S Parameter in the Holographic Walking/Conformal Technicolor

# S Parameter in the Holographic Walking/Conformal Technicolor

Kazumoto Haba Department of Physics, Nagoya University, Nagoya, 464-8602, Japan.    Shinya Matsuzaki Department of Physics and Astronomy, University of North Carolina, Chapel Hill 27599-3255.    Koichi Yamawaki Department of Physics, Nagoya University, Nagoya, 464-8602, Japan.
###### Abstract

We explicitly calculate the parameter in entire parameter space of the holographic walking/conformal technicolor (W/C TC), based on the deformation of the holographic QCD by varying the anomalous dimension from through continuously. The parameter is given as a positive monotonic function of which is fairly insensitive to and continuously vanishes as when , where is the vacuum expectation value of the bulk scalar field at the infrared boundary of the 5th dimension and is related to the mass of (techni-) meson () and the decay constant () as for . However, although is related to the techni-fermion condensate , we find no particular suppression of and hence of due to large , based on the correct identification of the renormalization-point dependence of in contrast to the literature. Then we argue possible behaviors of as near the conformal window characterized by the Banks-Zaks infrared fixed point in more explicit dynamics with . It is a curious coincidence that the result from ladder Schwinger-Dyson and Bethe-Salpeter equations well fits in the parameter space obtained in this paper. When is realized, the holography suggests a novel possibility that vanishes much faster than the dynamical mass does.

thanks: haba@eken.phys.nagoya-u.ac.jpthanks: synya@physics.unc.eduthanks: yamawaki@eken.phys.nagoya-u.ac.jp

## I Introduction

The origin of the electroweak symmetry breaking is the most urgent issue to be resolved at the LHC experiments. In the standard model, Higgs boson is introduced just as a phenomenological input only for the sake of making particles acquire masses. As such the standard model does not explain the origin of the electroweak symmetry breaking. With the existence of Higgs boson as an elementary particle, moreover, one necessarily faces with some problems such as naturalness, etc.

One of the candidates which resolve these problems is the technicolor (TC) model Weinberg:1975gm (). In the framework of TC, the origin of the electroweak symmetry breaking is explained dynamically without introduction of Higgs boson. The simplest model of TC, just a simple scale-up of QCD, however, does not pass the electroweak precision test, especially, suffers from a large contribution of to the Peskin-Takeuchi parameter Peskin:1990zt+ (), while the electroweak precision test shows that the value of the parameter is less than about 0.1.

There is an interesting possibility Appelquist:1991is (); Harada:2005ru () that contributions to the parameter can be reduced in the case of the walking/conformal TC (W/C TC) Holdom:1984sk (); Yamawaki:1985zg (); Akiba:1985rr (); Appelquist:1986an (), initially dubbed as “scale-invariant TC” Yamawaki:1985zg (), with almost non-running (walking) gauge coupling near the conformal fixed point, which produces a large anomalous dimension of the techni-fermion condensate operator  Yamawaki:1985zg () (for reviews, see Miransky:vk ()). A salient feature of this theory is the appearance of a composite Higgs boson as a (massive) techni-dilaton Yamawaki:1985zg () associated with the spontaneous breaking of the scale invariance (as well as the explicit breaking due to the scale anomaly) . A typical example Appelquist:1996dq (); Miransky:1996pd () of such a W/C TC is based on the Banks-Zaks infrared fixed point Banks:1981nn () (BZ-IRFP) in the large QCD, QCD with many massless flavors (). Looking at the region , we note that when , and hence there exists a certain region (“conformal window”) such that , where is the critical coupling for the spontaneous chiral symmetry breaking and hence the chiral symmetry gets restored in this region. Here may be evaluated as in the ladder approximation, in which case we have  Appelquist:1996dq () #1#1#1 In the case of , this value is somewhat different from the lattice value Iwasaki:2003de ()  , but is consistent with more recent lattice results Appelquist:2007hu ().  #2#2#2 There is another possibility for the W/C TC with much less based on the higher TC representation Hong:2004td (), although explicit ETC model building would be somewhat involved.. Related to the conformal symmetry, this phase transition (“conformal phase transition”Miransky:1996pd ()) has unusual nature that the order parameter changes continuously but the spectrum does discontinuously at the phase transition point.

When it is applied to TC, we set slightly larger than (slightly outside of the conformal window), with the running coupling becoming larger than the critical coupling only in the infrared region, we have a condensate or the dynamical mass of the techni-fermion of the order of such an infrared scale which is much smaller than the intrinsic scale of the theory . Although the BZ-IRFP actually disappears due to decoupling of massive fermion at the scale of , the coupling is still walking due to the remnant of the BZ-IRFP in a wide region . Then the theory develops a large anomalous dimension and enhanced condensate at the scale of which is usually identified with the ETC scale  Appelquist:1996dq (); Miransky:1996pd (). Note that as and the mass of techni-dilaton, #3#3#3 This estimate Shuto:1989te () is based on the ladder Schwinger-Dyson (SD) equation for the gauged Nambu-Jona-Lasinio model which well simulates Appelquist:1996dq (); Miransky:1996pd () the conformal phase transition in the large QCD. Actually, the result is consistent with the straightforward calculation Harada:2003dc () of scalar bound state mass, , through coupled use of the SD equation and (homogeneous) Bethe-Salpeter (BS) equation in the ladder approximation. also vanishes to be degenerate with the Nambu-Goldstone (NG) boson, although there is no light spectrum in the conformal window as a characteristic feature of the conformal phase transition.

The W/C TC, however, has a calculability problem, since its non-perturbative dynamics is not QCD-like at all, and hence no simple scaling of QCD results would be available. The best thing we could do so far has been a straightforward calculation based on the SD equation and (inhomogeneous) BS equation in the ladder approximation Harada:2005ru (), which is however not a systematic approximation and is not very reliable in the quantitative sense.

Of a late fashion, based on the so-called AdS/CFT correspondence, a duality of the string in the anti-de Sitter space background-conformal field theory Maldacena:1997re (), holography gives us a new method which may resolve the calculability problem of strongly coupled gauge theories Arkani-Hamed:2000ds (): Use of the holographic correspondence enables us to calculate Green functions in a four-dimensional strongly coupled theory from a five-dimensional weakly coupled theory. For instance, QCD can be reformulated based on the holographic correspondence either in the bottom-up approach DaRold:2005zs (); Erlich:2005qh () or in the top-down approach Sakai:2004cn (). In both approaches we end up with the five-dimensional gauge theory for the flavor symmetry, whose infinite tower of Kaluza-Klein modes describe nicely a set of the massive vector/axialvector mesons as the gauge bosons of Hidden Local Symmetries (HLS) Bando:1984ej (); Bando:1985rf (); Harada:2003jx (), or equivalently as the Moose Georgi:1985hf (). Although a holographic description is valid only for large limit, several observables of QCD have been reproduced within 30 % errors in both approaches. Moreover, through the high-energy behavior of current correlators in operator product expansion, some consistency with the QCD has been confirmed in the bottom-up approach.

Recently several authors Hong:2006si (); Piai:2006hy () calculated the parameter in the W/C TC as an application of the above technique of bottom-up holographic QCD DaRold:2005zs (); Erlich:2005qh () to the holographic W/C TC: They made some deformation adjusting a profile of a 5-dimensional bulk scalar field which is related to the anomalous dimension of techni-fermion condensate . They claimed that when , the parameter for certain parameter choices is substantially reduced compared to that of the QCD-like theory with . It is not clear, however, how the non-trivial feature of the dynamics of walking/conformal theory contributes to that reduction, since they discuss only specific parameter choices relevant to specific TC models. Actually, it is not but () that is needed for realistic model building of W/C TC where should be regarded an idealized limit of from the side of .

In this paper, based on the holographic correspondence in the bottom-up approach, we calculate the parameter in the W/C TC, treating the anomalous dimension as a free parameter as , varying continuously from the QCD monitor value through the one of the W/C TC . We calculate as an explicit function of in the entire region of , where is the radius, is the vacuum expectation value of the bulk scalar field and the 5th dimension has both infrared cutoff and ultraviolet cutoff : . This is in contrast to the previous authors Hong:2006si (); Piai:2006hy () whose discussions correspond to specific values of the parameter and are restricted to the case of as the W/C TC. Since the realistic model building of W/C TC is not for but () , the analysis of Ref. Hong:2006si (); Piai:2006hy () could be a too much idealization, unless their result is continuously connected with the limit from the side of . Actually, it turns out that the analysis of Ref. Hong:2006si () is not continuously connected with the limit of our result and thus would not precisely correspond to the realistic situation of the W/C TC we are interested in.

Then we find that is a positive function of in accord with the previous authors Hong:2006si (); Agashe:2007mc (). We also find that both and are monotonically increasing functions of which is related to chiral condensate . Noting that with being the techni- meson mass, we have an expression of as a function of .

Most remarkably, we find that continuously goes to zero as if (See Fig. 3), which is also in accord with the previous author Piai:2006hy () discussing the case of . We also find that the result is fairly insensitive to the value of unless we have substantial reduction of or for larger : Writing , we find for and for (See Fig. 5). Although the result of (slight) decreasing tendency for fixed is not inconsistent with Ref. Hong:2006si (), their value for is somewhat smaller than ours with .

Our result roughly coincides with the well-known fact (see e.g. Ref. Harada:2003jx ()) that in QCD case is given by the meson dominance as , with experimentally, where is the decay constant of the meson (or, of the fictitious Nambu-Goldstone boson absorbed into the meson in the language of HLS) and the gauge coupling of HLS. However, our result is highly nontrivial, since the holography includes an infinite tower of the vector and axialvector meson poles not just the lowest pole contribution. Note that the contributions of vector mesons are opposite in sign to those of the axialvector mesons and therefore the infinite sum of all contributions could in principle result in any functional form such as giving a non-vanishing constant in the limit .

It thus opens a novel possibility for having small parameter, if we find a mechanism of suppressing particularly near the conformal phase transition . Then the next issue is whether or not we can realize in the W/C TC.

It is also to be noted that the above continuous vanishing of in the case that is highly nontrivial in sharp contrast to the usual perturbative calculation where the parameter does not vanish even in the chiral restoration limit , i.e., for technicolors and techni-flavors, although it is identically zero when as it should, i.e. there is a discontinuity at the chiral phase transition.

Unfortunately, the holographic approach as it stands cannot decide whether or not : is given by a certain function of and which are both arbitrary parameters in this approach. Although is related with the chiral condensate which vanishes at the conformal phase transition point, we find no direct suppression of or and hence of due to the large in contrast to the previous authors Hong:2006si (); Piai:2006hy (). Based on the correct identification of the renormalization scale of , we have , independently of the non-physical renormalization point or , which may or may not be small even if , unless we know . Thus is not necessarily a small parameter in this framework even for , not to mention for , . Then the only possibility to realize would be to discuss more concrete dynamics approaching the conformal phase transition where we have (or ) not just a large anomalous dimension . Actually the effects of anomalous dimension are highly involved, combined with the scaling of as , as seen in the direct calculation based on the ladder SD and BS equations Harada:2005ru ().

We then discuss possible scaling behavior of near the conformal window . Although a simple large argument would always imply , the conformal phase transition takes place due to the Banks-Zaks infrared fixed point which only can be realized for large with fixed. Then the behavior of near the conformal phase transition is highly nontrivial. Obviously three possibilities in the limit of : i) , ii) , iii) .

We find that the case i) is realized only for , since is the monotonically increasing function of . In this case we have , which is the familiar scaling relation realized in QCD. Actually the case i) corresponds to the Vector Manifestation proposed in the HLS loop calculation Harada:2003jx (); Harada:2000kb ().

The case ii) where is realized only for the case and hence constant for . In this case , which is the same scaling relation as the case i). The case ii) actually corresponds to the straightforward calculation based on the ladder SD and BS equations Harada:2003dc (); Harada:2005ru (). It is amusing that a set of , obtained from homogeneous BS equation and from inhomogeneous BS equation both combined with SD equation, well coincides with a single point on the line of the -plane obtained in this paper.

The most interesting case for the TC is case iii) in which we have as . We find that the case iii) is realized only for , since is a monotonically increasing function of , although we have no explicit dynamics at this moment. We shall discuss some possible dynamics for this case which will be tested by future studies. In the case iii) we find a novel scaling property of vanishing much faster than near the conformal window, resulting in the form , which is quite different from the familiar one . This could be testable by lattice calculation for large QCD.

Although the bottom-up approach of the holography does not explicitly uses the large , the top-down approach needs that limit. Then the result here might be potentially valid only for large not near the conformal phase transition region where is large with fixed. Nevertheless, the result of this paper might be valid beyond the leading order of . Then it would be highly desired to investigate the possibility for in some explicit dynamics.

The paper is organized as follows:

In Sec. II we briefly review the framework of calculations in the holographic W/C TC model of Refs. Hong:2006si (); Piai:2006hy () based on the bottom-up holographic QCD DaRold:2005zs (); Erlich:2005qh ().

In Sec. III we calculate the parameter in models holographically dual to W/C TC allowing for varying values of the large anomalous dimension of techni-fermion condensation from the QCD monitor value to the W/C TC value .

In Sec. IV we identify the renormalization-point of the , based on which we find that there is no suppression factor solely due to large .

In Sec. V we classify holographic W/C TC models into three cases, i) , ii) constant , iii) as near the conformal window arising due to the Banks-Zaks infrared fixed point with . As an explicit dynamics for the case ii) we find a curious coincidence of the result of ladder SD and BS equations with the result in this paper. It is also shown that if the case iii) is realized, the parameter goes to zero at the edge of the conformal window in such a way that scales as much faster than the familiar form, .

Sec. VI is devoted to summary and discussion.

In Appendix A we discuss subtlety of the limit and .

In Appendix B we discuss the Pagels-Stokar formula in comparison with the holographic result.

## Ii Review of Holographic Calculations

In this section we briefly review the framework of calculations in the holographic W/C TC model of Refs. Hong:2006si (); Piai:2006hy () with which is the deformation of the the bottom-up holographic QCD DaRold:2005zs (); Erlich:2005qh () with by adjusting a profile of a 5-dimensional bulk scalar field. Here we consider a generic case with .

A holographic model DaRold:2005zs (); Erlich:2005qh (); Hong:2006si (); Piai:2006hy () is defined on the 5-dimensional anti-de Sitter space (AdS) with the metric,

 ds2=gMNdxMdxN=(Lz)2(ημνdxμdxν−dz2), (1)

where is a metric on 4-dimensional space-time spanned by the coordinate , and denotes the curvature radius of AdS. The fifth direction is compactified on the interval,

 ϵ≤z≤zm. (2)

A holographic action DaRold:2005zs (); Erlich:2005qh () possessing an gauge symmetry in 5 dimensions is constructed from gauge fields and , and a scalar field transforming under the gauge symmetry as a bi-fundamental representation. The action is given by #4#4#4 Here and . ,

 S5 = 1g25∫d4x∫zmϵdz √g (3) ×(−12Tr[LMNLMN+RMNRMN]+Tr[DMΦ†DMΦ−m25Φ†Φ]),

where denotes the gauge coupling in 5 dimensions and . The covariant derivative acting on the scalar field is defined as

 DMΦ=∂MΦ+iLMΦ−iΦRM. (4)

This may be parametrized by using scalar and pseudo-scalar fields, and , as

 Φ(x,z)=ϕ(x,z)exp[iP(x,z)/v(z)], (5)

with being the vacuum expectation value (VEV) of .

For later convenience, we introduce 5-dimensional vector and axialvector gauge fields and defined by

 VM=√12(LM+RM),AM=√12(LM−RM), (6)

and we choose a gauge,

 Vz(x,z)=Az(x,z)≡0. (7)

Based on AdS/CFT correspondence, boundary conditions for the bulk fields , , and are chosen so that their UV boundary values are related to the external sources in TC theories in the limit of : For the VEV of , the UV boundary value is related to the external source for the techni-fermion condensate , namely, the current mass of techni-fermion in such a way that

 M ≡ limϵ→0M, M = (Lϵ)γm(Lϵv(z))∣∣z=ϵ, (8)

where stands for the anomalous dimension of the techni-fermion condensate . The AdS/CFT correspondence makes it possible to associate , the mass of the scalar field , with the anomalous dimension :

 m25=−(3−γm)(γm+1)L2. (9)

We introduce the variable for the IR boundary value of VEV of ,

 ξ=Lv(z)∣∣z=zm, (10)

which corresponds to as will be seen later (Eq.(27)).

We shall later discuss and the corresponding are quantities renormalized at the scale (see Sec.IV.1), whereas is the quantity renormalized at .

As for the bulk gauge fields and , the UV boundary values play the role of the external sources (, ) for the vector and axialvector currents coupled to the holographic TC. Accordingly, under gauge, the boundary condition may be chosen,

 ∂zVμ(x,z)∣∣z=zm=∂zAμ(x,z)∣∣z=zm=0, Vμ(x,z)∣∣z=ϵ=vμ(x),Aμ(x,z)∣∣z=ϵ=aμ(x). (11)

With these boundary conditions (10) and (II), the equations of motion for the bulk gauge fields are completely solved at the classical level. By substituting those solutions into the action (3), the effective action is expressed as a functional of the UV boundary values/external sources, , , and , i.e., . The two-point Green functions are then readily calculated as

 δ2W[vμ]δ~vaμ(q)δ~vbν(0)∣∣∣vμ=0 = i∫xeiq⋅x⟨JaμV(x)JbνV(0)⟩=−δab(gμν−qμqνq2)ΠV(−q2), (12) δ2W[aμ]δ~aaμ(q)δ~abν(0)∣∣∣aμ=0 = i∫xeiq⋅x⟨JaμA(x)JbνA(0)⟩=−δab(gμν−qμqνq2)ΠA(−q2), (13) limϵ→0iiδW[M]δM∣∣∣M=0 ≡ limϵ→0iiδW[M]δM∣∣∣M=0=⟨¯TT⟩, (14)

where and respectively denote the Fourier component of and .

Once the current correlators are calculated, we can compute the parameter. We define as the parameter per each techni-fermion doublet,

 ^S=SNf/2, (15)

which is expressed by the vector and axialvector current correlators and as

 ^S=−4πddQ2[ΠV(Q2)−ΠA(Q2)]Q2=0, (16)

where . In the next subsections we shall calculate those current correlators, , and .

### ii.1 Generating Functional W[M] and ⟨¯TT⟩

Let us focus on a portion of the action (3) relevant for the VEV of , :

 S5∣v=∫d4x∫zmϵdzL32g25Tr[−1z3(∂zv(z))2+(3−γm)(1+γm)z5v2(z)], (17)

which leads to the following classical equation of motion for :

 ∂z(1z3∂zv(z))+(3−γm)(1+γm)z5v(z)=0. (18)

Solution for is given by #5#5#5 If we set in Eq.(18), we find a solution , which was used in analysis in Refs.Hong:2006si (); Piai:2006hy (). Here we understand as the limit which implies  Piai:2006hy (), namely, as seen from Eq.(25). The other choice was adopted in Ref.Hong:2006si (). See Appendix A for discussion on this point.

 v(z)(ϵ)=c1(zL)1+γm+c2(zL)3−γm, (19)

where and are determined by the boundary conditions (8) and (10) as

 c1 = M−(ϵzm)2−2γm(Lzm)1+γmξL(1−(ϵzm)2−2γm), (20) c2 = 1L(Lzm)3−γmξ−(Lzm)2−2γmLM(1−(ϵzm)2−2γm) (21) = (Lzm)3−γmξL−(Lzm)2−2γmc1.

In the continuum limit the solution takes the form

 v(z)=M(zL)1+γm+Σ(zL)3−γm, (22)

where and are quantities renormalized at the scale and we may write and . In terms of the renormalized quantities Eqs.(20) and (21) are rewritten as:

 M = M−(ϵzm)2−2γm(Lzm)1+γmξL, (23) Σ = (Lzm)3−γmξL−(Lzm)2−2γm1−(ϵzm)2−2γmM. (24)

In the chiral symmetric limit , we have

 Σ=(Lzm)3−γmξL. (25)

By substituting Eq.(22) into Eq.(17), the generating functional for is expressed as

 W[M]=∫d4xL2g25[−L2z3∂zv(z)⋅v(z)]zmϵ. (26)

From Eq.(14) we find the techni-fermion condensate :

 ⟨¯TT⟩=−1L3Lg25(3−γm)(Lzm)3−γmξ, (27)

where we have used Eq.(21) to rewrite in terms of and . From this form, we see that the IR value is actually associated with the techni-fermion condensate (in a combination with , however). From Eqs.(25) and (27) we see that is more directly related to (without combination with ) as

 Σ=−g25/L3−γm⋅⟨¯TT⟩L2. (28)

### ii.2 Generating Functional W[vμ,aμ] and ΠV,A

Under the gauge-fixing condition (7), we may find the equations of motion for the transversely polarized component of the gauge fields and ,

 [∂2−z∂z1z∂z]Vμ(x,z)=0, (29) [∂2−z∂z1z∂z+2L2v2(z)z2]Aμ(x,z)=0. (30)

In solving these equations, it is convenient to perform partially Fourier transformation on and with respect to ,

 Vμ(x,z)=∫qeiqxVμ(q,z),Aμ(x,z)=∫qeiqxAμ(q,z), (31)

where the Fourier components and may be decomposed as

 Vμ(q,z)=~vμ(q)V(q,z),Aμ(q,z)=~aμ(q)A(q,z). (32)

Putting these into Eqs.(29) and (30), we have

 [q2+z∂z1z∂z]V(q,z)=0, (33) [q2+z∂z1z∂z−2L2v2(z)z2]A(q,z)=0, (34)

with the boundary condition

 ∂zV(q,zm) = ∂zA(q,zm)=0, (35) V(q,ϵ) = A(q,ϵ)=1. (36)

The generating functional is now expressed in terms of and as follows:

 W[vμ,aμ]=12∫q−Lg25ϵTr% [~vμ(−q)∂zV(q,ϵ)⋅~vμ(q)+~aμ(−q)∂zA(q,ϵ)⋅~aμ(q)]. (37)

Accordingly, the vector and axialvector current correlators and , defined as in Eqs.(12) and (13), take the form:

 ΠV(Q2)=Lg25ϵ∂zV(Q2,ϵ),ΠA(Q2)=Lg25ϵ∂zA(Q2,ϵ), (38)

where we have rewritten and .

It is now obvious that the chiral symmetry breaking effects described by are related to and hence arise only from the term in Eq.(30) which is the unique origin of the -dependence in this approach. If the chiral symmetry gets restored such that , we should get , and hence at first glance, its derivative would also vanish. However, overall absolute value of is normalized by so that is realized even with . Of course, if we have , then as it should. The situation is very much like the perturbative calculation of : There could be discontinuity at the phase transition point. It should also be noted that although in the chiral limit, , implies and , it does not necessarily . This is because that is also possible in the expression of Eq.(27). The is given as a function of which is a combination of and , so that its behavior near the conformal phase transition is not directly connected with . In the following sections we shall discuss these points carefully.

#### ii.2.1 Vector Current Correlator ΠV

A solution of Eq.(33) with the boundary conditions (35) and (36) taken into account is given by the modified Bessel functions and ,

 V(Q2,z)=K0(Qzm)⋅I1(Qz)+I0(Qzm)⋅K1(Qz)I0(Qzm)⋅K1(Qϵ)+K0(Qzm)⋅I1(Qϵ), (39)

and hence in Eq.(38) is given by

 ΠV(Q2)=Lg25QϵK0(Qzm)⋅I0(Qϵ)−I0(Qzm)⋅K0(Qϵ)I0(Qzm)⋅K1(Qϵ)+K0(Qzm)⋅I1(Qϵ). (40)

In particular, for small , we may calculate approximately

 ΠV(Q2→0)∼−LQ2g25log(zmϵ). (41)

We will later come back to this expression in evaluating the parameter.

#### ii.2.2 Axialvector Current Correlator ΠA

To derive the solution for in an analytic manner Hong:2006si (), we may define the following quantity:

 P(Q2,z)=1z∂zlogA(Q2,z), (42)

in terms of which is expressed as

 ΠA(Q2)=Lg25P(Q2,ϵ). (43)

Equation of motion (34) is rewritten as

 z∂zP(Q2,z)+z2P(Q2,z)2−Q2−2(Lv(z)z)2=0, (44)

with the boundary condition

 P(Q2,zm)=0. (45)

We expand perturbatively in powers of as

 P(Q2,z)=P(0,z)+Q2P′(0,z)+⋯, (46)

where . These expansion coefficients are determined by solving the following equations order by order in :

 O(Q0) : z∂zP(0,z)+z2(P(0,z))2=2L2v2(z)z2, (47) O(Q2) : z∂zP′(0,z)+2z2P(0,z)P′(0,z)=1, (48)

which are derived from Eqs.(44) and (46). Inserting into Eq.(47) the solution of given in Eq.(22) with taken, we may find a solution of Eq.(47) so as to satisfy the boundary condition (45):

 P(0,z)=ΔX(z)zϵI1−ΔΔ(X(zm))⋅K1−ΔΔ(X(z))−K1−ΔΔ(X(zm))⋅I1−ΔΔ(X(z))I1−ΔΔ(X(zm))⋅K1Δ(X(ϵ))+K1−ΔΔ(X(zm))⋅I1Δ(X(ϵ)), (49)

with