- Email: [email protected]
Two-loop effective potential calculation of the lightest CP-even Higgs-boson mass in the MSSM
4 February 1999
Physics Letters B 447 Ž1999. 89–97
Two-loop effective potential calculation of the lightest CP-even Higgs-boson mass in the MSSM Ren-Jie Zhang
1
Department of Physics, UniÕersity of Wisconsin, 1150 UniÕersity AÕenue, Madison, WI 53706, USA Received 12 August 1998; revised 19 November 1998 Editor: M. Cveticˇ
Abstract We calculate a two-loop effective potential to the order of O Ž l2t a s . in the MSSM. We then study the corresponding two-loop corrections to the CP-even Higgs-boson mass for arbitrary tan b and left-right top-squark mixings. We find that the lightest Higgs-boson mass is changed by at most a few GeV. We also show the improved scale dependence and compare to previous two-loop analyses. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction In the minimal supersymmetric standard model ŽMSSM., the Higgs sector is composed of three neutral Žtwo CP-even, one CP-odd. and two charged scalar bosons. An important fact of the model is that the quartic self-coupling of the lightest CP-even Higgs boson h is not a free parameter, but related to the standard model gauge couplings g and g X ; as a result, the Higgs boson has an upper bounded tree-level mass, m h F MZ . This limit, however, is violated when the one-loop radiative corrections are included. The well-known dominant one-loop radiative correction comes from the incomplete cancellation of the virtual top-quark and top-squark loops w1x, it approximately has the size
D m2h ,
3 l2t m2t 4p 2
log Ž m 2t˜ rm2t . .
Taking m t˜ s 100–1000 GeV, one finds a large correction, D m h , a few y 50 GeV, due to the relatively large top-quark mass m pole s 175 GeV. The one-loop Higgs-boson mass sensitively depends on the top-quark mass, t and generally varies with the change of renormalization scale. So it remains a quite important problem to study the magnitude of two-loop radiative corrections and the scale dependence of the Higgs-boson mass after these corrections.
1
E-mail: [email protected]
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 5 7 5 - 5
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There are basically two approaches to calculate the two-loop radiative corrections. In the renormalization group equation ŽRGE.-improved effective potential approach w2–4x, one uses the one-loop effective potential, together with two-loop RGEs, then all leading-order and next-to-leading-order corrections can be calculated. The finite one-loop threshold corrections arising from the decouplings of the heavy top-squarks have also been included, both for the small and large left-right top-squark mixings. It is further observed that by judiciously setting the renormalization scale, one can use the one-loop renormalized Higgs-boson mass as a good approximation to the full two-loop results w3,5x. The second approach involves a two-loop effective potential. In the special case of tan b ™ ` and no left-right mixing in the top-squark sector, Hempfling and Hoang have calculated the upper bound to the two-loop Higgs-boson mass w6x. Their results qualitatively agree with the previous approach. Recently, two-loop corrections to m h have also been computed by an explicit diagrammatic method w7x. It is the purpose of this paper to generalize the two-loop effective potential calculation of w6x to the case of arbitrary tan b and left-right top-squark mixings. The effective potential method has an advantage over other approaches because it is simple and does not require complicate programming. In this paper we will study the improved scale dependence of the Higgs-boson mass m h and the size of two-loop corrections. We will also compare our results with previous two-loop calculations w4,7x. The rest of the paper is organized as follows: We first present a general formalism for calculating the CP-even Higgs-boson mass from the effective potential in Section 2, and compute the two-loop effective potential to the order of O Ž l2t a s .. We next show the results of our numerical analyses in Section 3; we find good agreements with previous two-loop calculations. Finally we conclude in Section 4. For completeness, some functions which appear in the two-loop calculation are given in the Appendix.
2. Effective potential and the CP-even Higgs-boson masses We start our analysis with the tree-level potential of the MSSM 2 , Vtree s Ž
m2H 1 q m2 g
. < H1 < 2 q Ž m2H 2 q m2 . < H2 < 2 q m B Ž H1 H2 q H.c.. q
g 2 q gX 2 8
Ž < H1 < 2 y < H 2 < 2 .
2
2
q 2
< H1†H2 < 2 ,
Ž 1.
where g, g X are the SUŽ2. and UŽ1. Y gauge couplings, m H 1,m H 2 and B are the soft-breaking Higgs-sector mass parameters, and m is the supersymmetric Higgs-boson mass parameter Žthe m-parameter.. We express H1 and H2 in terms of their component fields, H1 s
ž
Ž S1 q iP1 . r'2 Hy 1
/
,
H2 s
ž
Hq 2
Ž S2 q iP2 . r'2
/
,
Ž 2.
so the tree-level potential can be rewritten as a function of the CP-even fields, S1 and S2 . In general, the all-loop effective potential is a function of S1 and S2 , which are usually known as classical fields. The technique for calculating a higher loop effective potential was developed long ago by Jackiw w9x. First, the Higgs fields are expanded around the classical fields, in terms of which all the relevant particle masses and couplings are determined. One then calculates the higher loop effective potential by computing the corresponding zero-point function Feynman diagrams Žbubble diagrams..
2
We work in the modified DR-scheme of Ref. w8x.
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To be more specific, to the two-loop order that we consider in this paper, we write the effective potential as V Ž S1 ,S2 . s V0 q Vtree Ž S1 ,S2 . q V1loop Ž S1 ,S2 . q V2loop Ž S1 ,S2 . ,
Ž 3.
where V0 is a field-independent vacuum-energy term, and Vtree , V1loop and V2loop are the tree-level, one- and two-loop contributions respectively. V0 is necessary for the renormalization group invariance of the effective potential w10x. The one-loop effective potential in the DR scheme is well-known. It can be easily obtained by calculating the one-loop bubble diagrams with all kinds of Žs.particles running in the loop. We find that in the Landau gauge 2
V1loop Ž S1 ,S2 . s Ý
Ý Ncf G Ž f˜i . y 2 Ý Ncf G Ž f . q 3G Ž W . q 32 G Ž Z . q 12
f is1
G Ž H . q G Ž h. q G Ž A.
f 2
qG Ž G . q G Ž Hq . q G Ž Gq . y 2
4
Ý G Ž x˜qi . y Ý G Ž x˜ i0 . , is1
Ž 4.
is1
where f sums over all the Žs.quarks and Žs.leptons, Ncf is the color factor, 3 for Žs.quarks and 1 for Žs.leptons, and all the masses are implicitly S1 , S2-dependent. H, h, A and G Ž Hq and Gq . label the neutral Žcharged. Higgs and Goldstone bosons, x˜q ˜ i0 represent charginos and neutralinos. We have also used short-handed i and x 2 2 ˜ notations f i s m f˜i , W s mW , etc. The function GŽ x . is defined as x2
Ž 5. Ž ln x y 32 . , 32p 2 where ln x s lnŽ xrQ 2 ., with Q the renormalization scale. The two-loop effective potential can be derived similarly, the corresponding Feynman diagrams are plot in Fig. 1. To the order of O Ž l2t a s ., we have in the Landau gauge GŽ x . s
½
V2loop Ž S1 ,S2 . s 32pa s J Ž t ,t . y 2 t I Ž t ,t ,0 . q 12 Ž c t4 q st4 .
2
Ý
2
J Ž t˜i ,t˜i . q 2 st2 c t2 J Ž t˜1 ,t˜2 . q
is1
2
q Ý L Ž t˜i , g˜ ,t . y 4 m g˜ m t st c t Ž I Ž t˜1 , g˜ ,t . y I Ž t˜2 , g˜ ,t . . is1
Ý t˜i I Ž t˜i ,t˜i ,0 . is1
5
Ž 6.
where the Žminimally. subtracted functions I and J are defined in the Appendix, st is the top-squark mixing angle. This effective potential, in the limit of tan b ™ ` and no left-right squark-mixing Ž st s 0., agrees with that of Ref. w6x. As a good check, one can show that the effective potential V Ž S1 ,S2 . is invariant under the renormalization scale change, up to two-loop terms which are ignored in our approximation. The effective potential Eq. Ž6., as a generating functional, encodes the information of two-loop tadpoles and self-energies at zero external momentum. From this, one can find two-loop CP-even Higgs-boson masses by solving appropriate on-mass-shell conditions.
Fig. 1. Bubble diagrams for the two-loop effective potential to the order of O Ž l2t a s . in the MSSM.
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This proceeds as follows: We first minimize V Ž S1 ,S2 . at the Higgs vacuum expectation values S1 s Õ1 and S2 s Õ 2 ,
EV E S1
EV
s 0,
E S2
S1 sÕ 1 ,S 2 sÕ 2
s 0,
Ž 7.
S1 sÕ 1 ,S 2 sÕ 2
which is equivalent to the requirement that the tadpoles vanish, T1 Õ1 T2 Õ2
s 12 m2Z c2 b q m2H 1 q m2 q Bm tan b ,
Ž 8.
s y 12 m2Z c 2 b q m2H 2 q m2 q Bm cot b ,
Ž 9.
where tan b s Õ 2rÕ1 , m 2Z s Ž g 2 q g X 2 . Õ 2r4 and Õ 2 s Õ 12 q Õ 22 . To the two-loop order, the tadpoles Ti , i s 1,2 are given by Ti s Ti1loop q Ti2loop , where the one- and two-loop tadpoles are defined by Ti1loop sy
E V1loop
ž
E Si
/
,
Ti2loop sy
S1 sÕ 1 ,S 2 sÕ 2
ž
E V2loop E Si
/
.
Ž 10 .
S1 sÕ 1 ,S 2 sÕ 2
One can check that the one-loop tadpoles Ti1loop obtained in this way give the same results as in Ref. w11x 3, where they were explicitly calculated from the one-point function Feynman diagrams Žtadpole diagrams.. The CP-even Higgs-boson mass matrix, after some algebra, is M 2 Ž p2 . s
ž
m2Z cb2 q m2A sb2 y Re P 11 Ž p 2 . q T1rÕ1
y Ž m2Z q m2A . sb cb y Re P 12 Ž p 2 .
y Ž m2Z q m 2A . sb cb y Re P 12 Ž p 2 .
m 2Z sb2 q m 2A cb2 y Re P 22 Ž p 2 . q T2rÕ 2
/
,
Ž 11 .
where m2A s ym B Žtan b q cot b ., and both m Z and m A are DR running masses. P ’s represent two-point functions Žself-energies.. The radiatively corrected Higgs-boson masses can be found by computing the zeroes of the inverse propagator, p 2 y M 2 Ž p 2 .. The complete one-loop formulae for self-energies at nonzero external momentum can be found, e.g., in Ref. w11x. For the two-loop self-energies, we shall approximate them as follows:
P i 2loop Ž 0 . sy j
ž
E 2 V2loop E Si E S j
/
,
i , j s 1,2.
Ž 12 .
S1 sÕ 1 ,S 2 sÕ 2
Ž0. and P i 2loop Ž p 2 . is negligible. The difference of P i 2loop j j
3. Numerical procedure and results We shall first show the improvement of renormalization scale Ž Q . dependence of the lightest CP-even Higgs-boson mass. We start by solving the two-loop renormalization group equations ŽRGEs. of the MSSM. The boundary condition inputs for these RGEs are taken to be the observables a s , a em , GF , MZ , m t , m b and mt at the electro-weak scale, MZ , and the universal scalar soft mass M0 , gaugino soft mass M1r2 and trilinear scalar coupling A 0 at the unification scale, MGUT f 2 = 10 16 GeV ŽThis is sometimes called the minimal
3
In Ref. w11x, we used the ’t Hooft-Feynman gauge. To compare with the one-loop tadpoles obtained from Eq. Ž4., the Goldstone boson masses need to be set to zero.
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supergravity ŽmSUGRA. scenario... We have also properly taken into account the low energy threshold corrections to the gauge and Yukawa couplings, as in Ref. w11x. In this scenario, the m-parameter and the Žtree-level. mass m A are determined in terms of other variables through the minimization conditions Ž8. and Ž9., which ensure the correct electro-weak symmetry breaking,
m2 s 12 tan2 b m2H 2 tan b y m2H 1cot b y m2Z ,
ž
m2A s
1 cos2 b
žm
Ž 13 .
/
2 2 H 2 y m H1
/ ym
2 Z
,
Ž 14 .
where m 2H 1 s m2H 1 y T1rÕ1 and m2H 2 s m 2H 2 y T2rÕ 2 . We then calculate the lightest CP-even Higgs-boson mass from the mass-squared matrix Eq. Ž11.. We use the one-loop tadpole and self-energy formulae from Ref. w11x, while for the two-loop contributions, we compute the tadpoles and self-energies numerically according to Eqs. Ž10. and Ž12. by replacing the differentiation by a finite difference. The field-dependent masses in Eq. Ž6. are the top-quark mass m t s l t S2r '2 and the top-squark masses which are found from the following field-dependent mass-squared matrix:
MQ2 q 12 l2t S22 1
'2
l t Ž A t S2 q m S1 .
1
'2
l t Ž A t S2 q m S1 . MU2 q 12 l2t S22
0
,
Ž 15 .
where we have neglected the field-dependent D-term contributions and MQ , MU are squark soft masses at the low-energy scale. The field-dependent angle st in Eq. Ž6. is defined as the mixing angle of the above mass-squared matrix. In Fig. 2, we show the dependence of one- and two-loop radiatively corrected CP-even Higgs-boson masses m h on the renormalization scale Q. We choose the universal soft parameters M0 s 500 GeV, M1r2 s 200 GeV
Fig. 2. Renormalization-scale Ž Q . dependence of the lightest CP-even Higgs-boson mass m h . The dashed and solid lines correspond to the one- and two-loop masses respectively. We have fixed the universal boundary conditions, M0 s 500 GeV, M1r2 s 200 GeV, A 0 s 0, and chosen a negative m-parameter.
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Fig. 3. Higgs boson masses m h vs. the squark soft masses MS in the no-squark-mixing case A t s m s 0. The solid lines are results from the two-loop EP approach. For comparison, we also show the results from the RGE-improved one-loop EP approach in dashed lines.
and A 0 s 0 at the unification scale, and set the sign of m-parameter to be negative 4 . We plot two choices of tan b , 2 and 20. The one- and two-loop masses are shown in dashed and solid lines respectively. In the corrections to the Higgs-boson masses m h , we have used the DR running top-Yukawa coupling at the scale Q. The formulae which convert the top-quark pole mass m tpole to l t Ž Q . are given in Ref. w11x. We see that the one-loop radiatively corrected Higgs-boson masses vary by about 10 GeV as the renormalization scale Q varies from 100 to 500 GeV. However, once we properly include the two-loop radiative corrections, the scale dependence of m h becomes much milder, and we find for all ranges of the scale m h , 88 GeV and 110 GeV for tan b s 2 and 20 respectively. The two-loop calculation can only change m h by a few GeV. In Figs. 3 and 4, we compare our results with that of RGE-improved effective potential ŽEP. approach w4x. In that approach, the heavy particles are decoupled at m t˜ Žor m t˜1 and m t˜2 stepwisely if they are very different.. The two-loop RGEs of the effective field theory below the decoupling scale is then used to run the effective couplings to the scale where the Higgs-boson mass is evaluated, e.g. the on-shell top-quark mass. Since the next-to-leading-order corrections are negligible at this scale w3x, this allows an analytical solution to the two-loop RGEs w4x. In this part of numerical analysis, we do not impose the minimization conditions Eqs. Ž8. and Ž9., instead, the m-parameter and the CP-odd Higgs boson mass m A are taken as inputs. We further choose the squark soft masses MQ s MU s MS . In Fig. 3, we show the two-loop Higgs-boson masses m h versus MS , in the RGE-improved one-loop EP approach Ždashed lines. and in the two-loop EP approach Žsolid lines.. The parameters A t and m are set to zero; this corresponds to the no left-right squark-mixing limit. Other parameters are m A s M3 s MS . Results from the two approaches generally agree to Q 3 GeV, for both small and large tan b cases. The difference comes from the O Ž l4t . corrections which are neglected in this calculation w7x. In Fig. 4, we choose a nonzero m-parameter, m s y200 GeV, and plot m h versus X trMS , where X t s A t q mrtan b is related to the off-diagonal element of the squark-mass matrix. We see the results from the
4
Our convention of the sign of m-parameter is opposite to that of Refs. w4,7x.
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Fig. 4. Higgs-boson masses m h vs. X t r MS , where X t s A t q m rtan b . Both the results from the RGE-improved one-loop EP and two-loop EP approaches are shown.
two approaches still agree quite well except for the large X trMS . In particular, the curves for the two-loop EP approach peak at X trMS , "2, which is different from the RGE-improved one-loop EP approach. This however agrees with the results of a recent analysis w7x. The slight asymmetry of the solid curves with respect to the reflection, X t ™ yX t , originates from the last term in Eq. Ž6.. We found that for tan b s 40 the upper limit for the Higgs-boson mass is 125 GeV at the large squark-mixing Ž X trMS s "2..
4. Conclusions To conclude, we have used an effective potential method to calculate the two-loop corrections to the lightest CP-even Higgs-boson mass in the MSSM. Our approach is straightforward and easy to program, and can be extended to include two-loop corrections of order O Ž l4t .. We show that the renormalization scale dependence of m h improves after including the two-loop corrections, this largely reduces the uncertainty associated with one-loop calculations. We have shown that the two-loop correction is only about a few GeV with respect to the one-loop results Žwhere the DR running coupling l t is used.. We have also compared our results with some previous two-loop calculations. We found good agreements with the RGE-improved one-loop EP approach of Ref. w4x except for the case of large left-right squark mixing, where we obtained similar results as in Ref. w7x. The upper bound for the Higgs-boson mass m h Q 125 GeV is achieved at the region of parameter space for large tan b and left-right squark mixings.
Acknowledgements I would like to thank K. Matchev for participating in the early stage of this work, J. Bagger, T. Han and C. Kao for conversations and comments, and C. Wagner and G. Weiglein for communications and numerical comparisons. This work was supported in part by a DOE grant No. DE-FG02-95ER40896 and in part by the Wisconsin Alumni Research Foundation.
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Appendix A. The functions I( x, y, z ) and J( x, y ) Momentum integrals arising from the two-loop bubble diagrams have one-loop subdivergences which can be subtracted in the standard way w12x. Furthermore, in the DR scheme there is no complication associated with vector-boson loops 5, so all integrals can be expressed in terms of Žminimally. subtracted functions I, J which are w12x 2
Ž 16p 2 . J Ž x , y . s x y 1 yln x yln y qlnx ln y ,
Ž A.1 .
2
Ž 16p 2 . I Ž x , y, z . s y 12 Ž y q z y x . ln y ln z q Ž z q x y y . ln z ln x q Ž x q y y z . ln x ln y y4 Ž x ln x q y ln y q z ln z . q j Ž x , y, z . q 5 Ž x q y q z . ,
Ž A.2 .
where j is given by
p j Ž x , y, z . s 8 b L Ž ux . q L Ž u y . q L Ž uz . y
2
ln2
Ž A.3 .
when yb 2 s a2 s Ž x 2 q y 2 q z 2 y 2 xy y 2 xz y 2 yz .r4 F 0, and
j Ž x , y, z . s 8 a yM Ž yf x . q M Ž f y . q M Ž f z .
Ž A.4 .
2
when a ) 0. Here LŽ t . is Lobachevsky’s function, defined as t
LŽ t . s y
H0
`
dx lncos x s t ln2 y 12
Ý Ž y. ky 1 ks1
sin2 kt k2
,
Ž A.5 .
and the function M Ž t . is defined as t
MŽ t. sy
H0
p2 dx lnsinh x s
t2 y
12
2
q t ln2 y 12 Li 2 Ž ey2 t . ,
Li 2 is the dilogarithm function. The angles ux, y, z and f x, y, z are defined by yqzyx yqzyx ux s arctan , f x s arccoth , 2b 2a
ž
/
ž
/
etc.
Ž A.6 .
Ž A.7 .
Finally we have also used the following function in the two-loop effective potential: L Ž x , y, z . s J Ž y, z . y J Ž x , y . y J Ž x , z . y Ž x y y y z . I Ž x , y, z . .
Ž A.8 .
References w1x S.P. Li, M. Sher, Phys. Lett. B 140 Ž1984. 339; Y. Okada, M. Yamaguchi, T. Yanagida, Prog. Theor. Phys. 85 Ž1991. 1; Phys. Lett. B 262 Ž1991. 54; J. Ellis, G. Ridolfi, F. Zwirner, Phys. Lett. B 257 Ž1991. 83; H.E. Haber, R. Hempfling, Phys. Rev. Lett. 66 Ž1991. 1815; R. Barbieri, M. Frigeni, M. Caravaglios, Phys. Lett. B 263 Ž1991. 233; P.H. Chankowski, preprint IFT-7-91-WARSAW, 1991; J.R. Espinosa, M. Quiros, Phys. Lett. B 266 Ž1991. 389; J.L. Lopez, D.V. Nanopoulos, Phys. Lett. B 266 Ž1991. 397; D.M. Pierce, A. Papadopoulos, S.B. Johnson, Phys. Rev. Lett. 68 Ž1992. 3678; A. Brignole, Phys. Lett. B 281 Ž1992. 284; P.H. Chankowski, S. Pokorski, J. Rosiek, Nucl. Phys. B 423 Ž1994. 437; A. Dabelstein, Z. Phys. C 67 Ž1995. 495. w2x J. Kodaira, Y. Yasui, K. Sasaki, Phys. Rev. D 50 Ž1994. 7035; A.V. Gladyshev, D.I. Kazakov, Mod. Phys. Lett. A 10 Ž1995. 3129; M. Carena, J.R. Espinosa, M. Quiros, C.E.M. Wagner, Phys. Lett. B 355 Ž1995. 209.
5
In contrast, the vector bosons live in ds 4y2 e dimensions in the MS scheme, in reducing the two-loop integrals to the form of I and J, there will be extra finite terms from themselves and the associated one-loop subdiagrams.
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J.A. Casas, J.R. Espinosa, M. Quiros, A. Riotto, Nucl. Phys. B 436 Ž1995. 3. M. Carena, M. Quiros, C.E.M. Wagner, Nucl. Phys. B 461 Ž1996. 407. H.E. Haber, in: Brussels EPS HEP 1995, hep-phr9601330; H.E. Haber, R. Hempfling, A. Hoang, Z. Phys. C 75 Ž1997. 539. R. Hempfling, A.H. Hoang, Phys. Lett. B 331 Ž1994. 99. S. Heinemeyer, W. Hollik, G. Weiglein, hep-phr9803277; hep-phr9807423. I. Jack, D.R.T. Jones, S.P. Martin, M.T. Vaughn, Y. Yamada, Phys. Rev. D 50 Ž1994. 5481; The original DR scheme was discussed in W. Siegel, Phys. Lett. B 84 Ž1979. 193; D.M. Capper, D.R.T. Jones, P. van Nieuwenhuizen, Nucl. Phys. B 167 Ž1980. 479. R. Jackiw, Phys. Rev. D 9 Ž1974. 1686. B. Kastening, Phys. Lett. B 283 Ž1992. 287. D.M. Pierce, J.A. Bagger, K.T. Matchev, R.-J. Zhang, Nucl. Phys. B 491 Ž1997. 3. C. Ford, D.R.T. Jones, Phys. Lett. B 274 Ž1992. 409; C. Ford, I. Jack, D.R.T. Jones, Nucl. Phys. B 387 Ž1992. 373.
Physics Letters B 447 Ž1999. 89–97
Two-loop effective potential calculation of the lightest CP-even Higgs-boson mass in the MSSM Ren-Jie Zhang
1
Department of Physics, UniÕersity of Wisconsin, 1150 UniÕersity AÕenue, Madison, WI 53706, USA Received 12 August 1998; revised 19 November 1998 Editor: M. Cveticˇ
Abstract We calculate a two-loop effective potential to the order of O Ž l2t a s . in the MSSM. We then study the corresponding two-loop corrections to the CP-even Higgs-boson mass for arbitrary tan b and left-right top-squark mixings. We find that the lightest Higgs-boson mass is changed by at most a few GeV. We also show the improved scale dependence and compare to previous two-loop analyses. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction In the minimal supersymmetric standard model ŽMSSM., the Higgs sector is composed of three neutral Žtwo CP-even, one CP-odd. and two charged scalar bosons. An important fact of the model is that the quartic self-coupling of the lightest CP-even Higgs boson h is not a free parameter, but related to the standard model gauge couplings g and g X ; as a result, the Higgs boson has an upper bounded tree-level mass, m h F MZ . This limit, however, is violated when the one-loop radiative corrections are included. The well-known dominant one-loop radiative correction comes from the incomplete cancellation of the virtual top-quark and top-squark loops w1x, it approximately has the size
D m2h ,
3 l2t m2t 4p 2
log Ž m 2t˜ rm2t . .
Taking m t˜ s 100–1000 GeV, one finds a large correction, D m h , a few y 50 GeV, due to the relatively large top-quark mass m pole s 175 GeV. The one-loop Higgs-boson mass sensitively depends on the top-quark mass, t and generally varies with the change of renormalization scale. So it remains a quite important problem to study the magnitude of two-loop radiative corrections and the scale dependence of the Higgs-boson mass after these corrections.
1
E-mail: [email protected]
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 5 7 5 - 5
R.-J. Zhangr Physics Letters B 447 (1999) 89–97
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There are basically two approaches to calculate the two-loop radiative corrections. In the renormalization group equation ŽRGE.-improved effective potential approach w2–4x, one uses the one-loop effective potential, together with two-loop RGEs, then all leading-order and next-to-leading-order corrections can be calculated. The finite one-loop threshold corrections arising from the decouplings of the heavy top-squarks have also been included, both for the small and large left-right top-squark mixings. It is further observed that by judiciously setting the renormalization scale, one can use the one-loop renormalized Higgs-boson mass as a good approximation to the full two-loop results w3,5x. The second approach involves a two-loop effective potential. In the special case of tan b ™ ` and no left-right mixing in the top-squark sector, Hempfling and Hoang have calculated the upper bound to the two-loop Higgs-boson mass w6x. Their results qualitatively agree with the previous approach. Recently, two-loop corrections to m h have also been computed by an explicit diagrammatic method w7x. It is the purpose of this paper to generalize the two-loop effective potential calculation of w6x to the case of arbitrary tan b and left-right top-squark mixings. The effective potential method has an advantage over other approaches because it is simple and does not require complicate programming. In this paper we will study the improved scale dependence of the Higgs-boson mass m h and the size of two-loop corrections. We will also compare our results with previous two-loop calculations w4,7x. The rest of the paper is organized as follows: We first present a general formalism for calculating the CP-even Higgs-boson mass from the effective potential in Section 2, and compute the two-loop effective potential to the order of O Ž l2t a s .. We next show the results of our numerical analyses in Section 3; we find good agreements with previous two-loop calculations. Finally we conclude in Section 4. For completeness, some functions which appear in the two-loop calculation are given in the Appendix.
2. Effective potential and the CP-even Higgs-boson masses We start our analysis with the tree-level potential of the MSSM 2 , Vtree s Ž
m2H 1 q m2 g
. < H1 < 2 q Ž m2H 2 q m2 . < H2 < 2 q m B Ž H1 H2 q H.c.. q
g 2 q gX 2 8
Ž < H1 < 2 y < H 2 < 2 .
2
2
q 2
< H1†H2 < 2 ,
Ž 1.
where g, g X are the SUŽ2. and UŽ1. Y gauge couplings, m H 1,m H 2 and B are the soft-breaking Higgs-sector mass parameters, and m is the supersymmetric Higgs-boson mass parameter Žthe m-parameter.. We express H1 and H2 in terms of their component fields, H1 s
ž
Ž S1 q iP1 . r'2 Hy 1
/
,
H2 s
ž
Hq 2
Ž S2 q iP2 . r'2
/
,
Ž 2.
so the tree-level potential can be rewritten as a function of the CP-even fields, S1 and S2 . In general, the all-loop effective potential is a function of S1 and S2 , which are usually known as classical fields. The technique for calculating a higher loop effective potential was developed long ago by Jackiw w9x. First, the Higgs fields are expanded around the classical fields, in terms of which all the relevant particle masses and couplings are determined. One then calculates the higher loop effective potential by computing the corresponding zero-point function Feynman diagrams Žbubble diagrams..
2
We work in the modified DR-scheme of Ref. w8x.
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To be more specific, to the two-loop order that we consider in this paper, we write the effective potential as V Ž S1 ,S2 . s V0 q Vtree Ž S1 ,S2 . q V1loop Ž S1 ,S2 . q V2loop Ž S1 ,S2 . ,
Ž 3.
where V0 is a field-independent vacuum-energy term, and Vtree , V1loop and V2loop are the tree-level, one- and two-loop contributions respectively. V0 is necessary for the renormalization group invariance of the effective potential w10x. The one-loop effective potential in the DR scheme is well-known. It can be easily obtained by calculating the one-loop bubble diagrams with all kinds of Žs.particles running in the loop. We find that in the Landau gauge 2
V1loop Ž S1 ,S2 . s Ý
Ý Ncf G Ž f˜i . y 2 Ý Ncf G Ž f . q 3G Ž W . q 32 G Ž Z . q 12
f is1
G Ž H . q G Ž h. q G Ž A.
f 2
qG Ž G . q G Ž Hq . q G Ž Gq . y 2
4
Ý G Ž x˜qi . y Ý G Ž x˜ i0 . , is1
Ž 4.
is1
where f sums over all the Žs.quarks and Žs.leptons, Ncf is the color factor, 3 for Žs.quarks and 1 for Žs.leptons, and all the masses are implicitly S1 , S2-dependent. H, h, A and G Ž Hq and Gq . label the neutral Žcharged. Higgs and Goldstone bosons, x˜q ˜ i0 represent charginos and neutralinos. We have also used short-handed i and x 2 2 ˜ notations f i s m f˜i , W s mW , etc. The function GŽ x . is defined as x2
Ž 5. Ž ln x y 32 . , 32p 2 where ln x s lnŽ xrQ 2 ., with Q the renormalization scale. The two-loop effective potential can be derived similarly, the corresponding Feynman diagrams are plot in Fig. 1. To the order of O Ž l2t a s ., we have in the Landau gauge GŽ x . s
½
V2loop Ž S1 ,S2 . s 32pa s J Ž t ,t . y 2 t I Ž t ,t ,0 . q 12 Ž c t4 q st4 .
2
Ý
2
J Ž t˜i ,t˜i . q 2 st2 c t2 J Ž t˜1 ,t˜2 . q
is1
2
q Ý L Ž t˜i , g˜ ,t . y 4 m g˜ m t st c t Ž I Ž t˜1 , g˜ ,t . y I Ž t˜2 , g˜ ,t . . is1
Ý t˜i I Ž t˜i ,t˜i ,0 . is1
5
Ž 6.
where the Žminimally. subtracted functions I and J are defined in the Appendix, st is the top-squark mixing angle. This effective potential, in the limit of tan b ™ ` and no left-right squark-mixing Ž st s 0., agrees with that of Ref. w6x. As a good check, one can show that the effective potential V Ž S1 ,S2 . is invariant under the renormalization scale change, up to two-loop terms which are ignored in our approximation. The effective potential Eq. Ž6., as a generating functional, encodes the information of two-loop tadpoles and self-energies at zero external momentum. From this, one can find two-loop CP-even Higgs-boson masses by solving appropriate on-mass-shell conditions.
Fig. 1. Bubble diagrams for the two-loop effective potential to the order of O Ž l2t a s . in the MSSM.
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This proceeds as follows: We first minimize V Ž S1 ,S2 . at the Higgs vacuum expectation values S1 s Õ1 and S2 s Õ 2 ,
EV E S1
EV
s 0,
E S2
S1 sÕ 1 ,S 2 sÕ 2
s 0,
Ž 7.
S1 sÕ 1 ,S 2 sÕ 2
which is equivalent to the requirement that the tadpoles vanish, T1 Õ1 T2 Õ2
s 12 m2Z c2 b q m2H 1 q m2 q Bm tan b ,
Ž 8.
s y 12 m2Z c 2 b q m2H 2 q m2 q Bm cot b ,
Ž 9.
where tan b s Õ 2rÕ1 , m 2Z s Ž g 2 q g X 2 . Õ 2r4 and Õ 2 s Õ 12 q Õ 22 . To the two-loop order, the tadpoles Ti , i s 1,2 are given by Ti s Ti1loop q Ti2loop , where the one- and two-loop tadpoles are defined by Ti1loop sy
E V1loop
ž
E Si
/
,
Ti2loop sy
S1 sÕ 1 ,S 2 sÕ 2
ž
E V2loop E Si
/
.
Ž 10 .
S1 sÕ 1 ,S 2 sÕ 2
One can check that the one-loop tadpoles Ti1loop obtained in this way give the same results as in Ref. w11x 3, where they were explicitly calculated from the one-point function Feynman diagrams Žtadpole diagrams.. The CP-even Higgs-boson mass matrix, after some algebra, is M 2 Ž p2 . s
ž
m2Z cb2 q m2A sb2 y Re P 11 Ž p 2 . q T1rÕ1
y Ž m2Z q m2A . sb cb y Re P 12 Ž p 2 .
y Ž m2Z q m 2A . sb cb y Re P 12 Ž p 2 .
m 2Z sb2 q m 2A cb2 y Re P 22 Ž p 2 . q T2rÕ 2
/
,
Ž 11 .
where m2A s ym B Žtan b q cot b ., and both m Z and m A are DR running masses. P ’s represent two-point functions Žself-energies.. The radiatively corrected Higgs-boson masses can be found by computing the zeroes of the inverse propagator, p 2 y M 2 Ž p 2 .. The complete one-loop formulae for self-energies at nonzero external momentum can be found, e.g., in Ref. w11x. For the two-loop self-energies, we shall approximate them as follows:
P i 2loop Ž 0 . sy j
ž
E 2 V2loop E Si E S j
/
,
i , j s 1,2.
Ž 12 .
S1 sÕ 1 ,S 2 sÕ 2
Ž0. and P i 2loop Ž p 2 . is negligible. The difference of P i 2loop j j
3. Numerical procedure and results We shall first show the improvement of renormalization scale Ž Q . dependence of the lightest CP-even Higgs-boson mass. We start by solving the two-loop renormalization group equations ŽRGEs. of the MSSM. The boundary condition inputs for these RGEs are taken to be the observables a s , a em , GF , MZ , m t , m b and mt at the electro-weak scale, MZ , and the universal scalar soft mass M0 , gaugino soft mass M1r2 and trilinear scalar coupling A 0 at the unification scale, MGUT f 2 = 10 16 GeV ŽThis is sometimes called the minimal
3
In Ref. w11x, we used the ’t Hooft-Feynman gauge. To compare with the one-loop tadpoles obtained from Eq. Ž4., the Goldstone boson masses need to be set to zero.
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supergravity ŽmSUGRA. scenario... We have also properly taken into account the low energy threshold corrections to the gauge and Yukawa couplings, as in Ref. w11x. In this scenario, the m-parameter and the Žtree-level. mass m A are determined in terms of other variables through the minimization conditions Ž8. and Ž9., which ensure the correct electro-weak symmetry breaking,
m2 s 12 tan2 b m2H 2 tan b y m2H 1cot b y m2Z ,
ž
m2A s
1 cos2 b
žm
Ž 13 .
/
2 2 H 2 y m H1
/ ym
2 Z
,
Ž 14 .
where m 2H 1 s m2H 1 y T1rÕ1 and m2H 2 s m 2H 2 y T2rÕ 2 . We then calculate the lightest CP-even Higgs-boson mass from the mass-squared matrix Eq. Ž11.. We use the one-loop tadpole and self-energy formulae from Ref. w11x, while for the two-loop contributions, we compute the tadpoles and self-energies numerically according to Eqs. Ž10. and Ž12. by replacing the differentiation by a finite difference. The field-dependent masses in Eq. Ž6. are the top-quark mass m t s l t S2r '2 and the top-squark masses which are found from the following field-dependent mass-squared matrix:
MQ2 q 12 l2t S22 1
'2
l t Ž A t S2 q m S1 .
1
'2
l t Ž A t S2 q m S1 . MU2 q 12 l2t S22
0
,
Ž 15 .
where we have neglected the field-dependent D-term contributions and MQ , MU are squark soft masses at the low-energy scale. The field-dependent angle st in Eq. Ž6. is defined as the mixing angle of the above mass-squared matrix. In Fig. 2, we show the dependence of one- and two-loop radiatively corrected CP-even Higgs-boson masses m h on the renormalization scale Q. We choose the universal soft parameters M0 s 500 GeV, M1r2 s 200 GeV
Fig. 2. Renormalization-scale Ž Q . dependence of the lightest CP-even Higgs-boson mass m h . The dashed and solid lines correspond to the one- and two-loop masses respectively. We have fixed the universal boundary conditions, M0 s 500 GeV, M1r2 s 200 GeV, A 0 s 0, and chosen a negative m-parameter.
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Fig. 3. Higgs boson masses m h vs. the squark soft masses MS in the no-squark-mixing case A t s m s 0. The solid lines are results from the two-loop EP approach. For comparison, we also show the results from the RGE-improved one-loop EP approach in dashed lines.
and A 0 s 0 at the unification scale, and set the sign of m-parameter to be negative 4 . We plot two choices of tan b , 2 and 20. The one- and two-loop masses are shown in dashed and solid lines respectively. In the corrections to the Higgs-boson masses m h , we have used the DR running top-Yukawa coupling at the scale Q. The formulae which convert the top-quark pole mass m tpole to l t Ž Q . are given in Ref. w11x. We see that the one-loop radiatively corrected Higgs-boson masses vary by about 10 GeV as the renormalization scale Q varies from 100 to 500 GeV. However, once we properly include the two-loop radiative corrections, the scale dependence of m h becomes much milder, and we find for all ranges of the scale m h , 88 GeV and 110 GeV for tan b s 2 and 20 respectively. The two-loop calculation can only change m h by a few GeV. In Figs. 3 and 4, we compare our results with that of RGE-improved effective potential ŽEP. approach w4x. In that approach, the heavy particles are decoupled at m t˜ Žor m t˜1 and m t˜2 stepwisely if they are very different.. The two-loop RGEs of the effective field theory below the decoupling scale is then used to run the effective couplings to the scale where the Higgs-boson mass is evaluated, e.g. the on-shell top-quark mass. Since the next-to-leading-order corrections are negligible at this scale w3x, this allows an analytical solution to the two-loop RGEs w4x. In this part of numerical analysis, we do not impose the minimization conditions Eqs. Ž8. and Ž9., instead, the m-parameter and the CP-odd Higgs boson mass m A are taken as inputs. We further choose the squark soft masses MQ s MU s MS . In Fig. 3, we show the two-loop Higgs-boson masses m h versus MS , in the RGE-improved one-loop EP approach Ždashed lines. and in the two-loop EP approach Žsolid lines.. The parameters A t and m are set to zero; this corresponds to the no left-right squark-mixing limit. Other parameters are m A s M3 s MS . Results from the two approaches generally agree to Q 3 GeV, for both small and large tan b cases. The difference comes from the O Ž l4t . corrections which are neglected in this calculation w7x. In Fig. 4, we choose a nonzero m-parameter, m s y200 GeV, and plot m h versus X trMS , where X t s A t q mrtan b is related to the off-diagonal element of the squark-mass matrix. We see the results from the
4
Our convention of the sign of m-parameter is opposite to that of Refs. w4,7x.
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Fig. 4. Higgs-boson masses m h vs. X t r MS , where X t s A t q m rtan b . Both the results from the RGE-improved one-loop EP and two-loop EP approaches are shown.
two approaches still agree quite well except for the large X trMS . In particular, the curves for the two-loop EP approach peak at X trMS , "2, which is different from the RGE-improved one-loop EP approach. This however agrees with the results of a recent analysis w7x. The slight asymmetry of the solid curves with respect to the reflection, X t ™ yX t , originates from the last term in Eq. Ž6.. We found that for tan b s 40 the upper limit for the Higgs-boson mass is 125 GeV at the large squark-mixing Ž X trMS s "2..
4. Conclusions To conclude, we have used an effective potential method to calculate the two-loop corrections to the lightest CP-even Higgs-boson mass in the MSSM. Our approach is straightforward and easy to program, and can be extended to include two-loop corrections of order O Ž l4t .. We show that the renormalization scale dependence of m h improves after including the two-loop corrections, this largely reduces the uncertainty associated with one-loop calculations. We have shown that the two-loop correction is only about a few GeV with respect to the one-loop results Žwhere the DR running coupling l t is used.. We have also compared our results with some previous two-loop calculations. We found good agreements with the RGE-improved one-loop EP approach of Ref. w4x except for the case of large left-right squark mixing, where we obtained similar results as in Ref. w7x. The upper bound for the Higgs-boson mass m h Q 125 GeV is achieved at the region of parameter space for large tan b and left-right squark mixings.
Acknowledgements I would like to thank K. Matchev for participating in the early stage of this work, J. Bagger, T. Han and C. Kao for conversations and comments, and C. Wagner and G. Weiglein for communications and numerical comparisons. This work was supported in part by a DOE grant No. DE-FG02-95ER40896 and in part by the Wisconsin Alumni Research Foundation.
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Appendix A. The functions I( x, y, z ) and J( x, y ) Momentum integrals arising from the two-loop bubble diagrams have one-loop subdivergences which can be subtracted in the standard way w12x. Furthermore, in the DR scheme there is no complication associated with vector-boson loops 5, so all integrals can be expressed in terms of Žminimally. subtracted functions I, J which are w12x 2
Ž 16p 2 . J Ž x , y . s x y 1 yln x yln y qlnx ln y ,
Ž A.1 .
2
Ž 16p 2 . I Ž x , y, z . s y 12 Ž y q z y x . ln y ln z q Ž z q x y y . ln z ln x q Ž x q y y z . ln x ln y y4 Ž x ln x q y ln y q z ln z . q j Ž x , y, z . q 5 Ž x q y q z . ,
Ž A.2 .
where j is given by
p j Ž x , y, z . s 8 b L Ž ux . q L Ž u y . q L Ž uz . y
2
ln2
Ž A.3 .
when yb 2 s a2 s Ž x 2 q y 2 q z 2 y 2 xy y 2 xz y 2 yz .r4 F 0, and
j Ž x , y, z . s 8 a yM Ž yf x . q M Ž f y . q M Ž f z .
Ž A.4 .
2
when a ) 0. Here LŽ t . is Lobachevsky’s function, defined as t
LŽ t . s y
H0
`
dx lncos x s t ln2 y 12
Ý Ž y. ky 1 ks1
sin2 kt k2
,
Ž A.5 .
and the function M Ž t . is defined as t
MŽ t. sy
H0
p2 dx lnsinh x s
t2 y
12
2
q t ln2 y 12 Li 2 Ž ey2 t . ,
Li 2 is the dilogarithm function. The angles ux, y, z and f x, y, z are defined by yqzyx yqzyx ux s arctan , f x s arccoth , 2b 2a
ž
/
ž
/
etc.
Ž A.6 .
Ž A.7 .
Finally we have also used the following function in the two-loop effective potential: L Ž x , y, z . s J Ž y, z . y J Ž x , y . y J Ž x , z . y Ž x y y y z . I Ž x , y, z . .
Ž A.8 .
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In contrast, the vector bosons live in ds 4y2 e dimensions in the MS scheme, in reducing the two-loop integrals to the form of I and J, there will be extra finite terms from themselves and the associated one-loop subdiagrams.
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