Analytical Soft SUSY Spectrum in Mirage-Type Mediation Scenarios
aa r X i v : . [ h e p - ph ] A ug Prepared for submission to JHEP
Analytical Soft SUSY Spectrum in Mirage-TypeMediation Scenarios
Fei Wang, a,b a Department of Physics and Engineering, Zhengzhou University, Zhengzhou 450000, P. R. China b State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy ofSciences, Beijing 100190, P. R. China
E-mail: [email protected]
Abstract:
We derive explicitly the soft SUSY breaking parameters at arbitrary low en-ergy scale in the (deflected) mirage type mediation scenarios with possible gauge or Yukawamediation contributions. Based on the Wilsonian effective action after integrating out themessengers, we obtain analytically the boundary value (at the GUT scale) dependencies of theeffective wavefunctions and gauge kinetic terms. Note that the messenger scale dependenciesof the effective wavefunctions and gauge kinetic terms had already been discussed in GMSB.The RGE boundary value dependencies, which is a special feature in (deflected) mirage typemediation, is the key new ingredients in this study. The appearance of ′ mirage ′ unificationscale in mirage mediation is proved rigorously with our analytical results. We also discussbriefly the new features in deflected mirage mediation scenario in the case the deflection comespurely from the Kahler potential and the case with messenger-matter interactions. ontents After the discovery of the 125 GeV Higgs boson in 2012 at the CERN LHC[1, 2], the longmissing particle content of the Standard Model(SM) has finally been verified. In spite of theimpressive triumph of SM, many physicists still believe that new physics may be revealed atLHC. Among the many new physics models that can solve the fine-tuning problem, the mostelegant and compelling resolution is low energy supersymmetry. Augmented with weak scalesoft SUSY breaking terms, the quadratic cutoff dependence is absent, leaving only relativelymild but intertwined logarthmic sensitivity to high scale physics. As such soft SUSY breakingspectrum is determined by the SUSY breaking mechanism, it is interesting to survey the thephenomenology related to supersymmetry breaking mechanism.In Type IIB string theory compactified on a Calabi-Yau (CY) orientifold, the presence ofNS and RR 3-form background fluxes can fix the dilaton and the complex structure moduli,leaving only the Kahler moduli in the Wilsonian effective supergravity action after integrat-ing out the superheavy complex structure moduli and dilaton. The remaining Kahler modulifields could be stabilized by non-perturbative effects, such as instanton or gaugino condensa-tion. In order to generates SUSY breaking in the observable sector and obtain a very tinypositive cosmological constant, Kachru-Kallosh-Linde- Trivedi (KKLT)[3] propose to add ananti-D3 brane at the tip of the Klebanov-Strassler throat (or adding F-term, D-term SUSYbreaking contributions[4]) to explicitly break SUSY and lift the AdS universe to obtain a dS– 1 –ne. In addition to the anomaly mediation contributions, SUSY breaking effects from thelight Kahler moduli fields could also be mediated to the visible sector and result in a mixedmodulus-anomaly mediation SUSY breaking scenario [5, 6]. It is interesting to note that theinvolved modulus mediated SUSY breaking contributions can be comparable to that of theanomaly mediation [7]. With certain assumptions on the Yukawa couplings and the modu-lar weights, the SUSY breaking contributions from the renormalization group running andanomaly mediation could cancel each other at a ′ mirage ′ unification scale, leading to a com-pressed low energy SUSY breaking spectrum [8]. Such a mixed modulus-anomaly mediationSUSY breaking mechanism is dubbed as ′ mirage mediation ′ .Anomaly mediation contribution is a crucial ingredient of such a mixed modulus-anomalymediation. It is well known that the pure anomaly mediation is bothered by the tachyonicslepton problem [9]. One of its non-trivial extensions with messenger sectors, namely thedeflected anomaly mediated SUSY breaking (AMSB), can elegantly solve such a tachyonicslepton problem through the deflection of the renormalization group equation (RGE) trajec-tory [10–12]. Such a messenger sector can also be present in the mirage mediation so thatadditional gauge contributions by the messengers[13] can deflect the RGE trajectory andchange the low energy soft SUSY predictions. Additional deflection in mirage mediation canbe advantageous in phenomenological aspect. For example, apparent gaugino mass unifica-tion at TeV scale could still be realized with the simplest ′ no scale ′ Kahler potential, which,in ordinary mirage mediation, can only be possible with the not UV-preferable α = 2 case.Relevant discussions on mirage-type mediation scenarios can be seen, for example, in[14–17].In mirage type mediation scenarios, analytical expressions for the soft SUSY breakingparameters are no not given at the messenger scale M (or scale below M ), but given atthe GUT scale instead. One needs to numerically evolve the spectrum with GUT scaleinput to obtain the low energy SUSY spectrum. This procedure obscures the appearanceof ′ mirage ′ unification scale from the input. In mirage mediation scenarios with deflectionfrom Kahler potential, analytical results of mirage mediation are necessary to predict the lowenergy SUSY spectrum. So it is preferable to give the analytical expressions for the soft SUSYbreaking parameters in mirage type mediation scenarios at arbitrary low energy scale. Besides,possible new Yukawa-type interactions involving the messengers may give additional Yukawamediation contributions to the low energy soft SUSY spectrum (See [18] for example). Such ageneralization of deflected mirage mediation scenario shows new features in phenomenologicalstudies. The inclusion of Yukawa mediation contributions at (or below) the messenger scale M are non-trivial and again prefer analytical expressions near the messenger scale.This paper is organized as follows. We briefly review the mirage type mediation scenariosin Sec.2. A general discussion on the analytical expressions for the soft SUSY parameters inthe generalized deflected mirage mediation is given in Sec.3. We discuss some applications ofour analytical results in Sec.4, including the proof of the ′ mirage ′ unification scale in miragemediation with our analytical results and the discussions on deflection from Kahler potential.Sec.5 contains our conclusions. – 2 – Brief Review of the Mirage Type Mediation Scenarios
Inspired by string-motivated KKLT approach to moduli stabilization within Type IIB stringtheory, mirage mediation supersymmetry breaking is proposed, in which the modulus medi-ated supersymmetry breaking terms are suppressed by numerically a loop factor so that theanomaly mediated terms can be competitive.After fixing and integrating out the dilaton and the complex structure moduli, the four-dimensional Wilsonian effective supergravity action (defined at the boundary scale Λ) interms of compensator field and a single Kahler modulus parameterizing the overall size of thecompact space[8] is given as e − L = Z d θ h φ † φ (cid:16) − e − K/ (cid:17) − ( φ † φ ) ¯ θ θ P lift i + Z d θφ W + Z d θ f i W ai W ai (2.1) with a holomorphic gauge kinetic term f i = 1 g i + i θ π . (2.2)The Kahler potential takes the form K = − T + T † ) + Z X ( T † , T ) X † X + Z Φ ( T † , T )Φ † Φ+ X i Z P i , ¯ P i ( T † , T ) h P † i P i + ¯ P † i ¯ P i i , (2.3)with the ′ no − scale ′ kinetic term for the Kahler modulus T . The gauge kinetic term f i , themessenger superfields P i , the MSSM superfields Φ and the pseudo-moduli superfields are allassumed to depend non-trivially on the Kahler moduli T as Z X ( T † , T ) = 1( T † + T ) n X , Z Φ ( T † , T ) = 1( T † + T ) n Φ ,f i ( T ) = T l i , Z P i , ¯ P i ( T † , T ) = 1( T † + T ) n P . (2.4)Choices of n X , n Φ , n P , l i depend on the location of the fields on the D3/D7 branes. Besides,universal l i = 1 are adopted in our scenario to keep gauge coupling unification, so the gaugefields should reside on the D7 brane.The superpotential takes the most general form involving the KKLT setup[3], the mes-senger sectors W M and visible sector W MSSM W = (cid:0) ω − Ae − aT (cid:1) + W M + W MSSM , (2.5)where the first term is generated from the fluxes and the second term from non-perturbativeeffects, such as gaugino condensation or D3-instanton. Within W M , interactions betweenmessengers and MSSM fields can possibly arise which will be discussed subsequently. The– 3 –odulus T , which is not fixed by the background flux, can be stabilized by non-perturbativegaugino condensation with its VEV satisfying a ℜh T i ≈ ln (cid:18) Aω (cid:19) ≈ ln (cid:18) M P l m / (cid:19) ≈ π (2.6)up to O (ln[ M P l /m / ] − ). Boundary value of the soft SUSY breaking parameters at the GUTscale can be seen in [8]. Mirage mediation can be seen as a typical mixed modulus-anomaly mediation SUSY breakingmechanism with each contribution of similar size. Adding a messenger sector will add addi-tional gauge mediation contributions. Besides, upon the messenger thresholds, new Yukawainteractions involving the messengers could arise. Such interactions may cause new contribu-tions to trilinear couplings and sfermion masses (As an example, see our previous work [18]).Additional deflection with Yukawa mediation can be advantageous in several aspects. • The value of trilinear coupling | A t | can be increased by additional contributions involv-ing the new Yukawa interactions. Larger value of A t is always welcome in MSSM andNMSSM not only to accommodate the 125 GeV Higgs but also to reduce[19] the EWfine tuning[20] involved. • As noted in [18, 21, 22], pure gauge mediation contributions are not viable to generateeither trilinear couplings A κ , A λ or soft scalar masses m S for singlet superfields S whichare crucial to solve the mu-problem of NMSSM. Deflection with Yukawa interactionswill readily solve such difficulty.To take into account such Yukawa mediation contributions in soft SUSY breaking param-eters, it is better to derive the most general results involving the deflection. There are twoapproaches to obtain the low energy SUSY spectrum in the (deflected) mirage type mediationscenario: • In the first approach, the mixed modulus-anomaly mediation soft SUSY spectrum isgiven by their boundary values at the GUT scale[8]. Such a spectrum will receiveadditional contributions towards its RGE running to low energy scale, especially thethreshold corrections related to the appearance of messengers[23, 24]. The soft SUSYbreaking parameters are obtained by combing numerical RGE evolutions with thresholdcorrections. In [23], following this approach, some analytical expressions of the softSUSY spectrum, for example the gaugino masses, are given. General expressions ofthe soft scalar masses and trilinear couplings are not given explicitly except for somesimplified cases. – 4 –
In the second approach which we will adopt, the soft SUSY spectrum at low energy scaleis derived directly from the low energy effective action. We know that the SUGRA de-scription in eq.(2.1) can be seen as a Wilsonian effective action after integrating out thecomplex structure moduli and dilaton field. After the pseudo-modulus acquires a VEVand determines the messenger threshold, the messenger sector can be integrated out toobtain a low energy effective action below the messenger threshold. So we anticipatethe Kahler metric Z Φ and gauge kinetic f i will depend non-trivially on the messengerthreshold M mess /φ † φ and M mess /φ , respectively. The resulting soft SUSY spectrumbelow the messenger threshold can be derived from the wavefunction renormalizationapproach [25]. The main difficulty here is to find the boundary value dependencies ofthe wavefunction and gauge kinetic term.In this approach, the most general expressions for soft SUSY breaking parameters indeflected modulus-anomaly (mirage) mediation SUSY breaking mechanism are derivedbelow. Ordinary mirage mediation results can be obtained by setting the deflectionparameter ′ d ′ to zero. – The gaugino masses are given by M i = − g i (cid:18) F T ∂∂T − F φ ∂∂ ln µ + dF φ ∂∂ ln | X | (cid:19) f a ( T, µφ , s X † Xφ † φ ) , (3.1) – The trilinear terms are given by A Y abc ≡ A abc /y abc (3.2)= 12 X i = a,b,c (cid:18) F T ∂∂T − F φ ∂∂ ln µ + dF φ ∂∂ ln | X | (cid:19) ln h e − K / Z i ( µ, X, T ) i . – The soft sfermion masses are given by − m soft ( µ ) = (cid:12)(cid:12)(cid:12)(cid:12) F T ∂∂T − F φ ∂∂ ln µ + d F φ ∂∂ ln | X | (cid:12)(cid:12)(cid:12)(cid:12) ln h e − K / Z i ( µ, X, T ) i (3.3)= | F T | ∂ ∂T ∂T ∗ + F φ ∂ ∂ (ln µ ) + d F φ ∂ ∂ (ln | X | ) − F T F φ ∂ ∂T ∂ ln µ + dF T F φ ∂ ∂T ∂ ln | X | − dF φ ∂ ∂ ln | X | ∂ ln µ ! ln h e − K / Z i ( µ, X, T ) i , From the previous general expressions, we can deduce the concrete analytical results forsoft SUSY parameters. In our notation, we define the modulus mediation part M ≡ F T T + T ∗ , q Y ijk ≡ − ( n i + n j + n k ) . (3.4)The gauge and Yukawa couplings are used in the form α i = g i π , α λ ijk = λ ijk π . (3.5)– 5 – .1 Gaugino Mass The gaugino mass below the messenger scale can be obtained from Eqn.(3.1). At the GUT(compactification scale) M G , the gauge coupling unification requires T l a = 1 g ( GU T ) , (3.6)The gauge coupling at scale µ just below the messenger threshold M is given as1 g i ( µ ) = 1 g i ( GU T ) + b i + ∆ b i π ln M G | X | + b i π ln | X | µ , = T l a + b i + ∆ b i π ln M G M + b i π ln Mµ . (3.7)The derivatives are given as ∂∂ ln µ (cid:18) g i ( µ ) (cid:19) = − b i π , ∂∂ ln M (cid:18) g i ( µ ) (cid:19) = − ∆ b i π , (3.8)and ∂∂T (cid:18) g a ( µ ) (cid:19) = l a T l a − = ⇒ − g a ∂g a ( µ ) ∂T = l a T l a − , (3.9)So we can obtain the analytical results for gaugino mass M i ( µ ) = g i ( µ ) (cid:20) l a F T T g a ( GU T ) + F φ b i π − d F φ ∆ b i π (cid:21) . (3.10)with ∆ b i ≡ b ′ i − b i and b ′ i , b i the gauge beta function upon and below the messenger thresh-olds, respectively. This results can coincide with the gaugino masses predicted from RGErunning with threshold corrections at the messenger scale. Following the approach in [13],the gaugino mass at the scale µ slightly below the messenger scale M will receive additionalgauge mediation contributions M i ( µ . M ) = g i ( M ) g i ( GU T ) M i ( GU T ) − F φ g i ( M )16 π ( d + 1)∆ b i , = g i ( M ) (cid:20) l a F T T g a ( GU T ) + F φ b i + ∆ b i π (cid:21) − F φ g i ( M )16 π ( d + 1)∆ b i , (3.11)with M i ( GU T ) = g i ( GU T ) (cid:20) l a F T T g a ( GU T ) + F φ b i + ∆ b i π (cid:21) . (3.12)Then we can obtain the gaugino mass at scale µ < M from one-loop RGE M i ( µ ) = g i ( µ ) g i ( M ) M i ( µ . ln M ) , = g i ( µ ) (cid:20) l a F T T g a ( GU T ) + F φ b i π (cid:21) − F φ g i ( µ )16 π d ∆ b i , (3.13)So we can see that the two results agree with each other.– 6 – .2 Trilinear Terms From the form of wavefunction Z i ( µ ) = Z i (Λ) Y l = y t ,y b ,y τ (cid:18) y l ( µ ) y l (Λ) (cid:19) A l Y k =1 , , (cid:18) g k ( µ ) g k (Λ) (cid:19) B k (3.14)we can obtain the trilinear terms for scales below the messenger M from Eqn.(3.2). The mainchallenge is the calculation of ∂Z i /∂T .Before we derive the final results involving all y t , y b , y τ and g , g , g , we will study firstthe simplest case in which only the top Yukawa α t ≡ y t / π and α s ≡ g / π are kept in theanomalous dimension. The RGE equation for α t and α s takes the form ddt ln α t = 1 π (cid:18) α t − α s (cid:19) , ddt ln α s = − π b α s , (3.15)Note the definition b differs by a minus sign. Define A = ln (cid:18) α t α − b s (cid:19) , the equation can bewritten as ddt e − A = − π α b s , (3.16)So we can exactly solve the differential equation to get " α t ( µ ) α t (Λ) (cid:18) α s ( µ ) α s (Λ) (cid:19) − b − = 1 − α t (Λ) π π − b " α s (Λ) − − (cid:18) α s ( µ ) α s (Λ) (cid:19) b α − s ( µ ) . (3.17)Expanding the expressions and neglect high order terms, we finally have ∂∂T [ln α t ( µ ) − ln α t (Λ)] ≈ ∂∂T (cid:20) − π α s ( µ ) + 3 π α t ( µ ) (cid:21) ln (cid:18) Λ µ (cid:19) . (3.18)after calculations. It can be observed that the expression within the square bracket is justthe beta function of top Yukawa coupling.Now we will calculate ∂Z i /∂T with all y t , y b , y τ and g , g , g taking into account in theexpression. • Deduction of ∂Z i /∂T without messenger deflections:From the form of wavefunction Z i ( µ ) = Z i ( M G ) Y l = y t ,y b ,y τ (cid:18) y l ( µ ) y l ( M G ) (cid:19) A l Y k =1 , , (cid:18) g k ( µ ) g k ( M G ) (cid:19) B k (3.19)and renormalizatoin Z = Z (1 + δZ ), we have ∂ ln e − K / Z i ∂T = ∂∂T ln e − K / Z i ( M G ) + ∂∂T δZ i , = 1 − n i T + X m =1 , X a ∂g a ; m ∂T ∂δZ i ∂g a ; m + X a,b,c ∂ ln y abc ; m ∂T ∂δZ i ∂ ln y abc ; m , – 7 –ith m = 1 , µ and the GU T scale, respectively.The derivative with respect to g m gives ∂ ln e − K / Z i ∂g m ( µ ) = B m g m ( µ ) , ∂ ln e − K / Z i ∂g m ( M G ) = − B m g m ( M G ) , (3.20)and ∂g i ( µ ) ∂T = − l a T l a − g i ( µ ) , ∂g i ( M G ) ∂T = − l a T l a − g i ( M G ) . (3.21)The derivative with respect to y l gives ∂ ln e − K / Z i ∂y l ( µ ) = A l y l ( µ ) , ∂ ln e − K / Z i ∂y l ( M G ) = − A l y l ( M G ) , (3.22)and ∂y l ( M G ) ∂T = − y l ( M G )2 (cid:20) − a ijk T (cid:21) , ∂ ln α Y abc ( µ ) ∂T = − (cid:20) − a ijk T (cid:21) . (3.23)From the beta function of the Yukawa couplings, we have ∂ ln α Y abc ( µ ) ∂T = ∂ ln α Y abc ( M G ) ∂T − ∂∂T M G Z µ (cid:18) dd ln µ ′ ln α Y abc (cid:19) d ln µ ′ , = − − a abc T − π M G Z µ d ln µ ′ X Y lmn c lmn ∂∂T α Y lmn ( µ ′ ) + X m d m ∂∂T α m ( µ ′ ) , ≈ − − a abc T + 12 π X Y lmn c lmn − a lmn T α Y lmn ( µ ) + X m d m l a T α m ( µ ) α m ( M G ) ln (cid:18) M G µ (cid:19) , with a abc = n a + n b + n c .So the derivative with respect to T is given by ∂∂T δZ i = X m =1 , X a ∂ ln α a ; m ∂T ∂δZ i ∂ ln α a ; m + X Y abc ∂ ln α Y abc ; m ∂T ∂δZ i ∂ ln α Y abc ; m , = X a B a (cid:20) ∂∂T ln (cid:18) α a ( µ ) α a (Λ) (cid:19)(cid:21) + X Y abc A Y abc (cid:20) ∂∂T ln (cid:18) α Y abc ( µ ) α Y abc (Λ) (cid:19)(cid:21) , ≈ X a B a (cid:20) ∂∂T (cid:18) − b a π α a ( µ ) ln (cid:18) Λ µ (cid:19)(cid:19)(cid:21) + X Y abc A Y abc π X Y lmn c lmn − a lmn T α Y lmn ( µ ) − X m d m ∂∂T α m ( µ ) ln (cid:18) Λ µ (cid:19) , (3.24)– 8 –e know from the expression of the wavefunction, the coefficients satisfy X Y abc A Y abc d m + b m B m − ∂G Z i ∂α m , (3.25)for coefficients of α m . While the coefficients for Yukawa couplings Y lmn within Z i satisfy X Y abc A Y abc c lmn = − ∂G Z i ∂α Y lmn , (3.26)So the final results reduces to ∂∂T ln e − K / Z i ≈ − π " d ijk − a Y ijk T α Y ijk ( µ ) − C a ( i ) l a T α a ( µ ) ln (cid:18) GU Tµ (cid:19) + 1 − n i T , = 12 π ∂∂T " d ijk α Y ijk ( µ ) − C a ( i ) α a ( µ ) ln (cid:18) GU Tµ (cid:19) + 1 − n i T (3.27)with the expressions in the second square bracket being the anomalous dimension of Z i . • Deduction of ∂Z i /∂T with messenger deflections:From the form of wavefunction Z i ( µ ) = Z i ( M G ) Y l = y t ,y b ,y τ (cid:18) y l ( M ) y l ( M G ) (cid:19) A l Y k =1 , , (cid:18) g k ( M ) g k ( M G ) (cid:19) B k Y k = y U (cid:18) y k ( M ) y k ( M G ) (cid:19) C k Y l = y t ,y b ,y τ (cid:18) y l ( µ ) y l ( M ) (cid:19) A ′ l Y k =1 , , (cid:18) g k ( µ ) g k ( M ) (cid:19) B ′ k , (3.28)with y U the interactions involving the messengers which will be integrated below themessenger scale.We have ∂ ln e − K / Z i ∂T = ∂∂T ln e − K / Z i + ∂∂T δZ i , = X g a ∂ ln (cid:16) g a ( µ ) g a ( M ) (cid:17) ∂T ∂δZ i ∂ ln (cid:16) g a ( µ ) g a ( M ) (cid:17) + X y abc ∂ ln (cid:16) y abc ( µ ) y abc ( M ) (cid:17) ∂T ∂δZ i ∂ ln (cid:16) y abc ( µ ) y abc ( M ) (cid:17) , + X g a ∂ ln (cid:16) g a ( M ) g a ( M G ) (cid:17) ∂T ∂δZ i ∂ ln (cid:16) g a ( M ) g a ( M G ) (cid:17) + X y abc ∂ ln (cid:16) y abc ( M ) y abc ( M G ) (cid:17) ∂T ∂δZ i ∂ ln (cid:16) y abc ( M ) y abc ( M G ) (cid:17) , + X y U ∂ ln (cid:16) y U ( M ) y U ( M G ) (cid:17) ∂T ∂δZ i ∂ ln (cid:16) y U ( M ) y U ( M G ) (cid:17) + 1 − n i T , (3.29) with m = 1 , µ and the GU T scale, respectively.– 9 –sing similar deductions for Yukawa couplings, we can obtain ∂ ln e − K / Z i ∂T = 1 − n i T − π X g a (cid:18) B a b ′ a ∂α a ( M ) ∂T ln (cid:18) M G M (cid:19) + B ′ a b a ∂α a ( M ) ∂T ln (cid:18) Mµ (cid:19)(cid:19) + X y abc ∈ y t ,y b ,y τ A y abc ln (cid:18) M G M (cid:19) π X Y lmn ∈ y t ,y b ,y τ c lmn − a lmn T α Y lmn ( M ) , + X Y lmn ∈ y U ˜ c lmn − a U T α Y U ( M ) − X g m d m ∂∂T α m ( M ) . + X y abc ∈ y t ,y b ,y τ A ′ y abc ln (cid:16) µM (cid:17) π X Y lmn ∈ y t ,y b ,y τ c lmn − a lmn T α Y lmn ( M ) − X g m d m ∂∂T α m ( M ) . + X y U C y U ln (cid:18) MM G (cid:19) π X ˜ Y lmn ∈ y t ,y b ,y τ d lmn − a lmn T α Y lmn ( M ) − X g m f m ∂∂T α m ( M ) , + X ˜ Y lmn ∈ y U ˜ d lmn − a U T α Y U ( M ) . (3.30) with the beta function for y t , y b , y τ Yukawa couplings16 π β Y abc ( µ ) = P Y lmn ∈ y t ,y b ,y τ c lmn α Y lmn + P Y lmn ∈ y U ˜ c lmn α Y U − P g m d m α m , µ & M, P Y lmn ∈ y t ,y b ,y τ c lmn α Y lmn − P g m d m α m , µ . M, and the beta function for new messenger-matter y U Yukawa couplings16 π β Y U = X ˜ Y lmn ∈ y t ,y b ,y τ d lmn α ˜ Y lmn + X ˜ Y lmn ∈ y U ˜ d lmn α Y U − X m f m α m . (3.31)The coefficients satisfy X Y abc (cid:0) A y abc − A ′ y abc (cid:1) c lmn + X Y U C y U d lmn = 0 , (for y t , y b , y τ coefficients) X Y abc A y abc ˜ c lmn + X Y U C y U ˜ d lmn = 0 , (for y U coefficients) B m b ′ m + X Y abc A y abc d m + X Y U C Y U f m = B ′ m b m + X Y abc A ′ y abc d m (3.32)and similarly for g m , the sum then reduces to the previous case. So we have for µ < M∂∂T ln (cid:16) e − K / Z i ( µ ) (cid:17) − − n i T , (3.33) ≈ − π (cid:20) d ijk − a Y ijk T α Y ijk ( µ ) − C a ( i ) l a T α a ( µ ) (cid:21) ln (cid:18) M G µ (cid:19) . – 10 –ote that the expressions within the square bracket agree with the anomalous dimensionof Z − i below the messenger threshold MG − i ≡ dZ − i d ln µ ≡ − π (cid:18) d ikl λ ikl − c ir g r (cid:19) . (3.34)The G + i , which is the anomalous dimension of Z i upon the messenger threshold M , donot appear in the final expressions.The dependence of Z i on messenger scale M can be derived following the techniques [26, 27]developed in gauge mediated SUSY breaking (GMSB)[28] scenarios. From the expressions ofthe wavefunction, we can obtain ∂∂ ln M ln h e − K / Z i i = 14 π X g k (cid:2) ( B k − B ′ k )( b k + N F ) α k ( M ) + B ′ k N F α k ( µ ) (cid:3) (3.35)+ X Y l (cid:20) ( A l − A ′ l ) G + Y l (ln M ) + A ′ l ∂Y l (ln µ, M ) ∂ ln M (cid:21) + X Y U h C l G + Y U (ln M ) i . So the main challenge is to calculate ∂ ln Y a ( µ, ln M ) /∂ ln M .From the beta functions for Yukawa couplings upon and below the messenger thresholds,the Yukawa couplings at scale µ < M is given asln Y a ( µ, ln M ) = ln Y a ( M G ) + ln M Z M G G + Y a ( t ′ ) dt ′ + ln µ Z ln M G − Y a ( t ′ , ln M ) dt ′ , (3.36)with the Yukawa beta functions expressed as β Y a ≡ G Y a ≡ − X i ∈ a G i ≡ π (cid:18)
12 ˜ d ikl α λ ikl − c r α r (cid:19) ,G i = d ln Z i d ln µ ≡ − π (cid:18) d ikl α λ ikl − c ir α r (cid:19) . (3.37)We can derive the Yukawa couplings dependence on ′ ln M ′ at scale µ < M∂∂ ln M ln Y a ( µ, ln M ) = (cid:2) G + Y a (ln M ) − G − Y a (ln M, ln M ) (cid:3) + ln µ Z ln M ∂∂ ln M G − Y a ( t ′ , ln M ) dt ′ , ≈ ∆ G Y a (ln M ) − π (cid:20) ˜ d ikl λ ikl ( µ )∆ G λ ikl − c r ∆ b r π g r ( µ ) (cid:21) ln (cid:18) Mµ (cid:19) , In the case ∆ G = 0 in which no additional Yukawa couplings involving the messengers arepresent, we have ∂∂ ln M ln Y a ( µ, ln M ) ≈ ˜ c r π ∆ b r α r ( µ ) ln (cid:18) Mµ (cid:19) . (3.38)– 11 –ote that at the messenger scale ∂∂ ln M ln Y a (ln M, ln M ) = ∆ G a (ln M ) . (3.39)The expressions takes a simple form at the scale µ slightly below the messenger scale MA Y abc ( µ . M ) − (3 − a abc ) F T T + T ∗ = X l = a,b,c (cid:26) − F T T + T ∗ π (cid:20) d ijk (3 − a Y ijk ) α Y ijk ( µ ) − C a ( i ) l a α a ( µ ) (cid:21) ln (cid:18) GU Tµ (cid:19) + dF φ ∆ G i − F φ G − i (cid:27) , with ∆ G i ≡ G + i − G − i [here ′ G + i ( G − i ) ′ denotes respectively the anomalous dimension of Z i upon (below) the messenger threshold] the discontinuity of anomalous dimension across themessenger threshold. The soft scalar masses are given as − m soft = (cid:12)(cid:12)(cid:12)(cid:12) F T ∂∂T − F φ ∂∂ ln µ + dF φ ∂∂ ln X (cid:12)(cid:12)(cid:12)(cid:12) ln h e − K / Z i ( µ, X, T ) i , = | F T | ∂ ∂T ∂T ∗ + F φ ∂ ∂ (ln µ ) + d F φ ∂∂ (ln | X | ) − F T F φ ∂ ∂T ∂ ln µ + dF T F φ ∂ ∂T ∂ ln | X | − dF φ ∂ ∂ ln | X | ∂ ln µ ! ln h e − K / Z i ( µ, X, T ) i , (3.40)The new ingredients are the second derivative of Z i with respect to T∂ ∂T ln (cid:2) e − K Z i (cid:3) = − π ∂∂T (cid:20) d ijk − a Y ijk T α Y ijk ( µ ) − C a ( i ) l a T α a ( µ ) (cid:21) ln (cid:18) GU Tµ (cid:19) − − n i T , = − π " d ijk − a Y ijk T α Y ijk ( µ ) " − − a Y ijk T + 12 π ˜ d pmn − a Y ijk T α Y mnp − c r l a T α a ! ln (cid:18) GU Tµ (cid:19) − d ijk (3 − a Y ijk ) T α Y ijk ( µ ) − C a ( i ) (cid:18) − l a T α a ( µ ) − l a T α a ( µ ) α a ( GU T ) (cid:19)(cid:21) ln (cid:18) GU Tµ (cid:19) − − n i T . (3.41)with the beta function of Y ijk given by d ln Y ijk d ln µ = 116 π " ˜ d pmn α Y mnp − c ir α i . (3.42)– 12 –nd α a ( µ ) α a ( GU T ) = 1 − b a π α a ( µ ) ln (cid:18) GU Tµ (cid:19) . (3.43)The other terms within Eqn.(3.40) can be found in GMSB (not involving ∂T ) or calcu-lated directly using Eqn.(3.21) and Eqn.(3.23) (involving ∂T ). We list the analytical resultsof deflected mirage mediation in Appendix B. Equipped with the previous deduction, we can readily reproduce the ordinary mirage media-tion results by setting d →
0. As the visible gauge fields originate from D7 branes and gaugecoupling unification is always assumed, we adopt l a = 1. The following definitions are used M ≡ F T T ≡ F φ α ln (cid:16) M Pl m / (cid:17) ≈ F φ π α . (4.1)with the parameter α defined as the ratio between the anomaly mediation and modulusmediation contributions and the approximation ln( M P l /m / ) ≈ π . We have • Gaugino mass: M i ( µ ) = l a M g i ( µ ) g a ( GU T ) + F φ π b i g i ( µ ) , = l a M (cid:20) − b i π g i ( µ ) ln GU Tµ (cid:21) + M αb i g i ( µ ) . (4.2)So we can see that at the scale µ Mi which satisfies18 π ln (cid:18) M GUT µ Mi (cid:19) = α . (4.3)the gaugino masses unify at such ′ mirage ′ unification scale µ Mi = M GUT e − απ ≈ M GUT (cid:18) m / M P l (cid:19) α . (4.4) • Trilinear Term: A Y abc ( µ . M )= X l = a,b,c (cid:26) − M π (cid:20) d ijk (3 − a Y ijk ) α Y ijk ( µ ) − C a ( i ) l a α a ( µ ) (cid:21) ln (cid:18) GU Tµ (cid:19) + F φ π (cid:20) d ijk α Y ijk ( µ ) − C a ( i ) α a ( µ ) (cid:21) + (3 − a abc ) M , – 13 –n case the effect of Yukawa couplings are negligible or a Y ijk = 2, the trilinear term also” unif y ” at a mirage scale at which the last two terms cancel12 π ln (cid:18) M GUT µ Mi (cid:19) = πα. (4.5)which is just the mirage scale for gaugino mass ” unif ication ”. • Soft Scalar Masses: − m i = M π ln (cid:18) M GUT µ (cid:19) ( d ijk (cid:16) q Y ijk + q Y ijk (cid:17) α Y ijk ( µ ) − C a ( i ) (cid:0) l a + l a (cid:1) α a + 12 π " d ijk α Y ijk ( µ ) − ˜ d pmn q Y mnp α Y mnp + 2 c r l a α a ! + 2 C a ( i ) b a α a ln (cid:18) GU Tµ (cid:19)) + M F φ π " d ijk α Y ijk − q Y ikl + 12 π " ˜ d pmn q Y mnp α Y mnp − c r l r α r ln (cid:18) M GUT µ (cid:19)! + 2 C a ( i ) l a α a α a ( GU T ) − F φ π " d ijk π ˜ d pmn α Y mnp − c r α r ! α Y ijk − C a ( i ) b a π α a − (1 − n i ) M . (4.6) with q Y ijk ≡ − ( n i + n j + n k ) = 3 − a ijk . Again, we can check that for q Y ijk = 1 ornegligible Yukawa couplings, the soft scalar masses apparent unify at µ Mi defined above − m i + (1 − n i ) M = παM ( d ijk α Y ijk ( µ ) − C a ( i ) α a + 12 π " d ijk α Y ijk − ˜ d pmn α Y mnp + 2 c r α a ! + 2 C a ( i ) b a α a π α ) + 2 παM " d ijk α Y ijk − π " ˜ d pmn α Y mnp − c ir α i π α ! + 2 C a ( i ) (cid:18) α a − π b a α a π α (cid:19) − π α M " d ijk π ˜ d pmn α Y mnp − c ir α i ! α Y ijk − C a ( i ) b a π α a = 0 . (4.7) The subleading terms within ∂ Z i /∂T are crucial for the exact cancelation of anomalymediation and RGE effects.So the numerical results of ′ mirage ′ unification can be proved rigourously with our analyticalexpressions. It is well known that AMSB is bothered by tachyonic slepton problems. Such a problem inAMSB can be solved by the deflection of RGE trajectory with the introduction of messen-ger sector. There are two possible ways to deflect the AMSB trajectory with the presenceof messengers, either by pseudo-moduli field[10] or holomorphic terms (for messengers) inthe Kahler potential[29]. Mirage mediation is a typical mixed modulus-anomaly mediation– 14 –cenario. So the messenger sector, which can give additional gauge or Yukawa mediationcontributions, can also be added in the Kahler potential.The Kahler potential involving the vector-like messengers ¯ P i , P i contain the ordinarykinetic terms as well as new holomorphic terms K ⊇ φ † φ h Z P i , ¯ P i ( T † , T ) (cid:16) P † i P i + ¯ P † i ¯ P i (cid:17) + (cid:16) ˜ Z P i , ¯ P i ( T † , T ) c P ¯ P i P i + h.c. (cid:17)i , (4.8)with Z P i , ¯ P i ( T † , T ) = 1( T + T † ) n P , ˜ Z P i , ¯ P i ( T † , T ) = 1( T + T † ) ˜ n P . (4.9)After normalizing and rescaling each superfield with the compensator field Φ → φ Φ andsubstituting the F-term VEVs of the compensator field φ = 1 + F φ θ , the relevant Kahlerpotential reduces to W = Z d θ φ † φ T + T † ) ˜ n P − n P (cid:0) c P R ¯ P P (cid:1) , (4.10)For simply, we define ˜ n P − n P ≡ a P . Especially, a P = n P for ˜ n P = 0.The SUSY breaking effects can be taken into account by introducing a spurion superfields R with with the spurion VEV as R ≡ M R + θ F R = 1(2 T ) a P (cid:16) F φ − a P T F T (cid:17) + θ (cid:20) a P ( a P + 1) | F T | T − | F φ | (cid:21) . (4.11)with the value of the deflection parameter d ≡ F R M R F φ − , (4.12)depending on the choice of a P and α which gives d = − a P = 0. We can see that addingmessenger sector in the Kahler potential within mirage mediation will display a new featurein contrast to the AMSB case which always predicts d = − ′ d ′ into the general formula given in the appendix. Note that wecan derive the final results directly with its low energy analytical expressions. Besides, we canalso add messenger-matter mixing to induce new Yukawa couplings between the messengersand the MSSM fields. In this case, new Yukawa mediated contributions will also contributeto the low energy soft SUSY parameters (See Ref.[31] for an example in AMSB). In ordinary deflected mirage mediation SUSY breaking scenarios, additional messengers areintroduced merely to amend the gauge beta functions which will subsequently feed into thelow energy soft SUSY breaking parameters. In general, it is possible that the messengers– 15 –ill share some new Yukawa-type interactions with the visible (N)MSSM superfields, whichsubsequently will appear in the anomalous dimension of the superfields and contribute to thelow energy soft SUSY breaking parameters. Such realizations have analogs in AMSB (see[31]) and can be readily extended to include the modulus mediation contributions.Similar to the deflected mirage mediation scenarios, the superpotential include possiblepseudo-modulus superfields X , the relevant nearly flat superpotential W ( X ) to determinethe deflection and a new part that includes messenger-matter interactions W mm = λ φij XQ i Q j + y ijk Q i Q j Q k + W ( X ) . (4.13)with the Kahler potential K m = Z U T + T † , µ p φ † φ ! T + T † ) n Qi Q † i Q i , (4.14)Here ′ φ ′ denotes the compensator field with Weyl weight 1. The indices ′ i, j ′ run over allMSSM and messenger fields and the subscripts ′ U, D ′ denote the case upon and below themessenger threshold, respectively.After integrating out the heavy messenger fields, the visible sector superfields Q a willreceive wavefunction normalization L = Z d θQ † a Z abD ( T + T † , µ p φ † φ , s X † Xφ † φ ) Q b + Z d θy abc Q a Q b Q c , (4.15)which can give additional contributions to soft supersymmetry breaking parameters. Herethe analytic continuing threshold superfield ′ X ′ will trigger SUSY breaking mainly from theanomaly induced SUSY breaking effects with the form < X > = M + θ F X . So we have˜ X ≡ Xφ = M + F X θ F φ θ ≡ M (1 + dF φ θ ) , (4.16)with the value of the deflection parameter ′ d ′ determined by the form of superpotential W ( X ).Integrating out the messengers, the messenger-matter interactions will cause the discon-tinuity of the anomalous dimension upon and below the threshold. Such discontinuity willappear not only directly in the expressions for the trilinear couplings but also indirectly inthe soft scalar masses. For example, the trilinear couplings at the messenger scale receiveadditional contributions∆ A ijk | µ = M = X a = i,j,k d F φ ∂∂ ln | X | ln h e − K / Z a ( µ, X, T ) i(cid:12)(cid:12)(cid:12) µ = M = d F φ X a = i,j,k ∆ G i | µ = M . (4.17)We know that large trilinear couplings, especially A t , is welcome in low energy phenomeno-logical studies to reduce fine tuning and increase the Higgs mass. So the introduction ofmessenger-matter interactions can open new possibilities for mirage phenomenology.– 16 – Conclusions
We derive explicitly the soft SUSY breaking parameters at arbitrary low energy scale inthe (deflected) mirage type mediation scenarios with possible gauge or Yukawa mediationcontributions. Based on the Wilsonian effective action after integrating out the messengers,we obtain analytically the boundary value (at the GUT scale) dependencies of the effectivewavefunctions and gauge kinetic terms. Note that the messenger scale dependencies of theeffective wavefunctions and gauge kinetic terms had already been discussed in GMSB. TheRGE boundary value dependencies, which is a special feature in (deflected) mirage typemediation, is the key new ingredients in this study. The appearance of ′ mirage ′ unificationscale in mirage mediation is proved rigorously with our analytical results. We also discussbriefly the new features in deflected mirage mediation scenario in the case the deflection comespurely from the Kahler potential and the case with messenger-matter interactions.We should note that our approach is in principle different from that of Ref.[23] in whichthe soft SUSY breaking parameters are obtained by numerical RGE evolution, matching andthreshold corrections. For example, mixed gauge-modulus mediation contributions, whichwill not appear in previous approach, will be necessarily present for the soft scalar masses inour approach. Acknowledgement
This work was supported by the Natural Science Foundation of China under grant numbers11675147,11775012; by the Innovation Talent project of Henan Province under grant number15HASTIT017.
Appendix A: Coefficients In Wavefunction Expansion
We can construct the RGE invariants ddt ln Z i = X l = y t ,y b ,y τ A l d ln y l dt + X l = g ,g ,g B l d ln g l dt , (5.1)by solving the equation in the basis of ( y t , y b , y τ , g , g , g ) − − b − − − b − − − b A t A b A τ B B B = − c − c − c − d − d − d (5.2)– 17 –ith c , c , c , d , d , d the relevant coefficients of ( y t , y b , y τ , g , g , g ) within the anomalousdimension. So from ddt Z i ( µ ) Y l = y t ,y b ,y τ [ y l ( µ )] − A l Y k =1 , , [ g k ( µ )] − B k = 0 , (5.3)we have Z i ( µ ) = Z i (Λ) Y l = y t ,y b ,y τ (cid:18) y l ( µ ) y l (Λ) (cid:19) A l Y k =1 , , (cid:18) g k ( µ ) g k (Λ) (cid:19) B k (5.4)The general expressions of wavefunction at ordinary scale µ below the messenger scale M are given as Z i ( µ ) = Z i (Λ) Y l = y t ,y b ,y τ (cid:18) y l ( M ) y l (Λ) (cid:19) A l Y k =1 , , (cid:18) g k ( M ) g k (Λ) (cid:19) B k Y k = y U (cid:18) y k ( M ) y k (Λ) (cid:19) C k Y l = y t ,y b ,y τ (cid:18) y l ( µ ) y l ( M ) (cid:19) A ′ l Y k =1 , , (cid:18) g k ( µ ) g k ( M ) (cid:19) B ′ k , (5.5)with y U the interactions involving the messengers which will be integrated below the messen-ger scale. The coefficients are listed in Table.1 and Table.2. Table 1 . Relevant coefficients in wavefunction expansion with N F = 0 messengers. A ( y t ) A ( y b ) A ( y τ ) B ( g ) B ( g ) B ( g ) Q − - - - U − - - − D - - − L −
361 1861 − - - E −
661 3661 − - − H u − −
361 272183 2161 - H d - − − - Q U D L E – 18 – able 2 . The coefficients with N F = 0 , , , y t , y b , y τ , namely A ( y t ), A ( y b ), A ( y τ ), arethe same as the case N F = 1. B g ( i ) B g ( i ) B g ( i ) Q ( − , − , − , ) ( , , , ) ( − , − , − , − ) U ( − , − , − , ) ( − , − , − , − ) ( , , , ) D ( − , − , − , ) ( − , − , − , − ) ( , , , ) L ( − , − , − , ) ( , , , ) ( − , − , − , − ) E ( − , − , − , ) ( − , − , − , − ) ( , , , ) H u ( , , , − ) ( , , , ) ( − , − , − , − ) H d ( , , , − ) ( − , − , − , − ) ( − , − , − , − ) Q ( − , − , − , ) (3 , , , ) ( , , , ) U ( − , − , − , ) (0 , , ,
0) ( , , , ) D ( − , − , − , ) (0 , , ,
0) ( , , , ) L ( 0 , , ,
0) (3 , , , ) ( , , , ) E ( 0 , , ,
0) (0 , , ,
0) ( , , , ) Appendix B: Low Energy Spectrum in Deflected Mirage Mediation
In order to show some essential features of our effective theory results, we list the predictedsoft SUSY breaking parameters in deflected mirage mediation mechanism with N F messengersin ⊕ ¯ representations of SU(5).At energy µ below the messenger thresholds, we have • The gaugino masses: M i ( µ ) = l a M g i ( µ ) g a ( GU T ) + F φ π b i g i ( µ ) − d F φ π N F g i ( µ ) . (5.6)with ( b , b , b ) = ( − , ,
335 ) . (5.7) • The trilinear couplings A t , A b and A τ :Note that at the messenger scale, the third contribution ∂ X Z i vanishes. The trilinear A t term is given at arbitrary low energy scale µ < MA t ( µ ) − q y t M (5.8)= M π (cid:20) q y t α y t ( µ ) + q y b α y b ( µ ) − l α ( µ ) − l α ( µ ) − l α ( µ ) (cid:21) ln (cid:18) M GUT µ (cid:19) + F φ π (cid:20) α y t ( µ ) + α y b ( µ ) − α ( µ ) − α ( µ ) − α ( µ ) (cid:21) + δ G , (5.9)– 19 –ote that additional GMSB-type contributions are δ G = d F φ π X k =1 , , X F = Q L ,U ,H u h ( B k ( F ) − B ′ k ( F ))( b k + N F ) α k ( M ) + B ′ k ( F ) N F α k ( µ ) i + d F φ π X y l = y t ,y b ,y τ X k =1 , , A ′ l π N F ˜ c r ( y l ) α r ( µ ) ln (cid:18) Mµ (cid:19) = d F φ π (cid:20) − π N F (cid:18) α ( µ ) + 3 α ( µ ) + 1315 α ( µ ) (cid:19) ln (cid:18) Mµ (cid:19)(cid:21) , (5.10)with 2˜ c r ( y l ) the coefficients of g r within − π β y l and X F = Q L ,U ,H u B k ( F ) = X F = Q L ,U ,H u B ′ k ( F ) = 0 . (5.11)The trilinear A b term is A b ( µ ) − q y b M (5.12)= M π (cid:20) q y t α y t ( µ ) + 6 q y b α y b ( µ ) + q y τ α y τ ( µ ) − l α ( µ ) − l α ( µ ) − l α ( µ ) (cid:21) ln (cid:18) M GUT µ (cid:19) + F φ π (cid:20) α y t ( µ ) + 6 α y b ( µ ) + α y τ ( µ ) − α ( µ ) − α ( µ ) − α ( µ ) (cid:21) + d F φ π (cid:20) − π N F (cid:18) α ( µ ) + 3 α ( µ ) + 715 α ( µ ) (cid:19) ln (cid:18) Mµ (cid:19)(cid:21) . (5.13)The trilinear A τ term is A τ ( µ ) − q y τ M (5.14)= M π (cid:20) q y b α y b ( µ ) + 4 q y τ α y τ ( µ ) − l α ( µ ) − l α ( µ ) (cid:21) ln (cid:18) M GUT µ (cid:19) + F φ π (cid:20) α y b ( µ ) + 4 α y τ ( µ ) − α ( µ ) − α ( µ ) (cid:21) + d F φ π (cid:20) − π N F (cid:18) α ( µ ) + 95 α ( µ ) (cid:19) ln (cid:18) Mµ (cid:19)(cid:21) . (5.15) • The soft SUSY breaking scalar masses are parameterized by several terms: − m = − (1 − n i ) M + δ I + δ II + δ III + δ IV + δ V . (5.16)The anomalous dimension of Z i is supposed to take the form G i ≡ d ln Z i d ln µ = − π (cid:18) d ikl α λ ikl − C a ( i ) α a (cid:19) . (5.17)with α λ ikl = λ ikl / π and α a = g a / π .– 20 – Pure modulus mediation contributions δ I = M π ln (cid:18) M GUT µ (cid:19) ( d ijk (cid:16) q Y ijk + q Y ijk (cid:17) α Y ijk ( µ ) − C a ( i ) (cid:0) l a + l a (cid:1) α a + 12 π " d ijk α Y ijk ( µ ) − ˜ d pmn q Y mnp α Y mnp + 2 c r l a α a ! + 2 C a ( i ) b a α a ln (cid:18) GU Tµ (cid:19)) (5.18) – Pure anomaly mediation contributions δ II = F φ ∂ ∂ (ln µ ) ln h e − K / Z i i (5.19)= − F φ π ∂∂ (ln µ ) (cid:20) d ikl α λ ikl − C a ( i ) α a (cid:21) , = − F φ π (cid:20) d ikl α λ ikl G − λ ikl − C a ( i ) 12 π b a α a (cid:21) . (5.20)with the beta function for Yukawa coupling λ ikl being d ln λ ikl d ln µ = G λ ikl = 14 π (cid:20) d pmn α λ mnp − c r α r (cid:21) . (5.21) – Pure gauge mediation contributionsAs no new interactions involving the messengers are present, we have ∂∂ ln M ln h e − K / Z i i = 14 π X g k (cid:2) ( B k − B ′ k )( b k + N F ) α k ( M ) + B ′ k N F α k ( µ ) (cid:3) + X Y l A ′ l ˜ c r π ∆ b r α r ( µ ) ln (cid:18) Mµ (cid:19) , (5.22)So δ III = d F φ ∂ ∂ (ln M ) ln h e − K / Z i i (5.23)= d F φ π X g k (cid:2) ( B k − B ′ k )( b k + N F ) α k ( M ) + B ′ k N F α ( µ ) (cid:3) + X Y l A ′ l ˜ c r π ∆ b r α r ( µ ) . Here ∂∂ ln M α k ( M ) = b + k π α k ( M ) ,∂∂ ln M α k ( µ, M ) = b + k − b − k π α k ( µ, M ) ≡ ∆ b k π α k ( µ, M ) , (5.24)– 21 – The gauge-anomaly interference term δ IV = − dF φ ∂ ∂ ln M ∂ ln µ ln h e − K / Z i ( µ, X, T ) i , = − dF φ ∂∂ ln µ ( π X g k (cid:2) ( B k − B ′ k )( b k + N F ) α k ( M ) + B ′ k N F α k ( µ ) (cid:3) + X Y l A ′ l ˜ c r π ∆ b r α r ( µ ) ln (cid:18) Mµ (cid:19) , = − dF φ π B ′ k b k N F α k ( µ ) − dF φ ˜ c r π A ′ l ∆ b r α r ( µ ) . (5.25) – The modulus-anomaly and modulus-gauge interference terms are given as δ V = − F T F φ ∂ ∂T ∂ ln µ ln h e − K / Z i i + dF T F φ ∂ ∂T ∂ ln | X | ln h e − K / Z i i , = F T F φ π ∂∂T (cid:20) d ikl α λ ikl − C a ( i ) α a (cid:21) + dF T F φ ∂∂T ( π X g k (cid:2) ( B k − B ′ k )( b k + N F ) α k ( M ) + B ′ k N F α k ( µ ) (cid:3) + X Y l A ′ l ˜ c r π ∆ b r α r ( µ ) ln (cid:18) Mµ (cid:19) , = M F φ π (cid:20) d ikl α λ ikl (cid:18) − q y λikl + 12 π (cid:20) d pmn q y λmnp α λ mnp − c r l r α r (cid:21) ln (cid:20) M G µ (cid:21)(cid:19) +2 C a ( i ) l a T α a α a ( GU T ) (cid:21) − dF φ M π X g k (cid:20) ( B k − B ′ k )( b k + N F ) l k α k ( M ) α k ( GU T ) + B ′ k N F l k α k ( µ ) α k ( GU T ) (cid:21) − X Y l A ′ l dM F φ π ˜ c r ∆ b r l r α r ( µ ) α r ( GU T ) ln (cid:18) Mµ (cid:19) , (5.26)with ∂∂T α k ( µ ) = − l k T α k ( µ ) α k ( GU T ) α k ( µ ) , (5.27) References [1] G. Aad et al.(ATLAS Collaboration), Phys. Lett. B710, 49 (2012).[2] S. Chatrachyan et al.(CMS Collaboration), Phys. Lett.B710, 26 (2012). – 22 –
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