =X

n

8n(x) fn(y), (43)

and in the last step, we have applied the Pole function technique discussed in the previous section: P₈(q) =1 2 X n 1 q − mn(8)+ 1 q+ mn(8) , (44)

where the summation also includes the zero modes.

It is straightforward to perform the integration over 4D loop momentum with dimensional regularization. We then find that

Tr ln 1 + q2 = i Z d4p (2π)4G 8 a(q, p)A a µ(−p)(p2ηµν− pµpν)Aaν(p) + · · · , where G_a8(q, p) = 1 8π2Tr(Ta(8) 2₎ Z 1 0 dx 1 2d( j8)(1 − 2x) 2_{− 2C( j8)} × 2 4 − D4 + ln(4πe −γ_{) − ln(q}2_{+ x}_{(1 − x)p}2₎ . (45)

Here d( j8) = (1, 2, 2, 4) and C( j8) = (0, 1/2, 1/2, 2) for j8 = (0, 1/2L, 1/2R, 1) denoting the 4D spin and chirality8. The one-loop correction induced by 8 can be expressed as

1 8π21 8 a(p) = −1 2 j8 Z dD5_q 2πi P8(q)G 8 a(q, p), (46)

where the dependence on the 4D spin and unbroken gauge charges of 8 is encoded in G_a8, whereas the dependence on various mass parameters is encoded in the pole function P₈, which contains full information on the KK spectrum. As explained in section 2, we can deform the integration contour appropriately to simplify the integration over q (see figure 1). Then, following the method discussed in the previous section, we find that

1 8π21 8 a(p) = −12 j8 8π2 Tr(T 2 a(8)) 1 6d( j8) − 2C( j8) _A 8 4 − D4+ O(1) + Z 1 0 dx C( j₈) −1 4d( j8)(1 − 2x) 2 ln N8(ipx(1 − x)p2) , (47)

where the Pole function has the following asymptotic behavior at |q| → ∞:

P₈(q) = A8

q + B8(Im q) + O q

−2

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and N8 is a holomorphic even function defined as N8_{= C8}Y n m2n(8) − q2 P₈(q) =1 2 d dqln N 8_(q)_, 1 2ln N 8_{(i|q|) = A} 8ln |q| + iB8|q| + O(|q|−1) at |q| → ∞. (49) Since A₈/(4 − D4) is associated with the logarithmic divergence of the fixed point gauge couplings, we have A₈∝ (z + z0), where z and z0 are the orbifold parities of8. (See equations (10) and (12).) In our convention, for8(x, y) = {φ, ψL, ψR, Aµ}, we have

A₈₌ z+ z

0

4 , iB8=

eπk R_{− 1}

2k , (50)

where 8(−y) = z8(y) and 8(−y + π) = z08(y + π). Note that here φ can be a 5D scalar, or the 5th component of a 5D vector, or a ghost field. Also a 5D Dirac fermion ψ with orbifold parities z and z0 _{consists of} ψL _{with orbifold parities z and z}0 _and ψR _{with orbifold} parities ¯z = −z and ¯z0_{= −z}0_{, and thus A}

ψ = AψL+ AψR = 0. As was noted in the previous

section, in warped spacetime, the renormalized fixed point gauge couplings at the cutoff scale 3 are obtained by subtracting the pole divergence (z + z0)/(4 − D4) with a counterterm proportional to

δ(y)z ln 3 + δ(y − π)z0_ln_(e−kπ R_3).

We are now ready to present the one-loop corrections to low-energy gauge couplings, induced by generic 5D fields on a slice of AdS5. For this, let Nzz80 denote the N -function of8(x, y) having

a definite value of 4D spin j₈, of 4D chirality and of orbifold parities z and z0_{. Explicit forms} of N_zz80(q) and their limiting behaviors at |q| → 0, ∞ for 8’s with generic bulk and boundary

masses are presented in appendix A. Also let {8} denote a set of 8’s having the same j8 and unbroken gauge charges, but not necessarily the same orbifold parities, which generically have a mixing with each other in the quadratic action (39) of quantum fluctuations, and let N{8}_denote the N -function of this set of8’s. Then the full one-loop corrections are summarized as

1 8π21a= 1 8π2 1{A} a +1{B}a +1{ψ L} a +1{ψ R} a +1{ϕ}a , (51) where 1{A} a = 1{Aa µ}(p) + 1a{A5}(p) − 21{caA}(p), 1{B} a = 1{Ba µ}(p) + 1{B 5,π} a (p) − 21{c B} a (p) (52) for1{8}_a (p) given by 1{8} a (p) = a( j8) Tr(T 2 a(8)) n{8}₀ _{ln p −}1 2 n{8}₊₊ _{− n}{8}₋₋ln3 +1 2ln N {8} 0 + 1 4(n {8} ++ − n {8} +− + n{8}−+ − n{8}−−)πk R + O (1) . (53)

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Here n{8}₀ denotes the number of zero modes in {8}, n{8}zz0 is the number of 8’s with orbifold

parities z, z0_{defined as}

8(−y) = z8(y), 8(−y + π) = z0_{8(y + π),} and a( j8) = −1 6, − 2 3, 10 3 for j₈= 0,1 2, 1 , (54) N{8}(q) = (−q2)n{8}0 N₀{8}+ O(q2/m2_{K K}) ,

where mKK denotes the lightest KK mass of {8}. The above result shows that the model-parameter dependence of one-loop gauge couplings is determined mostly by the N -functions near q = 0, i.e. by N0{8}.

The one-loop corrections induced by 5D Dirac fermions {ψp

} take a simpler form. As the equation of motion forψ involves γ5, it is convenient to split eachψ into two chiral fermions: ψ = ψL+ψRwithγ5ψL_,R_{= ±ψL}_,R, and then we always have

n{ψL} zz0 = n{ψ R} ¯z ¯z0 , n{ψ L} 0 − n {ψR} 0 = n{ψ++L}− n{ψ++R}, (55)

regardless of the bulk and boundary fermion masses MF pq, µpq and ˜µpq. (If there is no mass mixing betweenψp _{with different orbifold parities, n}

0L = nψ++ and n0R = nψ−−.) Since {ψL} and

{ψR} have the same KK mass spectrum, we also have

N{ψL}(q) = (−q2)n{ψL}0 −n {ψR} 0 N{ψR}(q) ₍₅₆₎ and thus N{ψ}(q) = N{ψL}(q)N{ψR}(q) = (−q2)n{ψL}0 +n {ψR} 0 N{ψL} 0 2 + O(q2/m2_KK) , (57) where N{ψL}(q) = (−q2)n{ψL}0 N{ψL} 0 + O(q 2_/m2 KK)

in the limit |q| mKK. It is then found that the one-loop gauge couplings induced by {ψ} are given by 1 8π21 {ψ} a (p) = 1 8π2 1 {ψL} a (p) + 1{ψa R}(p) = − 1 12π2Tr(T 2 a(ψ)) h n{ψL} 0 + n {ψR} 0 ln p + ln N{ψL} 0 + O(1)i . (58) In the above, the external momentum p of the gauge boson zero modes is assumed to be smaller than the lowest KK mass, justifying the use of the N -function at q → 0. However, in a certain parameter limit, there might be KK states having a particularly light mass. For instance, the lightest KK state of a Dirac fermion ψ+− with bulk mass MF > k has a 4D mass

mψ_KK_{∼ k e}−(k+2MF)π R/2_{, which can be much smaller than 1 TeV even when k e}−kπ R_{& O}(1) TeV.

In such a case, one needs to consider the gauge couplings at p> mψ_KK, which can be easily obtained from (53). To see this, let us consider the case with

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in which there are n8₀ + n light modes with a mass smaller than p. One can then consider the

N-function at m2

n < q2< m 2

n+1, which can be expressed as

N8(q) = (−q2)n80 n Y l=1 (m2 l − q 2₎ ! N_n8+ O(q2/m2_n+1) , (60) where N_n8_{= N}₀8 ._Yn l=1 m2_l, (61)

and find that the one-loop gauge couplings at mn< p < mn+1are given by 1 8π21 8 a(p) = a( j₈) 8π2 Tr(T 2 a(8)) n8₀ + n ln p −1 2 n 8 ++− n8−− ln3 + 1 2ln N 8 n +1 4(n 8 ++− n8+−+ n8−+− n8−−)πk R + O (1) . (62)

As the N -functions play a crucial role in our analysis, let us discuss some relevant features of

N{8} _{here. More complete discussions will be given in appendix} _A_{. First, for A}σ

M = (Aσµ, Aσ5) and cA with orbifold parities z and z0_{, which do not get a mass from the Higgs vacuum values,} we have NA σ µ z_σz_0σ(q) = N cσ_A z_σz_0σ(q) = (−q 2₎(z_σ+z0 σ)/2_NAσ5 ¯zσ¯z0σ(q), (63)

where ¯zσ = −zσ and ¯z0σ = −zσ0. This relation simply means that Aσµ, Aσ5 and cσA have the same KK mass spectra, which explains the form of1{A}_a in (52). Here, the factor qzσ+z0

σ _{represents the}

zero mode of Aσ_µ with z_σ = z0_σ = 1 or of Aσ5 with zσ = zσ0 = −1. In the quadratic action (39),

Aσ_M does not have any mixing with other fields and therefore

N{Aµ}₌Y σ NA σ µ z_σz_0σ, N{A5}= Y σ NAσ5 ¯zσ¯z0σ, N{cA}= Y σ NcσA z_σz_0σ. (64)

On the other hand, for BMα = (Bµα, B5α), which obtain a nonzero mass from the Higgs vacuum values, and the associated Goldstone and ghost fields πα, cα_B, there is a mass mixing between

B₅αandπα, which have opposite parities. We still have

N_zBµα αz_0α(q) = N cα_B z_αz_0α(q), N{B α 5,πα}(q) = NB α µ z_αz_0α(q)N ˜Bα µ ¯zα¯z_0α(q), (65)

where ˜B_µα is an artificial vector field that has the same bulk mass as B_µα and also the boundary masses given by − Z d4xdye −2k R|y| R 2kδ(y) − 2kδ(y − π)ηµν˜B_µα˜B_να. We then find that

N{Bµ}₌Y α N_zBµα αz_0α, N{cB}= Y α NcαB z_αz_0α, N{B5,π}(q) =Y α N_zBµα αz_0α(q)N ˜Bα µ ¯zα¯z0α(q) . (66)

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In the presence of mixing between fields with different orbifold parities, N{ψ} _{and N}{ϕ} generically take a highly complicated form. Here we present the results for relatively simple cases: (i) two Dirac fermions with generic bulk and boundary masses and (ii) two scalar fields with just bulk masses, leaving the discussion of the more general case to appendixA. Let us first consider the case of two Dirac fermionsψ_zp

pz0p ( p = 1, 2) with the following bulk and boundary

masses:

MF pq, µ12, µ12._˜ (67)

Note that µpp= ˜µpp= 0, and the Dirac fermion ψ p

zpz0p consists of ψ

p

L with orbifold parities

zp and z0p and ψ p

R with orbifold parities ¯zp= −zp and ¯z0p= −z0p. In the fundamental domain 0< y < π, the 2 × 2 bulk mass matrix can be described by two mass eigenvalues MF p ( p = 1, 2) and a mixing angle θF:

U MF11 MF12 M_F21 M_F22 U†₌ MF1 0 0 M_F2 , U = cosθF sinθF − sin θF cosθF . (68)

Let N_zzψL,R0 (M) denote the N -function ofψL,Rwith orbifold parities z and z0 and a bulk mass M.

We then find that the N -function of the above two Dirac fermions is given by

N{ψ1,ψ2}(q) = N{ψL1,ψL2}(q)N{ψR1,ψR2}(q) = (−q2)−(z1+z01+z2+z02)/2 N{ψL1,ψL2}(q) 2 , (69) where N{ψL1,ψ2L}(q) = c0c_π∗N_zψ₁L_z(M0 F1) 1 + s0s ∗ πNzψ1Lz01(MF2) s₀∗s_πNψL(MF1) z2z02 + c ∗ 0cπN ψL(MF2) z2z02 − c0sπN_zψ₁L_z(M0 F1) 2 − s0 c_πNψL(MF2) z1z0₂ s₀∗c∗_πNψL(MF1) z2z0₁ − c ∗ 0s ∗ πNzψ2Lz0₁(MF2) (70) for c₀₌cosθ_pF− z1µ12sinθF 1 + |µ12|2 , cπ = cosθF− z0₁µ12˜ sinθF p 1 + | ˜µ12|2 , s₀ ₌sinθ_pF+ z1µ12cosθF 1 + |µ12|2 , sπ = sinθF+ z0₁µ12˜ cosθF p 1 + | ˜µ12|2 . (71)

Note that this N -function takes a factorized form, N{ψL1,ψL2} _{= N}ψL(MF1)

zz0 Nψ

L(MF2)

zz0 , ifψ1andψ2

have the same orbifold parities. One can similarly obtain the N -function of a two-scalar-field system {ϕi

ziz0i} (i = 1, 2) with bulk masses M

2

Si j only. Again, M2Si j can be described by two mass-square eigenvalues M_Si2 (i = 1, 2) and a mixing angle θS. Then the N -function of {ϕi} is given by N{ϕ1,ϕ2}(q) = c2Nϕ(MS1) z1z01 + s 2_Nϕ(MS2) z1z01 s2Nϕ(MS1) z2z02 + c 2_Nϕ(MS2) z2z02 −c2s2 Nϕ(MS1) z1z02 − N ϕ(MS2) z1z02 Nϕ(MS1) z2z01 − N ϕ(MS2) z2z01 , (72)

where c = cos θS, s = sin θS and N_zzϕ(M)0 is the N -function of a 5D scalar with orbifold parities z

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Table 1. One-loop gauge couplings induced by 5D vector fields Aσ_M in the limit p mKK where mKK is the lowest nonzero KK mass. Here ca(A) = Tr(T2 a(AM)). (zz0₎ ₁{A} a (++) ca(A) 12 [22πk R − 23 ln(3π R) + 44 ln(pπ R)] (+−) ca(A) 12 (−22πk R) (−+) ca(A) 12 (22πk R) (−−) ca(A) 12 21 ln sinhπk R πk R − πk R + 23 ln(3π R) − 2 ln(pπ R)

Table 2. One-loop gauge couplings induced by the 5D vector fields Bα_M and Goldstone bosons πα for the range of MB, that does not give any zero mode lighter than p. Here, ca(B) = Tr(T_a2(BM)), MB = g5αλαandαB =

p

1 + M2_B/k2_, where MB is the canonical 5D mass of BαM.

(zz0₎ ₁{B} a (++) ca(B) 12 20 ln sinhαBπk R αBπk R + 42 ln(MBπ R) − 22 ln (3π R) (+−) ca(B) 12 20 ln α BcoshαBπk R − sinh αBπk R αB (−+) ca(B) 12 20 ln α BcoshαBπk R + sinh αBπk R αB (−−) ca(B) 12 20 ln sinhαBπk R αBπk R − 2 ln(MBπ R) + 22 ln(3π R)

explicit expression of N_zz80 for8 with various 4D spin and orbifold parities, as well as its limiting

behaviors at |q| → 0, ∞. Once the N -functions are obtained, one can examine the behavior at

q → 0 to find N0{8} and finally apply (53) to obtain the one-loop corrections 1a. Using the properties of N -functions described above and also in appendix A, the expressions of 1{A}

a and 1{B}

a presented in tables1 and2, respectively are founded. (See (52) for the definition of 1{A},{B}

a .) For the one-loop corrections1{ϕ},{ψ}a induced by scalar and fermion fields, we consider two cases: the case that there is no mixing between matter fields with different orbifold parities and the case that two scalars or two Dirac fermions can have such a mixing. For the first case, one can simply consider a single scalar or a single fermion with definite orbifold parities, and the results are summarized in tables 3and4. For the second case, one can use the N -functions (70) and (72) to obtain the results presented in tables5and6. A prescription for1{ϕ},{ψ}

a in the

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Table 3.One-loop gauge couplings induced by a 5D real scalarϕ with definite orbifold parities z and z0_{. Here ca}(ϕ) = Tr(T2

a(ϕ)), α = q

4 + M2

S/k2, where MS,

mSand ˜mS are the bulk and boundary masses ofϕ. ϕ(0)denotes a particular type of 5D scalar field with (zz0) = (++), mS = ˜mS and M2_S= mS(mS− 4k), which has a zero mode lighter than p.

(zz0₎ ₁{ϕ} a (++) −1 12ca(ϕ (0)₎_ln sinh(mS− k)π R (mS− k)π R +πk R − ln(3π R) + 2 ln(pπ R) −1 12ca(ϕ) ln αk(m S− ˜mS) cosh απk R + (α2k2− (2k − mS)(2k − ˜mS)) sinh απk R αk − ln 3 (+−) −ca(ϕ) 12 ln αk cosh απk R − (2k − m S) sinh απk R αk (−+) −ca(ϕ) 12 ln αk cosh απk R + (2k − ˜m S) sinh απk R αk (−−) −ca(ϕ) 12 ln sinhαπk R απk R + ln(3π R)

Table 4.One-loop gauge couplings induced by a Dirac fermionψ with definite orbifold parities. Here ca(ψ) = Tr(T_a2(ψ)) and MF is the bulk mass ofψ.

(zz0₎ ₁{ψ} a (++) −2 3ca(ψ) ln sinh(MF− k/2)π R (MF− k/2)π R +1 2πk R + ln(pπ R) (+−) +2 3ca(ψ) MFπ R (−+) −2 3ca(ψ) MFπ R (−−) −2 3ca(ψ) ln sinh(MF+ k/2)π R (MF+ k/2)π R +1 2πk R + ln(pπ R) 4. Conclusion

Models with warped extra dimension might provide an explanation for various puzzles in particle physics, e.g. the weak scale to Planck scale hierarchy and the Yukawa coupling hierarchy, while implementing a breaking of unified gauge symmetry in bulk spacetime by boundary conditions, which would solve some of the naturalness problems in GUTs such as the doublet–triplet splitting problem. KK threshold corrections in such models are generically enhanced by the logarithm of an exponentially small warp factor and therefore can be crucial for successful gauge coupling unification in the framework of the warped unified model. In this paper, we discuss a novel method to compute one-loop gauge couplings in generic 5D gauge

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Table 5. _{One-loop corrections induced by two real scalars {ϕ}_z1

1z01, ϕ

2

z2z02}, which

have the same gauge charge, but can have different orbifold parities. Here αi =

q

4 + M_Si2/k2 _{(i = 1, 2) for the bulk mass eigenvalues MSi}_{, and s ≡ sin θS,}

c ≡ cos θS for the mixing angle θS. We are considering a generic bulk mass matrix that does not give any zero mode lighter than p, whereas the boundary masses are assumed to be zero for simplicity.

(z1z01) (z2z02) 1{ϕ1_,ϕ2_} a (++) (++) −ca(ϕ) 12 ln sinhα1πk R α1πk R sinhα2πk R α2πk R + 2 ln(MS1π R)(MS2π R) − 2 ln(3π R) (++) −ca(ϕ) 12 ln c2 (M S1π R)2sinhα1πk R α1πk R α 2coshα2πk R − 2 sinh α2πk R α2 (+−) _+s2 (M S2π R)2sinhα2πk R α2πk R α 1coshα1πk R − 2 sinh α1πk R α1 − ln(3π R) (++) −ca(ϕ) 12 ln c2 (M S1π R)2sinhα1πk R α1πk R α 2coshα2πk R + 2 sinh α2πk R α2 (−+) +s2 (M S2π R)2sinhα2πk R α2πk R α 1coshα1πk + 2 sinh α1πk R α1 − ln(3π R) (++) (−−) −ca(ϕ) 12 ln ((α1c2+α2s2)2− 4) sinhα1πk R α1 sinhα2πk R α2 + 4c2s2sinh2 (α1− α2)πk R 2 (+−) (+−) −ca(ϕ) 12 ln α 1coshα1πk R − 2 sinh α1πk R α1 + ln α 2coshα2πk R − 2 sinh α2πk R α2 −ca(ϕ) 12 ln α 1coshα1πk R − 2 sinh α1πk R α1 α 2coshα2πk R + 2 sinh α2πk R α2 (+−) (−+) +4 s2 α

2sinhα1πk R cosh α2πk R − α1sinhα2πk R cosh α1πk R

α1α2 +(α1− α2)2c2s2 sinhα1πk R α1 sinhα2πk R α2 − 4c2s2sinh2(α1− α2)πk R 2 (+−) −ca(ϕ) 12 ln c2 α 1coshα1πk R − 2 sinh α1πk R α1 sinhα2πk R α2πk R (−−) +s2 α 2coshα2πk R − 2 sinh α2πk R α2 sinhα1πk R α1πk R + ln(3π R) (−+) (−+) −ca(ϕ) 12 ln α 1coshα1πk R + 2 sinh α1πk R α1 + ln α 2coshα2πk R + 2 sinh α2πk R α2 (−+) −ca(ϕ) 12 ln c2 α 1coshα1πk R + 2 sinh α1πk R α1 sinhα2πk R α2πk R (−−) _+s2 α 2coshα2πk R + 2 sinh α2πk R α2 sinhα1πk R α1πk R + ln(3π R) (−−) (−−) −ca(ϕ) 12 ln sinhα1πk R α1πk R sinhα2πk R α2πk R + 2 ln(3π R)

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Table 6. _{One-loop corrections induced by two Dirac fermions {ψ}_z1

1z01, ψ

2 z2z02}

that have the same gauge charge, but can have different orbifold parities. Here

MF p(p = 1, 2) are the bulk mass eigenvalues, and c0_,π and s0,π are defined in

(71) in terms of the bulk mixing angle and the boundary mass mixings. We are considering a generic parameter range in which all nonzero KK masses are heavier than p. (z1z01) (z2z02) 1{ψ1_,ψ2_} a (++) (++) − 2 3ca(ψ) ln sinh(MF1− k/2)π R (MF1− k/2)π R + ln sinh(MF2− k/2)π R (MF2− k/2)π R +πk R + 2 ln(pπ R) (++) − 2 3ca(ψ) ln |cπ|2 sinh_(M(MF1− k/2)π R F1− k/2)π R e−MF2π R (+−) + |sπ|2 sinh(MF2− k/2)π R (MF2− k/2)π R e−MF1π R +1 2πk R + ln(pπ R) (++) − 2 3ca(ψ) ln |c0|2 sinh(MF1− k/2)π R (MF1− k/2)π R eMF2π R (−+) + |s0|2 sinh(MF2− k/2)π R (MF2− k/2)π R eMF1π R +1 2πk R + ln(pπ R) (++) (−−) − 2 3ca(ψ) ln c0sπe −(MF1−MF2)π R/2_{− c} πs0e(MF1−MF2)π R/2 2 (+−) (+−) + 2 3ca(ψ)(MF1π R + MF2π R) (+−) (−+) − 2 3ca(ψ) ln c0c ∗ πe−(MF1−MF2)π R/2+ s0s_π∗e(MF1−MF2)π R/2 2 (+−) − 2 3ca(ψ) ln |c0|2 sinh(MF2+ k/2)π R (MF2+ k/2)π R e−MF1π R (−−) + |s0|2 sinh(MF1+ k/2)π R (MF1+ k/2)π R e−MF2π R +1 2πk R + ln(pπ R) (−+) (−+) − 2 3ca(ψ)(MF1π R + MF2π R) (−+) − 2 3ca(ψ) ln |cπ|2 sinh_(M(MF2+ k/2)π R F2+ k/2)π R eMF1π R (−−) + |sπ|2 sinh(MF1+ k/2)π R (MF1+ k/2)π R eMF2π R +1 2πk R + ln(pπ R) (−−) (−−) − 2 3ca(ψ) ln sinh(MF1+ k/2)π R (MF1+ k/2)π R + ln sinh(MF2+ k/2)π R (MF2+ k/2)π R +πk R + 2 ln(pπ R)

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theory on a slice of AdS5, in which some of the bulk gauge symmetries are broken by orbifold boundary conditions and/or by bulk Higgs vacuum values, and also there can be nonzero mass mixings between the bulk fields with different orbifold parities. Explicit expressions of the KK thresholds as a function of various model parameters are derived, and our analysis can cover most of the warped GUT models that have been discussed so far in the literature.

Acknowledgments

KC and CSS were supported by KRF grants funded by the Korean Government (KRF-2007-341-C00010 and KRF-2008-314-C00064) and a KOSEF grant funded by the Korean Government (2009-0080844). IWK was supported by the US Department of Energy through grant no. DE-FG02-95ER40896.

Appendix A. The N-function

In this appendix, we discuss the N -function of the 5D field8 on a slice of AdS5, which has a definite value of 4D spin and 4D chirality as well as definite orbifold parities:

8(x, y) = {φ, e−2k R|y|_ψ

L, e−2k R|y|ψR, Aµ} (ψL,R = 1₂(1 ± γ5)ψ),

where the 4D scalar φ might be a 5D scalar, or the 5th component of a 5D vector, or a ghost field. A generic 5D field8 on a slice of AdS5can be decomposed as

8(x, y) =X 8n(x) fn(y), (A.1)

where the KK wavefunction fn satisfies −esk R|y|_∂y _e−sk R|y|_∂y_{+ R}2_M2

8 fn = R2e2k R|y|m2nfn (A.2)

for the KK mass eigenvalue mn. Here

M₈2 ₌ MS, MF2 (MF+ k) , MF(MF− k) , M_V2 ,

s = {4, 1, 1, 2} for 8 = {φ, e−2k R|y|ψL, e−2k R|y|ψR, Aµ}, (A.3) where MS, MF and MV denote the bulk masses of φ, ψ and A_µ, respectively. Here, we are using the mass parameter convention defined in (34) and (35), e.g. MS= 0 for φ = Aσ5 or cσA,

MS= g5αλαforφ = cαB, MV = 0 for Aµ= Aσµand MV = g5αλα for Aµ= Bµα. The generic solution of the above KK equation is given by

fn(y) = esk R|y|/2 h Aα(mn)Jα m_n k e k R|y|_{+ B}_α(mn_)Yαmn k e k R|y|i , _(A.4) where α = p(s/2)2_{+ M}2

8/k2, and Aα and Bα are determined by the boundary conditions at

y = 0 and π. To utilize those boundary conditions, it is convenient to introduce the following

functions: fJ0−(q) = Jα q k , fJ0+(q) = h r₀₋s 2 J_α q k −q kJ 0 α q ki , fY0−(q) = Yα q k , fY0+(q) = h r0− s 2 Y_α q k −q kY 0 α q ki , fJ_π−(q) = Jα q T , fJπ+(q) = hs 2− rπ J_αq T + q T J 0 α q Ti , fY_π−(q) = Yα q T , fYπ+(q) = hs 2− rπ Y_α q T + q TY 0 α q Ti , (A.5)