Running Top quark mass in the presence of light SM Higgs
aa r X i v : . [ h e p - ph ] O c t Running Top quark mass in the presence of light SM Higgs
V. ˇSauli CFTP and Departamento de F´ısica, Instituto Superior T´ecnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal
The running of the Top quark mass is considered in the nonperturbative framework of theSchwinger-Dyson equation. Based on the input of physical pole mass meassured at the Tevatron themethod provides the resulting mass function which is almost constant at low spacelike and timelikescales. The skeleton loops including Standard Model Higgs and gluons are taken into account. Thedominant two-loop skeleton contribution with triplet Higgs interaction is considered in addition toone loop dressed approximation of the top quark self-energy.
PACS numbers: 11.10.St, 11.15.Tk
I. INTRODUCTION
The quark masses are fundamental parameters of the Standard Model. The precise knowledge of quark masses atvarious scales is important for several reasons. In hadronic physics it is necessary for precise determination of CKMmatrix elements, while the theoretically extracted information about values of quark masses at very high momentacan be useful for model builders.In perturbation theory QCD approach the definition of the running mass is based on the renormgroup evolutionequations (RGEs). MS bar scheme represents short range mass definition and is commonly used due to its technicalsimplicity [1, 2]. On the other side the RGEs method cannot provide reliable results at low momenta where theperturbation method fails. A legitimate question is what is the relevant scale of the applicability of the perturbationtheory, when the corrections to the quark masses are evaluated. To determine this, the running masses calculationhas to rely on nonperturbative QCD techniques. So far, there are two methods that directly follow from the firstprinciples: the first is lattice theory (for a review see [3, 4, 5, 6, 7, 8]) which is based on the discretized Euclideanspace, the second is the functional method represented by a continuous framework of QCD Schwinger-Dyson equations[9, 10]. Within certain phenomenological assumptions the QCD sum rules [11, 12] are used to determine the quarkmasses.After the top quark discovery, the top quark mass value is obtained in the fairly limited regime, the CDF [13]and DO [14] collaborations measure the resonant top quark mass. The particle data group [15] quoted the value M t = 174 . ± . GeV as the pole mass of the top quark.The evolution of the Yukawa coupling has already been studied in [1]. However, the Yukawa interaction has not beenconsidered in RGEs for the top quark running mass. The contribution to the quark selfenergy due to the Higgs bosonhas been studied in SDE framework for the first time in [16]. This study has been performed with two approximatelyequivalent inputs: M (0) = 179 GeV and M ( M t ) = 174 GeV , noting that the later was defined incorrectly at spacelikescale M t . In the present paper we go beyond the one loop approximation and take into account the two loop Higgscontribution as well.There is a striking evidence that the RGE perturbation calculation overestimated low scale top quark mass fromthe very beginning (going from high spacelike Q to the infrared values). Recall that in the paper [1] the renormgroupequation for top quark running mass M ( µ ) has been solved in MS bar renormalization scheme with the followingresult (in GeV): M (180 ) = 170 . M ( M ) = 170 . M (91 . ) = 180; M (4 . ) = 253; M (1 . ) = 318; M (1) = 339 , (1.1)for spacelike arguments in the brackets and have been quoted within ≃ (12 − GeV error due to experimentallydetermined physical mass ( to that date, it was M t = 180 GeV ).Recall that the physical pole mass M t is determined in Minkowski space as the S − ( M t ) = 0, in other words M t ( − M t ) = M t . Assuming that the fit procedure of M ( M t ) from M t used in [1] and developed originally in [17] isreliable (note, the relation between MS mass and on shell mass is recently known to the order α ), one necessarilymust conclude that the renormgroup evaluation of masses becomes unreliably overestimated below the scale µ ≃ M z .While for leptons and light quarks perturbation QCD works perfectly at the M z scale, it appears that for an accurateestimate of the running top quark mass at M z mass scale might not be adequate. Technically this is because alreadyone loop correction to M is enhanced like δM ∼ α QCD
M . (1.2) S -1 S FIG. 1: Diagrammatic representation of top quark SDE. The solid line stands for quark and the dot (dashed) line stands forthe gluon (Higgs) respectively. The circles represent full vertices.
In other words, the exceptionally large mass of the top quark itself spoils the usual correctness of perturbative QCDat electroweak scale.This is one of the main reason of the present study to calculate the evolution of top quark mass in the wholemomentum of range, obtaining thus correct information for the low energy scales. In perturbative MS schemes therunning mass grows from MS value m ( m ) = 170 to µ = M Z about amount of 10 GEV and further blows up whenevolved to the infrared. We will argue that M ( M t ) and M ( M Z ) differs about tiny amount ≃ − GeV and the runningtop quark mass function remains stable when using selfconsistent framework of our SDE equations. The knowledgeof here observed infrared stability of the top quark mass should be useful whenever a selfconsistent treatment isrequired, i.e. for instance when one considers Higgsonia [18, 19] and the effect of top quark loop in the equations forHiggsonium bound states. Further, the top quark circulates in the loop of penguin diagrams describing rare mesonicelectroweak decays (see e.g. [20, 21]). In this case the typical energy of decaying B mesons is of the order m b , so thegood knowledge of the infrared value of the top quark mass is important for the description of heavy meson decays. Ofcourse, the knowledge of mass at high scales can useful for model builders. However, in the nonperturbative treatmenthere we are mainly for the physic not far above the electroweak scale, the knowledge of the quark mass at higherscales can be useful for model builders as well.The last but not least motivation is a direct check the effect of higher order corrections including Higgs trilinearcoupling on the solution. To do this the appropriate two loop skeleton diagram is calculated and included into thetop quark SDE. These, and other details of the model are described in the Section II. and Section III. of presentedpaper. To find the correct solution in the full Minkowski space is a problematic task for a strong coupling theory likeQCD. First we solve quark SDE in Euclidean space by standard numerical manner in the Section IV. In Section V.we continue the solution to the timelike axis in a way that experimentally known pole mass is achieved by the correctsolution. This is achieved by resolving of the SDE with renormalized mass adjusted to obtain the correct physicalpole mass at the end. II. SCHWINGER-DYSON EQUATION FOR TOP QUARK MASS FUNCTION
Neglecting the weak interaction, the quark propagator S can be conventionally characterized by two independentscalars, the mass function M and the renormalization wave function Z such that S ( p ) = Z ( p ) p − M ( p ) . (2.1)The SDE for the inverse of S reads S ( p ) − = p − g Y < φ > − Σ A ( p ) − Σ h ( p ) + ... (2.2)Σ A ( p ) = ig Z d q (2 π ) Γ α ( q, p ) G αβ ( p − q ) S ( q ) γ β Σ h ( p ) = ig Y Z d q (2 π ) Γ h ( q, p ) G h ( p − q ) S ( q )where g Y is the top Yukawa coupling, Higgs vev < φ > = 246 GeV / √ g is QCD gauge coupling. The dotsrepresent omitted contributions, e.g. W, Z, γ and related Goldstone exchanges. G, Γ stand for boson propagators andthe top quark-boson vertices and they satisfy their own SDEs.The knowledge about these Greens function is necessarily limited due to theoretical and experimental reasons. Theyneed to be approximated if they are not selfconsistently contained in a given truncation scheme of the SDEs system.A natural treatment of this problem is to make an expansion in the number of loops. Performing such an expansionsfor vertices Γ = P i Γ i and substituting this into the selfenergy (2.2), one gets the expansion for the mass function.Explicitly, the loop expansion for the selfenergy in (2.2) should readΣ A = Σ [1] A + X i Σ [ i +1] A (2.3)and similarly for Σ h .In the simplest approximation the first order estimates can be obtained by using the classical verticesΣ A ( p ) = ig Z d q (2 π ) γ α G αβ ( p − q ) S ( q ) γ β Σ h ( p ) = ig Y Z d q (2 π ) G h ( p − q ) S ( q ) (2.4)where the all propagator functions entering the Eqs. (2.4) are fully dressed.Including ”radiative corrections” to the SM model Higgs one should get coupled SDEs for G and S . In the case oflight Higgs, the top-antitop quark loop contribution would lead to the extremely large negative contribution to theHiggs boson mass. This mass hierarchy problem, although formally solved by renormalization, is one of the mainmotivation for extension of the Standard Model and the reason why the SM is regarded as an effective low energytheory. In the extensions of SM the mass hierarchy is stabilized by the introduction of the other scalars [24, 25, 26],SM doublets [27, 28], or is eliminated by supersymmetry or the Lee-Wick SM modification [29]. In all these models,a new particle content is expected at few TeV, the quadratic divergences to Higgs mass are reduced and the freepropagator could be a reasonable approximation of the exact Higgs propagator for a broad regime of scales. Thereforethe simplest -free Higgs boson propagator: G h ( p ) = 1 p − m (2.5)is used, where m is the physical Higgs boson mass (2.6).Following the recent precision test of the Standard Model [22]. the analysis of the radiative corrections favor a lightHiggs boson m ≃ GeV . Because of the lack of an experimentally observed Higgs particle, the mass of the Higgsboson could be rather close to the experimental lower bound m > . GeV [23]. In this paper the following valueof the Higgs mass is chosen m = 120 GeV, (2.6)as the input parameter in our model.At low scales, q ≃ Λ QCD , the running QCD coupling is large and the dressing of the gluon-quark-antiquark vertexcan play an important role in the description of light flavor dynamics [30]. However, in the case of the top quark, therunning coupling becomes quite small α QCD ( M ) ≃ . g G µν ( k )Γ ν ( q, p ) → πα ( k , Λ) − g µν + k µ k ν k k + iε γ ν . (2.7)where α represents the analytical running coupling [32, 33, 34, 35]. In the one loop approximation it is given bythe following expression: α ( q , Λ QCD ) = Z ∞ dω ρ g ( ω, Λ QCD ) q − ω , (2.8)where ρ g ( ω, Λ QCD ) = 4 π/βπ − ln ( ω/ Λ QCD ) . (2.9)Recall that the analytical running coupling is constructed in a simple way that avoids the unwanted artifact ofperturbation theory- the Landau pole at q = Λ QCD - which is subtracted away and thus the running coupling isfree of unphysical singularities. The procedure has been generalized to higher orders, provided that the ultravioletasymptotic behaviour of such running constant is identical with the perturbative result. In the numeric here the one
FIG. 2: Diagrammatic representation of selfenergy contribution due to the Higgs. The circle stands for the full vertex, therhs. is the approximation employed here. loop approximation (2.9) is used with the numerical value of Λ
QCD , Λ
QCD = 500
M eV for six active quarks. Thebeta function coefficient is thus β = 11 N c − N f N c = 3 , N f = 6.The computation is carried out in Landau gauge and the Z = 1 approximation is used. Whilst in pure gauge theorythe effect of the Z = 1 approximation can be minimized by proper adjustment of the gauge fixing, the importance of Z in the presence of the Higgs field is not explored and remains to be estimated in a future study. III. SOLVING TOP QUARK SDE IN EUCLIDEAN SPACE
Using the following formula Z π dηπ sin ηl − | l || p | cosη + p + m = − p − q − m + p ( p + q + m ) − p q − p , (3.1)the angular integrations in one loop skeleton diagram in (2.4) can be easily evaluated. After the explicit integrationthe Higgs-top loop contribution can be cast into the one dimensional integralΣ [1] h ( x ) = α Y π Z ∞ dy M ( y ) y + M ( y ) K ( x, y, m ) , (3.2)where α Y = g Y / π and the functions K is defined as K ( x, y, z ) = 2 yx + y + z + p ( x + y + z ) − xy . (3.3)Likewise, for the one loop QCD contribution we getΣ [1] A ( x ) = 1 π Z ∞ dy M ( y ) y + M ( y ) V ( x, y ) , (3.4)with the function V defined as V ( x, y ) = − Z ∞ dω ρ g ( ω ) ω [ K ( x, y, − K ( x, y, ω )] . (3.5)In addition, the one loop skeleton contribution to the Higgs-quark-antiquark proper vertex (see Fig. 2) is carefullyincluded. This is equivalent to the two loop 1PI contribution for the top quark dynamical mass function which reads:Σ [2] h ( p ) = λvg Y I ( p ) , (3.6)where λ/ I ( p ) is the following two loop integral: I ( p ) = T r i Z d l (2 π ) l − p ) − m l + M ( l ) l − M ( l ) × i Z d q (2 π ) q − p ) − m q + M ( q ) q − M ( q ) × q − l ) − m . (3.7)After the Wick rotation to Euclidean space, the integrations in (3.7) are not calculable directly, however most of themcan be calculated analytically by performing just one quite standard angular approximation (3.7)). This approximationeliminates the angle between the two internal loop momenta in the following manner:( l − q ) → l θ ( l − q ) + q θ ( q − l ) (3.8)and so writing also for l.q product (this stem from Dirac trace) l · q = l + q − ( l − q ) → q θ ( l − q ) + l θ ( q − l ) , (3.9)the expression for I can be recast as: I ( p ) = Z d l (2 π ) l − p ) + m Z d q (2 π ) q − p ) + m × M ( l ) M ( q )+ q / l + m θ ( l − q ) + M ( l ) M ( q )+ l / q + m θ ( q − l )( q + M ( q ))( l + M ( l )) . Here, it is an opportune point to remark that such an angular approximation has been extensively used in phenomeno-logical SDE studies of QCD and QED4 even at one loop level. In our case the coupling constant is small enough andfollowing the critical one loop analysis performed in [31], this must be a reliable approximation in our two loop case.In Euclidean domain it can lead to a few percent error in I . As we have estimated posterior, it entails only a tiny (afew promile) error in the total result for M .Using the formula (3.1) the remaining angular integrations can be performed, resulting the following expression for I : I ( p ) = Z dq π p + q + m − p ( p + q + m ) − p q p [ q + M ( q )] × Z dl π p + l + m − p ( p + l + m ) − p l p [ l + M ( l )] × M ( l ) M ( q )+ q / l + m θ ( l − q ) + M ( l ) M ( q )+ l / q + m θ ( q − l )( q + M ( q ))( l + M ( l ) ) . (3.10)In what follows we interchange of the variables l ↔ q in the second term of the third line of the Eq. (3.10). Consideringthe appropriate prefactors, I ( p ) can be finally written in the following way: I ( p ) = Z ∞ dq p + q + m − p ( p + q + m ) − p q p [ q + M ( q )] × Z q dl p + l + m − p ( p + l + m ) − p l p [ l + M ( l )] × π l / M ( l ) M ( q )( q + m )( q + M ( q ))( l + M ( l )) (3.11)Putting these all together, the SDE for top quark mass function that is to be solved reads M ( x ) = g Y < φ > +Σ [1] A ( x ) + Σ [1] h ( x ) + Σ [2] h ( x ) , (3.12)where the individual terms are given by (3.4), (3.2) and (3.6) wherein I is given by Rel. (3.11). As a consequence ofthe Z = 1 approximation the function M ( x ) is renormgroup invariant. After making a subtraction the ”renormalized”equation actually solved reads M ( x ) = M ( ξ ) + Σ( x ) − Σ( ξ )Σ( x ) = Σ [1] A ( x ) + Σ [1] h ( x ) + Σ [2] h ( x ) , (3.13)where the renormalized mass at the scale ξ is related to the bare top quark mass through the following rel.: M ( ξ ) = g Y < φ > +Σ( ξ ). -1 p E [100GeV] M [ G e V ] FIG. 3: Running Top quark mass as described in the text. The solid line represents the full solution, dot-dot-dashed linestands for the case when two loop skeleton is omitted. The dashed and dotted lines stand for pure QCD and Yukawa solutionsrespectively.
IV. RESULTS IN SPACELIKE REGIME
In this section, we discuss the numerical solution of the renormalized Euclidean SDE (3.13). The physical masspole being on the timelike axis cannot be directly used for the solution. The main purpose of this section is to exhibitthe importance of various contribution in the case of light Higgs exchanges.The SDE (3.13) has been solved by the method of iterations with high accuracy. For this purpose we have chosenthe (spacelike) renormalization scale to be ξ = 100 GeV (4.1)and fixed the renormalized mass M ( ξ ) through the Yukawa coupling.The two loop diagram depicted in Fig. 2 includes the triplet Higgs interaction constant, which is already determinedthrough the quartic one. In our numerical calculation the coupling constant actually used is read from the relation λ = m v (at the given scale ξ ).The resulting mass function is displayed in the Fig. 3. The presented calculations were performed with g Y = 0 . M (100 GeV ) = 169 . GeV . With these inputs we get thenumerical solution. As the presented solution is regularization independent, we used hard cutoff regulator Λ >> M (0)obtaining the same solution when Λ was varied through many orders. The mass function is increasing when going toinfrared, reaching its infrared value M (0) = 170 . GeV , being not far the experimental one. How to gain the solutionactually based on determined physical top quark mass will be discussed in the section. Before this we discus somegeneral features of the solution.In our presented framework of SDEs the dynamical mass function is slowly varying function in the infrared. Up tofew GeV contribution the infrared mass does not change drastically at the scale of 0-100 GeV.The Yukawa interaction between Higgs and top quark is quite strong even when comparing to the QCD interactionstrength. In Fig. 3. we show the comparison of solutions stemming purely from the Yukawa interaction and fromthe pure QCD (by switching off QCD or Yukawa interaction). The same value of the renormalized top quark mass iskept for this purpose. As expected, the QCD dominates in the infrared regime, while in high momenta, q >> M t both interactions are of the same magnitude.Interestingly, the two loop effect with Higgs trilinear coupling gives a marginal contribution for all p . Numerically,two loop skeleton effect is comparable with the one loop PT electroweak corrections. For a heavier Higgs the one loopHiggs contribution becomes less important, while the two loop contribution appears to be less affected since the tripletHiggs coupling is getting strong. We have also solved the SDE with different Higgs masses as well. For instance Higgsheavy as m = 0 . T eV , two loop contribution becomes more important giving rise a few
GeV negative contribution inthe infrared top quark mass. However, one should note that in this case the Higgs sector becomes strongly interactingwhat would require more careful reinvestigation due to the new nonperturbative dynamics [42, 43, 44].
V. SOLUTION FOR ALL MOMENTA
Experimentally the top quark mass is reconstructed by collecting jets and leptons. From the position of the bunchin cross section measured at the Tevatron the pole position is identified M t = 172 . ± .
4. The ambiguity anduncertainty of the full top propagator pole mass is affected by experimental methods and theoretical weaknesses ofperturbation theory description of jets, e.g. reconciling the contribution of soft and collinear particles. Furthermore,the correct identification of the mass requires nonperturbative technique at all. While, including perturbative 1-loop b, W electroweak correction this pole could only move into the second sheet complex plane giving rise top quark decaywidth Γ t = 1 . GeV , the perturbation theory cannot give nonambiguous result because of uncertainty proportionalto Λ
QCD [36, 37, 38]. The real pole of the (pure QCD) perturbation theory can turn to be complex because ofconfinement phenomena as recently observed in [39] by studying complex mass generation in temporal Euclideanspace.In this paper we do not solve the problem of confinement of th top quark in SDE framework, instead we showthat the running top quark mass function turns to be stable, very slightly varying, quantity when continued to thetimelike momenta. For this purpose the mass function M ( − x ) at timelike square of the fourmomenta t = − x > x → − t . We can write for the continuedsolution M ( t ) = M ( ξ ) + Σ( t ) − Σ( ξ ) (5.1)Σ( t ) = Σ [1] A ( t ) + Σ [1] h ( t ) + Σ [2] h ( t ) , Σ [1] h ( t ) = α Y π Z ∞ dy M ( y ) y + M ( y ) K ( − t, y, m ) , (5.2) K ( − t, y, z ) = 2 y − t + y + z + p ( − t + y + z ) + 4 ty , and similarly, the functions Σ [1] A , Σ [2] h are obtained by the substitution x → − t in their kernels.Since the mass function on the rhs. of Eq. (5.2) remains defined at the spacelike regime, the pole mass cannotbe used as the renormalized point directly. To achieve the solution of SDE with experimentally known value of topquark mass, we shift the renormalized mass in (3.13) and then have a look for the solution for M t by integrating theequation (5.1). With sufficient accuracy it is easily achieved by hand iteration process.The experimentally observed mass knowledge based solution is presented in Fig. 4. The numerical value M t =172 . GeV is obtained as the solution for pole mass. The timelike solution is plotted at the negative axis. The resultingYukawa coupling to our 120
GeV heavy Higgs field has been adjusted as g Y = 0 . M t . The experimentaluncertainty defines the errors repesented by narrow band of width ∼ − . GeV with presented solution inside. Wedo not display these.The other interesting values we can quote here are (in GeV): M (10 ) = 134 . M ( M t ) = 171 . M ( M Z ) = 172 . M (0) = 173 . M ( − M t ) = 172 . , M ( − ) = 151 . , . (5.3) VI. CONCLUSION
The SDE calculation of running top quark mass previously discussed in the literature [16] is presented in someextent. The obtained solution is based on the measured physical top quark mass. The top quark mass can be safelyevolved to small q when one avoids the pathology of perturbation theory, e.g. Landau pole in gluon propagator. Itexhibit great stability at all scales of spacelike and timelike domain as well. For the timelike domain the function issuch slowly varied that the top quark physical mass appears to be a rather good approximation at all low scales.At low scales, QCD contribution dominates over the one due to the Higgs loop(s), at large q both Higgs and QCDloops are comparable. In addition, we estimated the two loop Higgs contribution, which gives only tiny contribution p [10 GeV] M [ G e V ] M(x)M(-x)x
FIG. 4: Running top quark mass based on observed Tevatron pole mass. The solid line represents the spacelike and dashedthe timelike solution, dotted solid line represents the linear function f = p ( x ) ,which when cuts the dashed line, identifies thereal pole of the full top quark propagator). for the case of the light Higgs. The extension of presented technique to the more general models, e.g. with more Higgsdoublets and/or scalar singlets added to SM Higgs sector is straightforward. [1] H. Fusaoka, Y. Koide, Phys. Rev. D , 3986 (1998).[2] K.G. Chetyrkin, M. Steinhauser, Nucl. Phys. B , 617 (2000)[3] V. Lubicz, Nucl.Phys.Proc.Suppl. ,291 (1999).[4] Sinead Ryan, Nuc. Phys. B, Proc. Suppl. , 86 (2002).[5] L. Lellouch, Nucl.Phys.Proc.Suppl. , 127 (2003).[6] S. Hashimoto, T. Onogi, Ann. Rev. Nucl. Part. Sci. ,451 (2004).[7] P.E.L. Rakow, Plenary talk given at Lattice 04, arXiv:hep-lat/0411036v1.[8] B. Blossier et. all, arXiv:0709.4574v1.[9] R. Alkofer (1), L. von Smekal, Phys.Rept. , 281 (2001).[10] A. Hoell, C.D. Roberts, S.V. Wright, Hadron Physics and Dyson-Schwinger Equations, lecture notes contributed to theproceedings of the 20th Annual HUGS 2005, JLab.[11] L.J. Reinders, H. Rubinstein, S. Yazaki, Phys. Rept. , (1985).[12] M. Shifman, Nucl. Phys. B , 385 (1979).[13] By CDF Collaboration (F. Abe et al.), Phys. Rev. Lett. , 2626 (1995).[14] By D0 Collaboration (S. Abachi et al.), Phys. Rev. Lett. , 2632 (1995).[15] Particle Data Group 2006.[16] L.L. Smith, P. Jain, D.W. McKay, Mod.Phys.Lett. A ,773 (1995).[17] N. Gray, D.J. Broadhurt, W. Grafe and K. Schilcher, Z. Phys. C , 673 (1990).[18] B. Grinstein and M. Trott, arXiv:0704.1505 .[19] V. Sauli, Higgsonium in the Standard Model and beyond, Workshop on Scalar Mesons and Related Topics, 11-16 Feb.2008 at IST Lisbon.[20] A.J. Buras, R. Fleischer, Adv.Ser.Direct.High Energy Phys. B697 ,133 (2004).[22] LEP electroweak working group, http://lepewwg.web.cern.ch/LEPEWWG/.[23] R. Barate et al. , [LEP Working Group for Higgs boson searches], Phys. Lett.
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