- Email: [email protected]
Couplings of a light Higgs boson
Volume 222, number 2
PHYSICSLETTERSB
18 May 1989
COUPLINGS OF A L I G H T H I G G S BOSON R. Sekhar CHIVUKULA Department of Physws, Boston Umverszty, Boston, MA 02215, USA
Andrew COHEN 1, Howard GEORGI Lyman Laboratory of Physws, Harvard Umverstty, CambrMge, MA 02138, USA
and Aneesh V. MANOHAR Centerfor TheoretzcalPhysics, Laboratoryfor Nuclear Scwnce and Department of Physzcs, Massachusetts Institute of Technology, Cambrzdge, MA 02139, USA
Received 1 March 1989
We comment on the calculation of Voloshm and Zakharov of the couplingsof a light Hlggs boson to pions and extend their analysis to consider flavor-changingHiggs-ploncouphngs,includingK--,cpevand K~ cpn.
1. Introduction
2. Flavor-conserving couplings
The Higgs boson of the standard model may be very light, with a mass of 1 GeV or less, if there is a heavy weak doublet fermion (e.g. an 80 GeV quark or 105 GeV lepton) [ 1-3 ]. If this is the case, the couplings of the Higgs into mesons composed of light quarks become important. Recently Voloshin and Zakharov [4,5 ] have pointed out that some of the couplings of a light Higgs to pions are calculable to leading order in chiral perturbation theory. In this note, we review the Voloshin-Zaldaarov analysis in slightly different language, which, we hope, will convince the reader of the significance of their result. This note is intended to simplify the analysis previously presented in ref. [ 6 ]. In addition we will extend the Voloshin-Zakharov analysis so as to consider the flavor-changing Higgs production processes K--,~q~ and K~ev~p.
The couplings of the Higgs boson just below the SU(2) breaking scale may be conveniently summarized to lowest order in small couplings as [ 7 ]
Junior Fellow, Harvard Societyof Fellows. 258
-
v
m,--+
Ore,
+ self-couplings,
Z M ` t=W-+,Z
(2.1)
where the first sum runs over all the weak doublet fermions in the theory, 5¢ refers to the lagrangian describing the non-Higgs interactions, and v~ 250 GeV is the weak scale. At tree level, the correspondence of (2.1) to the usual Higgs couplings is straightforward, the first term accounts for the Yukawa couplings, while the second term accounts for the couplings of the Higgs to the weak gauge bosons. The self-couplings are characterized by the dimensionless coupling constant 2 which appears in the Higgs potential. In terms of physical constants, 2 is proportional to the ratio of the Higgs mass squared to the VEV squared, mZ/v 2. For the light Higgs which we wish to consider, mH ~<1 GeV, these couplings are corn-
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 222, number 2
PHYSICS LETTERSB
pletely negligible, and will be ignored in our subsequent analysis. The utility of (2.1) arises when we consider the couplings of the Higgs to the light pseudo-scalar mesons at energies below 1 GeV. To lowest order in weak couplings, the effective lagrangian for the Higgs couplings may be computed using (2.1), where the lagrangian that appears on the right-hand side of this equation is the effective lagrangian for the light degrees of freedom we are interested in. For the case of a light Higgs interacting with the light mesons, we may use an effective low-energy chiral lagrangian truncated to lowest order in momenta and symmetry breaking [ 8 ]: ~¢=iaf2TrOUXO~Xt+ l f ~ A z ( T r M X + h . c . ) ,
(2.2)
where 27(x)=exp(2izc/f~), re=zeST ~ are the pion fields, and the T ~ are the generators of SU (3). Here f~=93 MeV is the pion decay constant, Ax~ 1 GeV, and
M=
(i u0 00) ma 0
(2.3)
rn~
is the mass matrix of the light quarks. To the extent that the plons and kaons are Goldstone bosons, the u, d, and s quarks are light and, as assumed in (2.2), their masses may be treated as perturbations to the chirally symmetric lagrangian. The terms in the first sum in (2.1) involving these quarks may then be calculated directly from (2.2) and generate the effective Higgs couplings = ~ 2 ((p/v)Az(T r M S + h.c. ) .
(2.4)
It remains to compute the contributions from the heavy quarks c, b and t (and any other as yet undiscovered heavy quarks). Although a direct computation of the derivative in (2.1) with respect to a heavy quark mass is not practicable, usmg the effective field theory technology we can easily replace the derivative with one that is applicable at low energies. As we integrate out a heavy quark, we know that below the heavy quark threshold the effective lagrangian will involve all operators of dimension four or less with coefficients that are renormalized by heavy quark effects, as well as higher dimension operators that are suppressed by inverse powers of the heavy
18 May 1989
quark m a s s ~1. Ignonng these higher dimension terms, the derivative with respect to the heavy quark mass may be rewritten using the chain rule for partial derivatives 0 0g, 0 mh 0mh - ~, mh 0mh 0g, '
(2.5)
where the {g~} are the effective couplings in the theo131 with the heavy quark removed. For QCD, these couplings are g~, the strong interaction coupling constant, and the light quark masses. The derivative Og,/ 0mh is performed keeping all couplings of the high energy theory fixed. Since we are evaluating these derivatives near the heavy quark threshold, g~ is much less than 1, and we may employ QCD perturbation theory. The masses of quarks lighter than mh only receive a contribution from the heavy quarks at order ce~ and hence we may ignore these terms to lowest order 0ml ___O(o~). 0mh
(2.6)
There remains simply the effect on gs- To leading order in QCD the effective coupling near the quark threshold may be written as 6zc _ 3 b l O g A :c ~s(#) I) - ~ [ 2 + O ( a s ( r n h ) ) ] O ( # - - m h ) log # h mh +O(a~(#)),
(2.7)
where O is the Heaviside step function, b = 11 - J nl and nl ( = 3) is the number of light fermions. This equation comes from integrating the lowesi order renormalization group equation for o~s(#); the theta function terms anse from the discontinuity of the beta function at the heavy quark thresholds. Varying rnh in (2.7), holding o~s(#) (# >> rnt) fixed, we find that O~ 2 mh 0mh - ~
O~
AQC°0AQcD
+O(OZs(mh) ) ,
(2.8)
for each mh, h = c, b, t. Hence to lowest order m ors(me) the calculation of the couphngs of a light Hlggs boson ~1 See also ref. [9]. 259
Volume 222, number 2
PHYSICSLETTERSB
to pions reduces to calculatmg the response of the chtral lagrangian to changes in the QCD scale. The physics of (2.8) may be understood by comparing two different field theories which have identical short-distance behavior. For definiteness consider two QCD-like theories, one with a heavy quark of mass rn, the second with a slightly heavier quark of mass m + 8. In order to make the short-distance physics of the two theories agree, we adjust the color couphng constants in the two theories to agree at energies much greater than m + 8. At scales above m + 8 the color beta functions in the two theories are the same; at scales below the lighter quark mass m, the beta functions are still the same, since both quarks have been integrated out. However in the region between m + 8 and m, the heavier quark in the second theory has been integrated out, but the quark in the first theory remains. Consequently the second theory is more asymptotically free than the first since its beta function lacks the contribution from the heavy quark. the coupling in the second theory increases more rapidly in the region between rn and rn + 8. Thus AQCD, which characterizes the strength of the color interaction at scales below these heavy quark masses, will be larger in the second theory. Eq. (2.8) expresses this increase of AQci) with increasing heavy quark mass, when the short-distance physics is held fixed. The effective low momentum lagrangian involving pions is given by (2.2). Here 27 is a (dimensionless) field which implements the non-linear transformation law under the spontaneously broken chiral symmetry. This lagrangian is correct to lowest order in the light quark mass matrix - the parameters f= and A x are independent of M. By dimensional analysis, they must therefore be proportional to the only dimensionful parameter in the theory, AQCD. Consequently the derivative in (2.8) may be replaced by AQcD 05° 0£0 O~ 0AQcD -- f=-0-f= + AxO£xx" The derivative is easily evaluated to give 0£o
mh 0m----7= ( 2nh/3b ) [-]f ~ Tr 0u270uZ* +!2f2 A z (Tr M Z + h . c . ) ] ,
(2.10)
where n h is the total number of heavy quarks. Finally, 260
using (2.1) the effective Higgs-Goldstone boson lagrangian is
£P~ff= (nh/3b ) (tp/v)f 2 Tr OuX*Ou27 + ~-2f2Az(TrM27+h.c.)(l+2nh/b)tp/v.
(2.11)
For the standard six-quark model nh/b = 1/3. The formula (2.1) is only sufficient to calculate the couplings of the Higgs boson which do not vanish in the limit of zero Higgs momentum. The only other flavor symmetric chirally invariant coupling, to lowest order in chiral perturbation theory, is
v-lOUq~ Tr ZB~27t +h.c.
(2.12)
Since 27 is a matrix of determinant one, however, the trace in (2.12) is identically zero. Therefore, to lowest order in chiral perturbation theory and in as (me), (2.1 1 ) is the complete effective lagrangian describing the flavor symmetric couplings of a light Higgs boson to pions. From (2.1 1 ), we may compute the amplitude for the decay of a light Higgs into pions d ( 9 ~ a r ~ b) = - [ (2/3v) (nh/b)m 2
+ (1/v)m2a( 1 +2nh/3b) ]Sab,
(2.1 3)
in agreement with refs. [4,5 ]. There are high energy corrections to (2.13) suppressed by powers of as (me) / zr ( g 0.1 ) and low energy corrections which, if chiral perturbation theory is valid, are suppressed by powers of rn,/AzsB, 2 2 where AzsB ~ 1 GeV. For a Higgs mass of order 700 MeV or less, these corrections are expected to be relatively small. We may check, self consistently, that chiral perturbation theory is valid by computing the chiral logarithmic corrections to (2.1 3 ). Evaluating these in the chiral limit [ 10 ] we find d (tp--,~a~zb) = - [ (2/3v) (nh/b)m 2 ] × [l_3~(m,/Axs~)2 2 ln(m2/A2s,)]Sab.
(2.9)
18 May 1989
(2.14)
From (2.1 4) we see that final-state pion interactions tend to enhance the decay of a Higgs boson to two pions. This result is consistent with the dispersion theoretic analysis presented in ref. [ 1 1 ]. As the coefficient of the logarithm is not large, we expect that chiral perturbation theory is valid for this computation.
Volume 222, number 2
PHYSICS LETTERS B
3. A S = 1 couplings
Let us first consider the decay K + ~e-+vtO [ 12 ]. The effective lagrangian for the semileptonic AS= 1 decay in the quark model is
(gZ/2M2+) sin 0c (gLyuUL) (gLTUeL),
(3.1)
where g is the SU(2)w coupling constant and 0c is the Cabibbo angle. To lowest order in chiral perturbation theory this matches to the chiral lagrangian operator
18 May 1989
The corresponding calculation for K ~ to~tv may be found in ref. [ 13 ]. The non-leptonic processes such as the decay K + ~ rc-+to may be computed in the same way. The situation is slightly more complicated here since there are a number of operators in the effective theory that can contribute to the process in lowest order chiral perturbation theory. To lowest order in chiral perturbation theory the non-leptonic AS= 1 lagrangian without the Higgs is 2 2 2 2 £P=a(f ~g Az/Mw) Tr(h+h*)&ZO~Z*
(g2/4M2+) sin 0of2=(Tr e270uZ~) (gL~eL) ,
3 2 2 + b ( f 2,~Axg /Mw) [Tr h ( M Z t + S M *) +b.c.]
(3.2)
+ e (f 2A~g2/m2) T~I(ZOuZ t ) ,j (ZO~Z*) ~
where
+h.c., e=
0 0
.
(3.3)
We can calculate the corresponding Higgs coupling using (2.1). The terms in the first sum of (2.1) that come from heavy quarks are computed exactly as in the previous section, by converting to a derivative with respect tof~ (and the terms from the fight quarks vanish). The term in the second sum coming from the W can be computed explicitly. Combining both terms we get a AS= 1 semi-leptonic Higgs coupling £*~ff= ( 4 n ~ / 3 b - 2) (tO/v) (g2/4M2+) sin 0c × f 2 (Tr e270aX~) (gLy~eL)
.
(3.4)
While (2.1) is only sufficient to compute the couplings of the Higgs boson which do not vanish in the limit of vanishing Higgs momentum, to this order in chiral perturbation theory there are no other chirally invariant operators. The 4nh/3b term in (3.4) may be thought of as arising from the contribution of the first term in A°~ff to the appropriate chiral current. The 4nh/3b factor was neglected in ref. [ 2 ] and, in the six-quark model, this results in a multiplicative correction of 7/9 in the amplitude for K~tOev. Including this correction, we find from ref. [21 B R ( K ~ e v $ ) = 2 . 4 × 10-Sf(x),
(3.5)
where x=m2/m~: and
f(x)=(1--8x+x2)(1--x2)--12x21nx.
(3.6)
(3.7)
where h=
0
,
(3.8)
1
and T is a traceless tensor, symmetric in both upper and lower indices. The c term accounts for the M = 3/ 2 part of the weak nonqeptonic hamiltonian ~2. We may choose b = 0 by making an SUL(3) rotatmn which diagonalizes the mass matrix (including the AS= 1 term above) ~3. To include the Higgs our by now familiar procedure gives
£P~=a(8nh/3b--2)(to/v) (f=g22Ax/Mw)2 2 X Tr( h + h *) OuZO~ * +c(8nh/3b--2) (to~v) (f2A~g2/M2) × T~(XOuZ*),j(ZOuZ*) k%h.c. +dv-lOu9 Tr XOuZ*(h+h*) .
(3.9)
Note that the d term has no analog in the Higgs independent part of the lagrangian - without the Higgs field, this operator would be a total derivative and would not contribute to momentum-conserving processes involving pions only. Consequently its coeffi~2 The explicit form of the tensor, which we wall not need, may be found in ref. [8 ]. ~a In this basis, where the effective mass matrix as dmgonal, C P S symmetry places no further restrichons on the coefficients of
these operators [ 14]. 261
Volume 222, number 2
PHYSICS LETTERS B
cient is not d e t e r m i n e d by our procedure, which only gives the Higgs couplings which do not vanish at zero m o m e n t u m . The processes not involving the Higgs suffice to d e t e r m i n e only two o f these three coefficients ( a a n d c), a n d there remains one uncertain par a m e t e r in the A S = 1 non-leptonic couplings o f the Higgs. However, the p a r a m e t e r d is expected to be o f the same o r d e r as other A / = 1/2 amplitudes. This allows an estimate o f the decay K-->~0n as in ref. [2 ].
4. Conclusions Using the effective field theory f o r m a l i s m we have d e m o n s t r a t e d that light Higgs couplings at low m o m e n t u m can be separated into terms from three distinct sources: those from weak gauge boson couplings; those from light quark masses; a n d those from heavy quark masses. The first two terms are taken care o f in a straightforward fashion; however the heavy quark terms require a special technique. We have related the contribution from these terms to the dependence o f the effective theory on the Q C D scale,
AQCD. The discussion that is often given o f the contribution o f heavy quarks involves the q u a n t u m violation o f scale invariance, which implies the non-vanishing o f the divergence o f the scale current, s ~. To connect our current a p p r o a c h with this view, we need only note that in the low energy regime, although the theory is not invariant u n d e r the field t r a n s f o r m a t i o n corresponding to scale symmetry, it ts i n v a r i a n t under the c o m b i n e d scaling o f all dimensionful couplings along with this scale transformation. This includes all quark a n d gauge b o s o n masses, as well as AQCD. The result is that the divergence o f the scale current satisfies the equation ~LP
~SU+/QCD O/QCD +
262
2
m,
+
2
M,
/:W-+,Z
=0
(4.1)
18 May 1989
F r o m this we m a y i m m e d i a t e l y relate the discussion o f Higgs couplings v i a the derivative with respect to AQCD to discussions involving the divergence o f the scale current.
Acknowledgement We are grateful to Benjamin G r i n s t e i n for getting us started. This work was s u p p o r t e d in part b y N S F contracts PHY-82-15249, PHY87-14654, a n d D O E grants DE-AC02-76ER03069, and DE-AC0286ER40284. A.M. was s u p p o r t e d in part by a grant from the Alfred P. Sloan foundation.
References [ 1] H. Haber, A. Schwarz and A. Snyder, Nucl. Phys B 294 (1987) 301. [ 2 ] R.S. Chlvukula and A.V Manohar, Phys. Lett. B 207 (1988) 86, B 217 (1989) 568(E). [ 3 ] B. Gnnsteln, L. Hall and L. Randall, Phys Lett. B 211 (1988) 363. [4] M. Voloshm and V. Zakharov, Phys Rev Leit 45 (1980) 688. [5] M. Voloshln, Sov. J Nucl. Phys. 44 (1986) 478 [ 6 ] R S Cluvukula, A Cohen, H Geor~, B. Gnnstem and A.V. Manohar, Ann. Phys. (NY), m press. [7 ] J Elhs, M.K Galllard and D.V. Nanopoulos, Nucl. Phys. B 106 (1976) 292 [8 ] H. Georga, Weak interactions and modern particle theory (Benjamin/Cummings, Menlo Park, CA, 1984). [9 ] L.S Brown, Umverslty of Washington preprint (January 1989). [ 10 ] A. Cohen, in preparation. [ 11 ] S. Raby and G.B. West, Los Alamos preprmt LA-UR-882491 (July 1988). [12] T.N. Pham and D.G Sutherland, Phys. Rev D 34 (1986) 1634; A De Rtijula and R S. Chlvukula, unpublished. [ 13 ] S. Dawson, Brookhaven preprmt (January 1989). [ 14 ] C Bernard, T. Draper, A. Sore, H.D. Pohtzer and M.B. Wise, Phys Rev. D 32 (1985) 2343.
PHYSICSLETTERSB
18 May 1989
COUPLINGS OF A L I G H T H I G G S BOSON R. Sekhar CHIVUKULA Department of Physws, Boston Umverszty, Boston, MA 02215, USA
Andrew COHEN 1, Howard GEORGI Lyman Laboratory of Physws, Harvard Umverstty, CambrMge, MA 02138, USA
and Aneesh V. MANOHAR Centerfor TheoretzcalPhysics, Laboratoryfor Nuclear Scwnce and Department of Physzcs, Massachusetts Institute of Technology, Cambrzdge, MA 02139, USA
Received 1 March 1989
We comment on the calculation of Voloshm and Zakharov of the couplingsof a light Hlggs boson to pions and extend their analysis to consider flavor-changingHiggs-ploncouphngs,includingK--,cpevand K~ cpn.
1. Introduction
2. Flavor-conserving couplings
The Higgs boson of the standard model may be very light, with a mass of 1 GeV or less, if there is a heavy weak doublet fermion (e.g. an 80 GeV quark or 105 GeV lepton) [ 1-3 ]. If this is the case, the couplings of the Higgs into mesons composed of light quarks become important. Recently Voloshin and Zakharov [4,5 ] have pointed out that some of the couplings of a light Higgs to pions are calculable to leading order in chiral perturbation theory. In this note, we review the Voloshin-Zaldaarov analysis in slightly different language, which, we hope, will convince the reader of the significance of their result. This note is intended to simplify the analysis previously presented in ref. [ 6 ]. In addition we will extend the Voloshin-Zakharov analysis so as to consider the flavor-changing Higgs production processes K--,~q~ and K~ev~p.
The couplings of the Higgs boson just below the SU(2) breaking scale may be conveniently summarized to lowest order in small couplings as [ 7 ]
Junior Fellow, Harvard Societyof Fellows. 258
-
v
m,--+
Ore,
+ self-couplings,
Z M ` t=W-+,Z
(2.1)
where the first sum runs over all the weak doublet fermions in the theory, 5¢ refers to the lagrangian describing the non-Higgs interactions, and v~ 250 GeV is the weak scale. At tree level, the correspondence of (2.1) to the usual Higgs couplings is straightforward, the first term accounts for the Yukawa couplings, while the second term accounts for the couplings of the Higgs to the weak gauge bosons. The self-couplings are characterized by the dimensionless coupling constant 2 which appears in the Higgs potential. In terms of physical constants, 2 is proportional to the ratio of the Higgs mass squared to the VEV squared, mZ/v 2. For the light Higgs which we wish to consider, mH ~<1 GeV, these couplings are corn-
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 222, number 2
PHYSICS LETTERSB
pletely negligible, and will be ignored in our subsequent analysis. The utility of (2.1) arises when we consider the couplings of the Higgs to the light pseudo-scalar mesons at energies below 1 GeV. To lowest order in weak couplings, the effective lagrangian for the Higgs couplings may be computed using (2.1), where the lagrangian that appears on the right-hand side of this equation is the effective lagrangian for the light degrees of freedom we are interested in. For the case of a light Higgs interacting with the light mesons, we may use an effective low-energy chiral lagrangian truncated to lowest order in momenta and symmetry breaking [ 8 ]: ~¢=iaf2TrOUXO~Xt+ l f ~ A z ( T r M X + h . c . ) ,
(2.2)
where 27(x)=exp(2izc/f~), re=zeST ~ are the pion fields, and the T ~ are the generators of SU (3). Here f~=93 MeV is the pion decay constant, Ax~ 1 GeV, and
M=
(i u0 00) ma 0
(2.3)
rn~
is the mass matrix of the light quarks. To the extent that the plons and kaons are Goldstone bosons, the u, d, and s quarks are light and, as assumed in (2.2), their masses may be treated as perturbations to the chirally symmetric lagrangian. The terms in the first sum in (2.1) involving these quarks may then be calculated directly from (2.2) and generate the effective Higgs couplings = ~ 2 ((p/v)Az(T r M S + h.c. ) .
(2.4)
It remains to compute the contributions from the heavy quarks c, b and t (and any other as yet undiscovered heavy quarks). Although a direct computation of the derivative in (2.1) with respect to a heavy quark mass is not practicable, usmg the effective field theory technology we can easily replace the derivative with one that is applicable at low energies. As we integrate out a heavy quark, we know that below the heavy quark threshold the effective lagrangian will involve all operators of dimension four or less with coefficients that are renormalized by heavy quark effects, as well as higher dimension operators that are suppressed by inverse powers of the heavy
18 May 1989
quark m a s s ~1. Ignonng these higher dimension terms, the derivative with respect to the heavy quark mass may be rewritten using the chain rule for partial derivatives 0 0g, 0 mh 0mh - ~, mh 0mh 0g, '
(2.5)
where the {g~} are the effective couplings in the theo131 with the heavy quark removed. For QCD, these couplings are g~, the strong interaction coupling constant, and the light quark masses. The derivative Og,/ 0mh is performed keeping all couplings of the high energy theory fixed. Since we are evaluating these derivatives near the heavy quark threshold, g~ is much less than 1, and we may employ QCD perturbation theory. The masses of quarks lighter than mh only receive a contribution from the heavy quarks at order ce~ and hence we may ignore these terms to lowest order 0ml ___O(o~). 0mh
(2.6)
There remains simply the effect on gs- To leading order in QCD the effective coupling near the quark threshold may be written as 6zc _ 3 b l O g A :c ~s(#) I) - ~ [ 2 + O ( a s ( r n h ) ) ] O ( # - - m h ) log # h mh +O(a~(#)),
(2.7)
where O is the Heaviside step function, b = 11 - J nl and nl ( = 3) is the number of light fermions. This equation comes from integrating the lowesi order renormalization group equation for o~s(#); the theta function terms anse from the discontinuity of the beta function at the heavy quark thresholds. Varying rnh in (2.7), holding o~s(#) (# >> rnt) fixed, we find that O~ 2 mh 0mh - ~
O~
AQC°0AQcD
+O(OZs(mh) ) ,
(2.8)
for each mh, h = c, b, t. Hence to lowest order m ors(me) the calculation of the couphngs of a light Hlggs boson ~1 See also ref. [9]. 259
Volume 222, number 2
PHYSICSLETTERSB
to pions reduces to calculatmg the response of the chtral lagrangian to changes in the QCD scale. The physics of (2.8) may be understood by comparing two different field theories which have identical short-distance behavior. For definiteness consider two QCD-like theories, one with a heavy quark of mass rn, the second with a slightly heavier quark of mass m + 8. In order to make the short-distance physics of the two theories agree, we adjust the color couphng constants in the two theories to agree at energies much greater than m + 8. At scales above m + 8 the color beta functions in the two theories are the same; at scales below the lighter quark mass m, the beta functions are still the same, since both quarks have been integrated out. However in the region between m + 8 and m, the heavier quark in the second theory has been integrated out, but the quark in the first theory remains. Consequently the second theory is more asymptotically free than the first since its beta function lacks the contribution from the heavy quark. the coupling in the second theory increases more rapidly in the region between rn and rn + 8. Thus AQCD, which characterizes the strength of the color interaction at scales below these heavy quark masses, will be larger in the second theory. Eq. (2.8) expresses this increase of AQci) with increasing heavy quark mass, when the short-distance physics is held fixed. The effective low momentum lagrangian involving pions is given by (2.2). Here 27 is a (dimensionless) field which implements the non-linear transformation law under the spontaneously broken chiral symmetry. This lagrangian is correct to lowest order in the light quark mass matrix - the parameters f= and A x are independent of M. By dimensional analysis, they must therefore be proportional to the only dimensionful parameter in the theory, AQCD. Consequently the derivative in (2.8) may be replaced by AQcD 05° 0£0 O~ 0AQcD -- f=-0-f= + AxO£xx" The derivative is easily evaluated to give 0£o
mh 0m----7= ( 2nh/3b ) [-]f ~ Tr 0u270uZ* +!2f2 A z (Tr M Z + h . c . ) ] ,
(2.10)
where n h is the total number of heavy quarks. Finally, 260
using (2.1) the effective Higgs-Goldstone boson lagrangian is
£P~ff= (nh/3b ) (tp/v)f 2 Tr OuX*Ou27 + ~-2f2Az(TrM27+h.c.)(l+2nh/b)tp/v.
(2.11)
For the standard six-quark model nh/b = 1/3. The formula (2.1) is only sufficient to calculate the couplings of the Higgs boson which do not vanish in the limit of zero Higgs momentum. The only other flavor symmetric chirally invariant coupling, to lowest order in chiral perturbation theory, is
v-lOUq~ Tr ZB~27t +h.c.
(2.12)
Since 27 is a matrix of determinant one, however, the trace in (2.12) is identically zero. Therefore, to lowest order in chiral perturbation theory and in as (me), (2.1 1 ) is the complete effective lagrangian describing the flavor symmetric couplings of a light Higgs boson to pions. From (2.1 1 ), we may compute the amplitude for the decay of a light Higgs into pions d ( 9 ~ a r ~ b) = - [ (2/3v) (nh/b)m 2
+ (1/v)m2a( 1 +2nh/3b) ]Sab,
(2.1 3)
in agreement with refs. [4,5 ]. There are high energy corrections to (2.13) suppressed by powers of as (me) / zr ( g 0.1 ) and low energy corrections which, if chiral perturbation theory is valid, are suppressed by powers of rn,/AzsB, 2 2 where AzsB ~ 1 GeV. For a Higgs mass of order 700 MeV or less, these corrections are expected to be relatively small. We may check, self consistently, that chiral perturbation theory is valid by computing the chiral logarithmic corrections to (2.1 3 ). Evaluating these in the chiral limit [ 10 ] we find d (tp--,~a~zb) = - [ (2/3v) (nh/b)m 2 ] × [l_3~(m,/Axs~)2 2 ln(m2/A2s,)]Sab.
(2.9)
18 May 1989
(2.14)
From (2.1 4) we see that final-state pion interactions tend to enhance the decay of a Higgs boson to two pions. This result is consistent with the dispersion theoretic analysis presented in ref. [ 1 1 ]. As the coefficient of the logarithm is not large, we expect that chiral perturbation theory is valid for this computation.
Volume 222, number 2
PHYSICS LETTERS B
3. A S = 1 couplings
Let us first consider the decay K + ~e-+vtO [ 12 ]. The effective lagrangian for the semileptonic AS= 1 decay in the quark model is
(gZ/2M2+) sin 0c (gLyuUL) (gLTUeL),
(3.1)
where g is the SU(2)w coupling constant and 0c is the Cabibbo angle. To lowest order in chiral perturbation theory this matches to the chiral lagrangian operator
18 May 1989
The corresponding calculation for K ~ to~tv may be found in ref. [ 13 ]. The non-leptonic processes such as the decay K + ~ rc-+to may be computed in the same way. The situation is slightly more complicated here since there are a number of operators in the effective theory that can contribute to the process in lowest order chiral perturbation theory. To lowest order in chiral perturbation theory the non-leptonic AS= 1 lagrangian without the Higgs is 2 2 2 2 £P=a(f ~g Az/Mw) Tr(h+h*)&ZO~Z*
(g2/4M2+) sin 0of2=(Tr e270uZ~) (gL~eL) ,
3 2 2 + b ( f 2,~Axg /Mw) [Tr h ( M Z t + S M *) +b.c.]
(3.2)
+ e (f 2A~g2/m2) T~I(ZOuZ t ) ,j (ZO~Z*) ~
where
+h.c., e=
0 0
.
(3.3)
We can calculate the corresponding Higgs coupling using (2.1). The terms in the first sum of (2.1) that come from heavy quarks are computed exactly as in the previous section, by converting to a derivative with respect tof~ (and the terms from the fight quarks vanish). The term in the second sum coming from the W can be computed explicitly. Combining both terms we get a AS= 1 semi-leptonic Higgs coupling £*~ff= ( 4 n ~ / 3 b - 2) (tO/v) (g2/4M2+) sin 0c × f 2 (Tr e270aX~) (gLy~eL)
.
(3.4)
While (2.1) is only sufficient to compute the couplings of the Higgs boson which do not vanish in the limit of vanishing Higgs momentum, to this order in chiral perturbation theory there are no other chirally invariant operators. The 4nh/3b term in (3.4) may be thought of as arising from the contribution of the first term in A°~ff to the appropriate chiral current. The 4nh/3b factor was neglected in ref. [ 2 ] and, in the six-quark model, this results in a multiplicative correction of 7/9 in the amplitude for K~tOev. Including this correction, we find from ref. [21 B R ( K ~ e v $ ) = 2 . 4 × 10-Sf(x),
(3.5)
where x=m2/m~: and
f(x)=(1--8x+x2)(1--x2)--12x21nx.
(3.6)
(3.7)
where h=
0
,
(3.8)
1
and T is a traceless tensor, symmetric in both upper and lower indices. The c term accounts for the M = 3/ 2 part of the weak nonqeptonic hamiltonian ~2. We may choose b = 0 by making an SUL(3) rotatmn which diagonalizes the mass matrix (including the AS= 1 term above) ~3. To include the Higgs our by now familiar procedure gives
£P~=a(8nh/3b--2)(to/v) (f=g22Ax/Mw)2 2 X Tr( h + h *) OuZO~ * +c(8nh/3b--2) (to~v) (f2A~g2/M2) × T~(XOuZ*),j(ZOuZ*) k%h.c. +dv-lOu9 Tr XOuZ*(h+h*) .
(3.9)
Note that the d term has no analog in the Higgs independent part of the lagrangian - without the Higgs field, this operator would be a total derivative and would not contribute to momentum-conserving processes involving pions only. Consequently its coeffi~2 The explicit form of the tensor, which we wall not need, may be found in ref. [8 ]. ~a In this basis, where the effective mass matrix as dmgonal, C P S symmetry places no further restrichons on the coefficients of
these operators [ 14]. 261
Volume 222, number 2
PHYSICS LETTERS B
cient is not d e t e r m i n e d by our procedure, which only gives the Higgs couplings which do not vanish at zero m o m e n t u m . The processes not involving the Higgs suffice to d e t e r m i n e only two o f these three coefficients ( a a n d c), a n d there remains one uncertain par a m e t e r in the A S = 1 non-leptonic couplings o f the Higgs. However, the p a r a m e t e r d is expected to be o f the same o r d e r as other A / = 1/2 amplitudes. This allows an estimate o f the decay K-->~0n as in ref. [2 ].
4. Conclusions Using the effective field theory f o r m a l i s m we have d e m o n s t r a t e d that light Higgs couplings at low m o m e n t u m can be separated into terms from three distinct sources: those from weak gauge boson couplings; those from light quark masses; a n d those from heavy quark masses. The first two terms are taken care o f in a straightforward fashion; however the heavy quark terms require a special technique. We have related the contribution from these terms to the dependence o f the effective theory on the Q C D scale,
AQCD. The discussion that is often given o f the contribution o f heavy quarks involves the q u a n t u m violation o f scale invariance, which implies the non-vanishing o f the divergence o f the scale current, s ~. To connect our current a p p r o a c h with this view, we need only note that in the low energy regime, although the theory is not invariant u n d e r the field t r a n s f o r m a t i o n corresponding to scale symmetry, it ts i n v a r i a n t under the c o m b i n e d scaling o f all dimensionful couplings along with this scale transformation. This includes all quark a n d gauge b o s o n masses, as well as AQCD. The result is that the divergence o f the scale current satisfies the equation ~LP
~SU+/QCD O/QCD +
262
2
m,
+
2
M,
/:W-+,Z
=0
(4.1)
18 May 1989
F r o m this we m a y i m m e d i a t e l y relate the discussion o f Higgs couplings v i a the derivative with respect to AQCD to discussions involving the divergence o f the scale current.
Acknowledgement We are grateful to Benjamin G r i n s t e i n for getting us started. This work was s u p p o r t e d in part b y N S F contracts PHY-82-15249, PHY87-14654, a n d D O E grants DE-AC02-76ER03069, and DE-AC0286ER40284. A.M. was s u p p o r t e d in part by a grant from the Alfred P. Sloan foundation.
References [ 1] H. Haber, A. Schwarz and A. Snyder, Nucl. Phys B 294 (1987) 301. [ 2 ] R.S. Chlvukula and A.V Manohar, Phys. Lett. B 207 (1988) 86, B 217 (1989) 568(E). [ 3 ] B. Gnnsteln, L. Hall and L. Randall, Phys Lett. B 211 (1988) 363. [4] M. Voloshm and V. Zakharov, Phys Rev Leit 45 (1980) 688. [5] M. Voloshln, Sov. J Nucl. Phys. 44 (1986) 478 [ 6 ] R S Cluvukula, A Cohen, H Geor~, B. Gnnstem and A.V. Manohar, Ann. Phys. (NY), m press. [7 ] J Elhs, M.K Galllard and D.V. Nanopoulos, Nucl. Phys. B 106 (1976) 292 [8 ] H. Georga, Weak interactions and modern particle theory (Benjamin/Cummings, Menlo Park, CA, 1984). [9 ] L.S Brown, Umverslty of Washington preprint (January 1989). [ 10 ] A. Cohen, in preparation. [ 11 ] S. Raby and G.B. West, Los Alamos preprmt LA-UR-882491 (July 1988). [12] T.N. Pham and D.G Sutherland, Phys. Rev D 34 (1986) 1634; A De Rtijula and R S. Chlvukula, unpublished. [ 13 ] S. Dawson, Brookhaven preprmt (January 1989). [ 14 ] C Bernard, T. Draper, A. Sore, H.D. Pohtzer and M.B. Wise, Phys Rev. D 32 (1985) 2343.