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Electroweak corrections to reduction solutions in the standard model
Physics Letters B 311 ( 1993 ) 249-254 North-Holland
PHYSICS LETTERSB
Electroweak corrections to reduction solutions in the standard model W. Z i m m e r m a n n Max-Planck-lnstitutJ~r Physik- W_erner-Heisenberg-lnstitut, P.O. Box 40 12 12, W-8000 Munich, Germany Received 4 March 1993; revised manuscript received 18 May 1993 Editor: P.V. Landshoff
In the limit of vanishing electroweak interactions an asymptotic requirement is proposed which uniquely determines electroweak corrections to reduction solutions in the standard model. The precise form of these boundary conditions is worked out for the one- and two-loop approximation of QCD extended by the top and Higgs interactions. It is shown that the conditions are satisfied in Kubo's two-loop computation of the top and Higgs mass.
In ref. [ l ] the reduction method was used for deriving constraints on the Higgs and quark mass in the standard model. On this basis two-loop corrections were calculated by Kubo [2 ]. Generalizing the approach of ref. [3] a set of partial differential equations was set up in ref. [2 ] for the top and Higgs couplings G]
G~
Pt -- 4zcols -- g 2 ,
2
2
PH = 4~0~ s -- g~
( 1)
expressed as functionals p,=pt(u,
v, o~s) ,
p i 4 = p H ( U , v, o ~ )
(2)
ofc~ and the electroweak coupling ratios g.2
g2
u=47ra~ - g ~ '
5 g,2
5 g,2
v = 3 4 7 r o ~ - 3g2 •
(3)
The partial differential equations were solved in ref. [ 2 ] by expansions with respect to powers of u, v and % determined by the initial values p,=2,
pH=~(6x/~_25)
atu=v=C~s=0,
(4)
which correspond to the non-trivial case of reduction. With the current experimental values sin20MS= 0.2324 and c~s= 0.118 at M z the top and Higgs mass become [ 2 ] mt=100+6GeV,
mg=65+lGeV.
(5)
Using power series expansions for (2) is certainly an efficient method for obtaining solutions of the evolution equations which approach the values (4) characteristic of the non-trivial reduction. However, there are many other solutions with the same limit values (4) but involving arbitrary constants of integrations. The purpose of this note is to show that the power series solution used by Kubo in ref. [2] is indeed the only solution which satisfies the correct boundary conditions in the limit of vanishing electroweak interactions. We start from the evolution equations of the gauge couplings, the top Yukawa coupling and the Higgs coupling
Volume 311, number 1,2,3,4
PHYSICS LETTERS B
29 July 1993
neglecting all other Yukawa couplings. After eliminating the momentum scale variable in favor of ots the differential equations take the form
dv
asots~-~ = a i ,
du
dp,
asO%~-~ = a 2 ,
a s o t s=-a-t , d O t s
dpH
a ~ O t s ~ =crH
(6,7)
for the electroweak couplings (3), the top Yukawa and the Higgs couplings G,2 ,~ Pz - 47tOts ' Pn = 4zcOt~ "
(8)
The functions aj are given in perturbation theory as formal expansions with respect to fi:
~j= ~ ~°Ot,"o)"~.
(9)
n=O
• oj - ( n ) include the contributions from diagrams with n - 1 loops and depend on the coupling ratios The coefficients (3), (8)
a) n) = a ) m ( u , v,p, ptt) •
(10)
For the explicit form of the one- and two-loop coefficients see ref. [4]. We further expand the solutions of the differential equations (6), (7) with respect to powers of h:
n=0
n=0
n=0
n=0
For given n > I the coefficients u (n), v(,), p},) and p~/") satisfy decoupled linear differential equations which can be solved explicitly in terms of lower order coefficients. In the absence of electroweak interactions ( u = v = 0 ) the reduction solutions Pto(ots) = ~ ~.,~n,,(n) ,, ~sv,o , Puo(Ot~)= ~ h"Ot'~P~o) n=0
(13)
n=0
are determined uniquely as solutions of (7) by the condition that they approach limpto = 2 ,
limpuo=~(6x/c~-25)
Ots~O
Ots~0
(14)
in the ultraviolet limit Ots~0. This is called the nontrivial case of reduction in contradistinction to the trivial case where the ratios Pt and Pu vanish in the limit as--, 0. The first two terms of the expansions are [ 2 ]
p,o=-~ +fia~qo +O(h2Ot2) , p H o = ~ ( x / 6 8 9 - 2 5 ) +hOtsro +O(h2Ot~) ,
(15)
- -1 ~.6/'6~ q o - 207367r ( 1 0 4 5 3 + ' v - - ' ) '
(16)
ro=
1 27813x/6~-487613 31104n 21+ 6 ~ / ~
In order to include the electroweak interactions the full system of differential equations (6), (7) has to be solved. Extending the treatment of ref. [ 3 ] to higher order corrections we consider the electroweak interactions as perturbations of the system. In the differential equations of p} m , p}m the perturbation terms are of order u • ~(n) or v respectively. Accordingly we require that the solutxonsp t , p }g) approach the undisturbed reduction solutions p},) ~p}~),
p~)~p}go )
(17)
asymptotically in the limit of vanishing gauge coupling ratios g2
u= g~ --,o, 250
5 g'2
v= ~
~o.
(18)
Volume 311, number 1,2,3,4
29 July 1993
PHYSICS LETTERS B
The aim is to make the asymptotic connections (17), (18) mathematically precise in order to determine the correct solution pt , p}p) for given undisturbed solution p}o") , p}~"o) . In the one-loop approximation the solutions of the differential equations (6) are •
-(n)
u(O) = 42 a 42 ~ 1 9 o q + a - 19 1 + ~ '
a
v(o)
( = o~-~'
70
b
70
c(
- 41 o ~ - b - 41 1 - c ~ '
b c = -'a
(19)
The constants of integration a and b are determined experimentally. The parameter ( is small at the scale of the Z mass and u (o) = v(o) = 0 at ~= 0. A similar result holds in the two-loop approximation [ 4 ]. Including contributions of order h the leading terms of u and v in the infrared region are 42 u=]-~+h
~
~+
a~lg a
...,
70b~ v=~-]-~-
h 3040 b 2 ~+... 5043~ a ~ l g ~ ,
(20)
up to terms of order (2 apart from logarithmic factors. Hence in this approximation (--,0 corresponds to u, v-~0. Therefore, (--,0 may be used as the appropriate limit for formulating the asymptotic requirement (17), (18). The one- and two-loop contributions to the top and Higgs coupling satisfy the following differential equations: 14(@} 0) =9p}O)2_2p}O)
9...(o)
- - -~ u P t
d(
cdp}')
-~
17,,c~(0 )
- - TO " p t
(21)
,
=F,p} l) +G,,
(22)
Ft = 1 + 9 p}O) + fiB,, @ = p}O)~ ( I A B , + C t ) - ~ P } ° ) ( 9u(1)-* ~1,(') ~1o v
A=p}°) +~u (°)*~''(°) --
B t_- -O , ( ot ) lp,
v
--~--
20
v
9,,(0) 2 ~
]
,
13
--
177,(0 )
--i-'6~
Ct=3p~,)Z_3p}O)p~)6p(tO)2+17p}O)_41+9u(O)+lv(O)
:3,,(o)2
9 ,, ( 0 ) , , ( 0 ) ..[ 1 1 8 7 , , ( 0 ) 2
14(dP}~) = 6 p ~ )2 + 14p}g ) + 12p}°)p~ ) - 24p} °)2 _ 9u (O)p~) _ 9 v (°)p h°) + 9 u (o)2 + ~u (O)v(o) + ~ v (o)2 d~
(23) dp~ )
(24)
d~ =FHp~) +GI~,
F,, = 2 + ~p tg) + ~p (O) _ ~ u (o)_ ~o ~ (o) GH=~p(tl)p~I
O) - - T V24t)(O)n(1) t vt
_9u(l)p,
~ ) 9 v ( l ) p ~ o ) . . [ . . . ~ 4 b / ( O ) b / ( l ) _ [ _ ~ _ 70 9 U(I),,(O)_.L" 27 ~,t,(O),,(l) v J 350 v
i + ] ~ (~ABH+CH), .a_ 271,(0)2 Bn=6p~)2 + 14p~) + 12p}°)Pgg) --24p ~°)2-9u(°)p g°) _ ~v(°)P}4°)+9u(°)2+9u(°)v(°)5o~ ,
251
Volume 311, number
1,2,3,4
PHYSICS
LETTERS
CH = _ ~ p ~ ) 3 _ 8p}O)p~)+ ~U (O)p~)2 + ~2o~,,(o),, + t~9)2 ,, ~73u (0)2,~)__t, ~0~u (0)V(0)P~) - - ~
- - 1 (1_ 0v, ), (200) nH H
--
32p}0)2
--
B
29 J u l y 1 9 9 3
13p,~ ) + 19P}°)P~ ) _ ]p}O)2pff)- 9 u (O)p~)
V(0)2P~ ) + ~u(O)p}O)P~ ) + ~v(O)p}O)p~ ) + 30p} °)3
9u(O)p}O)Z__8v(O)p}O)2+63,,(O),,(O),~(O ) 171. (0)2_(0).305 .(0)3-- 28~uCO)2v(O)__1a6~u(O)v(O)2 10" ~ Yt -- "i-if6U Pt "Vq'g-U
3411
( = 0 is a singular point for these differential equations. For given initial values and ( = 0 there are no unique solutions. In order to remove this ambiguity we transform the functions p~), p~0) into new functions z~J) , z~0) for which the differential equations become regular at ~=0. Given initial values at ~=0 they then uniquely determine the functions z~ ) and Z~o) . Accordingly we formulate the asymptotic requirement ( 17 ), ( 18 ) by r~ ) = z ~ ) at ¢ = 0 ( j = 0 , 1 ; k = t , H) .
(25)
By this condition we have a one-to-one correspondence between the coupling ratios p~) and their approximations p~o) in the limit of electroweak interactions. The regularity transformations are p(0) 2 _1_/" 1/7,r (0) t -- 9 --~ ~t ~
(26)
,,~ (0) __ 2 .a_ r l / 7 , ~ ( 0 ) Ut0 -- 9 --~ ~tO
p ~) = ~ (.v/-6~- 25 ) + p~ + ~ ' , / ~ / z l r ~ ) , p ~o) = ~8 ( ~ p}~) =qo+q,~lg~+q~+~8/vCt~),
25 ) + Po~ + ~~ / z ' Z ~ o
),
p,~(t)=qo +~8/7z}~) ,
pff)=ro+r'l(lg(+rl(+r~(21g2(+r'E(21g(+r2(Z+(t+x/g~/zlr#
(27) (28)
) , phlo)=ro+(l+',//g-ffg/2'zff ) .
(29)
the form of these transformations is in agreement with the explicit solutions of the differential equations ( 2 1 ) - ( 2 4 ) . The coefficients qj, q~, r;, rj are taken from the formal expansions near ~=0. The limits l i m p } ° ) = ~2 ,
limp~)=~8(6~/6~-25)
~o
~o
(30)
represent the infrared fixed points of Pendleton and Ross [ 5 ]. They coincide with the ultraviolet limits of the reduction solutions in the absence of electroweak interactions. Likewise the limits limp} ~)=qo,
limp~ )=ro
~0
~0
(31)
are identical to the coefficients (16) occurring in the ultraviolet approximations (15). Due to the factor ~ , however, the original fixed points (30) are destroyed by diverging contributions of order h. The regularity transformations represent Taylor formulae modified by taking into account the occurrence of logarithmic terms and non-integral powers. In this sense the transformed functions z~), Z~o) may be interpreted as remainders of Taylor type formulae for the coupling ratios p~), p~o) . In a more detailed paper it will be shown that the coefficients of the differential equations for the functions z~), r ~ ) satisfy Lipschitz conditions at ~= 0 so that solutions are uniquely determined by their initial values (25) [ 6 ]. Combining the one- and two-loop approximations ( 2 6 ) - ( 2 9 ) we get
pt~p}O ) +hotsp(tl)= a_l +a,olg ~+ao +~l/7zt, z , = z t (°) +hoqr~ t) ,
pH.~p~)) +hoqp~) = ~-~ +b'olg(+bo+b'{(lg2(+b'~(lg~+bt~+(',/~/2~z~, ztz= z ~ ) + h a ~ z ~ ) .
(32)
In case of the non-trivial reduction the coupling ratios p~0) form constant solutions of the differential equations (see ( 1 8 ) ) 252
PHYSICS LETTERS B
Volume 311, number 1,2,3,4 p(O) ,o = ~2,
"(') =qo, v~ n( °o)-- ~ s ( 6 x / ~ - 25) , ~-,o
P})o)=ro,
29 July 1993
(33)
implying that the functions r~6) vanish identically z~ )=0,
j=0,1,
k=t,H.
(34)
According to (25) the correct solutions p~, of the full system (6), (7) of differential equations are then determined by the initial values r~)=0
at(=0
(35)
which represent the limit of vanishing electroweak couplings. It would be quite cumbersome to compute the couplings on the basis of this condition. Moreover, the solution would involve the constants a and b as given by (19) and two-loop corrections, a and b would have to be reexpressed by the values of the running coupling parameters at the Z mass. Instead, it is much more convenient to consider the top and Higgs coupling ratios as functionals (2) expressing them directly in terms of the experimental parameters gs, g and g'. It is now easy to show that Kubo's expansion ofpt and p, with respect to powers of u, v and c~s is the correct solution of the partial differential equations
Opt
0"2~
Op,." 1 - O ' s O gOpt s-
+0-I - 0~'
0~ s =0-t '
0"2 ~
"~- 0-1
0PH + a~C~ ~ s
=crH.
(36)
Any solution of (36) yields a solution of the original system (7) of ordinary differential equations provided the solutions u and v of (6) are substituted into the functionals (2). The power series solution 2 P'=9"~E CpqrupvqO~r' P - = ~ s ( 6 V / ~ - 2 5 )
+ ~ Cpqrupl)qOLr
(37)
is unique by the choice of the lowest order (4). Using the asymptotic behavior (20) of u and v it is seen that terms involving non-integral powers (~/7 and (.,/~/2~ as in the general form of (32), (33) are absent in the expansion (37). Therefore the requirement (35) is satisfied. This completes the proof that the power series expansion (37) of the functional (2) represents the correct solution of the evolution equations (7) which for vanishing electroweak interactions approaches the reduction solution with the ultraviolet limit (4). In Kubo's paper also the influence of the bottom coupling Gb was included and found negligible [2 ]. In that case the justification of the power series solution is different since the appropriate limits in lowest order are limp} °) = g , lmp}°) = ~ w i t h p b = G~ , ¢~o ~o 47~O~s
(38)
so that the form of the regularity transformations is changed completely. Nevertheless, it could be shown that the power series expansion is correct in the one-loop order [ 6 ]. So far applications of the reduction method have been restricted to the standard model as such. The modification with two Higgs doublets was treated by Denner [ 7 ]. Not included have been effects which originate in particles and interactions beyond the standard model. Substantial changes for the top and Higgs masses could be expected, for instance, in a supersymmetric extension of the standard model. I am grateful to Dr. J. Kubo and Dr. K. Sibold for helpful discussions.
References
[ 1] J. Kubo, K. Sibold and W. Zimmermann, Nucl. Phys. B 259 (1985) 331. 253
Volume 311, number 1,2,3,4
PHYSICS LETTERS B
[2] J. Kubo, Phys. Lett. B 262 ( 1991 ) 472. [3 ] J. Kubo, K. Sibold and W. Zimmermann, Phys. Lett. B 220 ( 1989 ) 185. [ 4 ] W. Zimmermann, Phys. Lett. B 308 (1993) 117. [5] B. Pendleton and G. Ross, Phys. Lett. B 98 ( 1981 ) 291. [6] W. Zimmermann, in preparation. [7] A. Denner, Nucl, Phys. B 347 (1990) 184.
254
29 July 1993
PHYSICS LETTERSB
Electroweak corrections to reduction solutions in the standard model W. Z i m m e r m a n n Max-Planck-lnstitutJ~r Physik- W_erner-Heisenberg-lnstitut, P.O. Box 40 12 12, W-8000 Munich, Germany Received 4 March 1993; revised manuscript received 18 May 1993 Editor: P.V. Landshoff
In the limit of vanishing electroweak interactions an asymptotic requirement is proposed which uniquely determines electroweak corrections to reduction solutions in the standard model. The precise form of these boundary conditions is worked out for the one- and two-loop approximation of QCD extended by the top and Higgs interactions. It is shown that the conditions are satisfied in Kubo's two-loop computation of the top and Higgs mass.
In ref. [ l ] the reduction method was used for deriving constraints on the Higgs and quark mass in the standard model. On this basis two-loop corrections were calculated by Kubo [2 ]. Generalizing the approach of ref. [3] a set of partial differential equations was set up in ref. [2 ] for the top and Higgs couplings G]
G~
Pt -- 4zcols -- g 2 ,
2
2
PH = 4~0~ s -- g~
( 1)
expressed as functionals p,=pt(u,
v, o~s) ,
p i 4 = p H ( U , v, o ~ )
(2)
ofc~ and the electroweak coupling ratios g.2
g2
u=47ra~ - g ~ '
5 g,2
5 g,2
v = 3 4 7 r o ~ - 3g2 •
(3)
The partial differential equations were solved in ref. [ 2 ] by expansions with respect to powers of u, v and % determined by the initial values p,=2,
pH=~(6x/~_25)
atu=v=C~s=0,
(4)
which correspond to the non-trivial case of reduction. With the current experimental values sin20MS= 0.2324 and c~s= 0.118 at M z the top and Higgs mass become [ 2 ] mt=100+6GeV,
mg=65+lGeV.
(5)
Using power series expansions for (2) is certainly an efficient method for obtaining solutions of the evolution equations which approach the values (4) characteristic of the non-trivial reduction. However, there are many other solutions with the same limit values (4) but involving arbitrary constants of integrations. The purpose of this note is to show that the power series solution used by Kubo in ref. [2] is indeed the only solution which satisfies the correct boundary conditions in the limit of vanishing electroweak interactions. We start from the evolution equations of the gauge couplings, the top Yukawa coupling and the Higgs coupling
Volume 311, number 1,2,3,4
PHYSICS LETTERS B
29 July 1993
neglecting all other Yukawa couplings. After eliminating the momentum scale variable in favor of ots the differential equations take the form
dv
asots~-~ = a i ,
du
dp,
asO%~-~ = a 2 ,
a s o t s=-a-t , d O t s
dpH
a ~ O t s ~ =crH
(6,7)
for the electroweak couplings (3), the top Yukawa and the Higgs couplings G,2 ,~ Pz - 47tOts ' Pn = 4zcOt~ "
(8)
The functions aj are given in perturbation theory as formal expansions with respect to fi:
~j= ~ ~°Ot,"o)"~.
(9)
n=O
• oj - ( n ) include the contributions from diagrams with n - 1 loops and depend on the coupling ratios The coefficients (3), (8)
a) n) = a ) m ( u , v,p, ptt) •
(10)
For the explicit form of the one- and two-loop coefficients see ref. [4]. We further expand the solutions of the differential equations (6), (7) with respect to powers of h:
n=0
n=0
n=0
n=0
For given n > I the coefficients u (n), v(,), p},) and p~/") satisfy decoupled linear differential equations which can be solved explicitly in terms of lower order coefficients. In the absence of electroweak interactions ( u = v = 0 ) the reduction solutions Pto(ots) = ~ ~.,~n,,(n) ,, ~sv,o , Puo(Ot~)= ~ h"Ot'~P~o) n=0
(13)
n=0
are determined uniquely as solutions of (7) by the condition that they approach limpto = 2 ,
limpuo=~(6x/c~-25)
Ots~O
Ots~0
(14)
in the ultraviolet limit Ots~0. This is called the nontrivial case of reduction in contradistinction to the trivial case where the ratios Pt and Pu vanish in the limit as--, 0. The first two terms of the expansions are [ 2 ]
p,o=-~ +fia~qo +O(h2Ot2) , p H o = ~ ( x / 6 8 9 - 2 5 ) +hOtsro +O(h2Ot~) ,
(15)
- -1 ~.6/'6~ q o - 207367r ( 1 0 4 5 3 + ' v - - ' ) '
(16)
ro=
1 27813x/6~-487613 31104n 21+ 6 ~ / ~
In order to include the electroweak interactions the full system of differential equations (6), (7) has to be solved. Extending the treatment of ref. [ 3 ] to higher order corrections we consider the electroweak interactions as perturbations of the system. In the differential equations of p} m , p}m the perturbation terms are of order u • ~(n) or v respectively. Accordingly we require that the solutxonsp t , p }g) approach the undisturbed reduction solutions p},) ~p}~),
p~)~p}go )
(17)
asymptotically in the limit of vanishing gauge coupling ratios g2
u= g~ --,o, 250
5 g'2
v= ~
~o.
(18)
Volume 311, number 1,2,3,4
29 July 1993
PHYSICS LETTERS B
The aim is to make the asymptotic connections (17), (18) mathematically precise in order to determine the correct solution pt , p}p) for given undisturbed solution p}o") , p}~"o) . In the one-loop approximation the solutions of the differential equations (6) are •
-(n)
u(O) = 42 a 42 ~ 1 9 o q + a - 19 1 + ~ '
a
v(o)
( = o~-~'
70
b
70
c(
- 41 o ~ - b - 41 1 - c ~ '
b c = -'a
(19)
The constants of integration a and b are determined experimentally. The parameter ( is small at the scale of the Z mass and u (o) = v(o) = 0 at ~= 0. A similar result holds in the two-loop approximation [ 4 ]. Including contributions of order h the leading terms of u and v in the infrared region are 42 u=]-~+h
~
~+
a~lg a
...,
70b~ v=~-]-~-
h 3040 b 2 ~+... 5043~ a ~ l g ~ ,
(20)
up to terms of order (2 apart from logarithmic factors. Hence in this approximation (--,0 corresponds to u, v-~0. Therefore, (--,0 may be used as the appropriate limit for formulating the asymptotic requirement (17), (18). The one- and two-loop contributions to the top and Higgs coupling satisfy the following differential equations: 14(@} 0) =9p}O)2_2p}O)
9...(o)
- - -~ u P t
d(
cdp}')
-~
17,,c~(0 )
- - TO " p t
(21)
,
=F,p} l) +G,,
(22)
Ft = 1 + 9 p}O) + fiB,, @ = p}O)~ ( I A B , + C t ) - ~ P } ° ) ( 9u(1)-* ~1,(') ~1o v
A=p}°) +~u (°)*~''(°) --
B t_- -O , ( ot ) lp,
v
--~--
20
v
9,,(0) 2 ~
]
,
13
--
177,(0 )
--i-'6~
Ct=3p~,)Z_3p}O)p~)6p(tO)2+17p}O)_41+9u(O)+lv(O)
:3,,(o)2
9 ,, ( 0 ) , , ( 0 ) ..[ 1 1 8 7 , , ( 0 ) 2
14(dP}~) = 6 p ~ )2 + 14p}g ) + 12p}°)p~ ) - 24p} °)2 _ 9u (O)p~) _ 9 v (°)p h°) + 9 u (o)2 + ~u (O)v(o) + ~ v (o)2 d~
(23) dp~ )
(24)
d~ =FHp~) +GI~,
F,, = 2 + ~p tg) + ~p (O) _ ~ u (o)_ ~o ~ (o) GH=~p(tl)p~I
O) - - T V24t)(O)n(1) t vt
_9u(l)p,
~ ) 9 v ( l ) p ~ o ) . . [ . . . ~ 4 b / ( O ) b / ( l ) _ [ _ ~ _ 70 9 U(I),,(O)_.L" 27 ~,t,(O),,(l) v J 350 v
i + ] ~ (~ABH+CH), .a_ 271,(0)2 Bn=6p~)2 + 14p~) + 12p}°)Pgg) --24p ~°)2-9u(°)p g°) _ ~v(°)P}4°)+9u(°)2+9u(°)v(°)5o~ ,
251
Volume 311, number
1,2,3,4
PHYSICS
LETTERS
CH = _ ~ p ~ ) 3 _ 8p}O)p~)+ ~U (O)p~)2 + ~2o~,,(o),, + t~9)2 ,, ~73u (0)2,~)__t, ~0~u (0)V(0)P~) - - ~
- - 1 (1_ 0v, ), (200) nH H
--
32p}0)2
--
B
29 J u l y 1 9 9 3
13p,~ ) + 19P}°)P~ ) _ ]p}O)2pff)- 9 u (O)p~)
V(0)2P~ ) + ~u(O)p}O)P~ ) + ~v(O)p}O)p~ ) + 30p} °)3
9u(O)p}O)Z__8v(O)p}O)2+63,,(O),,(O),~(O ) 171. (0)2_(0).305 .(0)3-- 28~uCO)2v(O)__1a6~u(O)v(O)2 10" ~ Yt -- "i-if6U Pt "Vq'g-U
3411
( = 0 is a singular point for these differential equations. For given initial values and ( = 0 there are no unique solutions. In order to remove this ambiguity we transform the functions p~), p~0) into new functions z~J) , z~0) for which the differential equations become regular at ~=0. Given initial values at ~=0 they then uniquely determine the functions z~ ) and Z~o) . Accordingly we formulate the asymptotic requirement ( 17 ), ( 18 ) by r~ ) = z ~ ) at ¢ = 0 ( j = 0 , 1 ; k = t , H) .
(25)
By this condition we have a one-to-one correspondence between the coupling ratios p~) and their approximations p~o) in the limit of electroweak interactions. The regularity transformations are p(0) 2 _1_/" 1/7,r (0) t -- 9 --~ ~t ~
(26)
,,~ (0) __ 2 .a_ r l / 7 , ~ ( 0 ) Ut0 -- 9 --~ ~tO
p ~) = ~ (.v/-6~- 25 ) + p~ + ~ ' , / ~ / z l r ~ ) , p ~o) = ~8 ( ~ p}~) =qo+q,~lg~+q~+~8/vCt~),
25 ) + Po~ + ~~ / z ' Z ~ o
),
p,~(t)=qo +~8/7z}~) ,
pff)=ro+r'l(lg(+rl(+r~(21g2(+r'E(21g(+r2(Z+(t+x/g~/zlr#
(27) (28)
) , phlo)=ro+(l+',//g-ffg/2'zff ) .
(29)
the form of these transformations is in agreement with the explicit solutions of the differential equations ( 2 1 ) - ( 2 4 ) . The coefficients qj, q~, r;, rj are taken from the formal expansions near ~=0. The limits l i m p } ° ) = ~2 ,
limp~)=~8(6~/6~-25)
~o
~o
(30)
represent the infrared fixed points of Pendleton and Ross [ 5 ]. They coincide with the ultraviolet limits of the reduction solutions in the absence of electroweak interactions. Likewise the limits limp} ~)=qo,
limp~ )=ro
~0
~0
(31)
are identical to the coefficients (16) occurring in the ultraviolet approximations (15). Due to the factor ~ , however, the original fixed points (30) are destroyed by diverging contributions of order h. The regularity transformations represent Taylor formulae modified by taking into account the occurrence of logarithmic terms and non-integral powers. In this sense the transformed functions z~), Z~o) may be interpreted as remainders of Taylor type formulae for the coupling ratios p~), p~o) . In a more detailed paper it will be shown that the coefficients of the differential equations for the functions z~), r ~ ) satisfy Lipschitz conditions at ~= 0 so that solutions are uniquely determined by their initial values (25) [ 6 ]. Combining the one- and two-loop approximations ( 2 6 ) - ( 2 9 ) we get
pt~p}O ) +hotsp(tl)= a_l +a,olg ~+ao +~l/7zt, z , = z t (°) +hoqr~ t) ,
pH.~p~)) +hoqp~) = ~-~ +b'olg(+bo+b'{(lg2(+b'~(lg~+bt~+(',/~/2~z~, ztz= z ~ ) + h a ~ z ~ ) .
(32)
In case of the non-trivial reduction the coupling ratios p~0) form constant solutions of the differential equations (see ( 1 8 ) ) 252
PHYSICS LETTERS B
Volume 311, number 1,2,3,4 p(O) ,o = ~2,
"(') =qo, v~ n( °o)-- ~ s ( 6 x / ~ - 25) , ~-,o
P})o)=ro,
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(33)
implying that the functions r~6) vanish identically z~ )=0,
j=0,1,
k=t,H.
(34)
According to (25) the correct solutions p~, of the full system (6), (7) of differential equations are then determined by the initial values r~)=0
at(=0
(35)
which represent the limit of vanishing electroweak couplings. It would be quite cumbersome to compute the couplings on the basis of this condition. Moreover, the solution would involve the constants a and b as given by (19) and two-loop corrections, a and b would have to be reexpressed by the values of the running coupling parameters at the Z mass. Instead, it is much more convenient to consider the top and Higgs coupling ratios as functionals (2) expressing them directly in terms of the experimental parameters gs, g and g'. It is now easy to show that Kubo's expansion ofpt and p, with respect to powers of u, v and c~s is the correct solution of the partial differential equations
Opt
0"2~
Op,." 1 - O ' s O gOpt s-
+0-I - 0~'
0~ s =0-t '
0"2 ~
"~- 0-1
0PH + a~C~ ~ s
=crH.
(36)
Any solution of (36) yields a solution of the original system (7) of ordinary differential equations provided the solutions u and v of (6) are substituted into the functionals (2). The power series solution 2 P'=9"~E CpqrupvqO~r' P - = ~ s ( 6 V / ~ - 2 5 )
+ ~ Cpqrupl)qOLr
(37)
is unique by the choice of the lowest order (4). Using the asymptotic behavior (20) of u and v it is seen that terms involving non-integral powers (~/7 and (.,/~/2~ as in the general form of (32), (33) are absent in the expansion (37). Therefore the requirement (35) is satisfied. This completes the proof that the power series expansion (37) of the functional (2) represents the correct solution of the evolution equations (7) which for vanishing electroweak interactions approaches the reduction solution with the ultraviolet limit (4). In Kubo's paper also the influence of the bottom coupling Gb was included and found negligible [2 ]. In that case the justification of the power series solution is different since the appropriate limits in lowest order are limp} °) = g , lmp}°) = ~ w i t h p b = G~ , ¢~o ~o 47~O~s
(38)
so that the form of the regularity transformations is changed completely. Nevertheless, it could be shown that the power series expansion is correct in the one-loop order [ 6 ]. So far applications of the reduction method have been restricted to the standard model as such. The modification with two Higgs doublets was treated by Denner [ 7 ]. Not included have been effects which originate in particles and interactions beyond the standard model. Substantial changes for the top and Higgs masses could be expected, for instance, in a supersymmetric extension of the standard model. I am grateful to Dr. J. Kubo and Dr. K. Sibold for helpful discussions.
References
[ 1] J. Kubo, K. Sibold and W. Zimmermann, Nucl. Phys. B 259 (1985) 331. 253
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PHYSICS LETTERS B
[2] J. Kubo, Phys. Lett. B 262 ( 1991 ) 472. [3 ] J. Kubo, K. Sibold and W. Zimmermann, Phys. Lett. B 220 ( 1989 ) 185. [ 4 ] W. Zimmermann, Phys. Lett. B 308 (1993) 117. [5] B. Pendleton and G. Ross, Phys. Lett. B 98 ( 1981 ) 291. [6] W. Zimmermann, in preparation. [7] A. Denner, Nucl, Phys. B 347 (1990) 184.
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