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Strong interaction corrections to semiweak decays: Calculation of the V → Hγ decay rate to order αS
Volume 97B, number 1
PIIYSICS LETTERS
17 November 1980
STRONG INTERACTION CORRECTIONS TO SEMIWEAK DECAYS: CALCULATION OF THE V --' Hy DECAY RATE TO ORDERer s M.I. VYSOTSKY
Institute of Theoretical and Experimental Physics, Moscow, USSR Received 2 June 1980
The decay of heavy onium V ~ [t3, was proposed by Wilczek as a source of Higgs bosons. The calculation of one-loop gluon corrections to this process is presented.
The Higgs boson [ 1] is one of the m o s t intriguing particles of modern physics. It is indispensable for the presentday theory o f electroweak interactions, the Weinberg-Salam model [2]. Peculiar features of the H-particle - an elementary scalar field whose coupling to the matter fields is proportional to their masses - makes the investigation of the properties of the Higgs boson interesting both in theoretical and experimental respects. It is extremely difficult to find them because: (1) the Higgs couplings to particles which can be accelerated to produce H-bosons (e, /a, d and g) are rather small; (2) even if the Higgs particle was Produced it would be difficult to detect it; (3) Higgs bosons are supposed to be rather heavy: Mtt ~> 10 GeV [3]. Recently Wilczek [4] has suggested detecting the H-boson in the decays V -+ H3,, where V is a vector particle consisting of a QQ pair (Q is a heavy quark). This way avoids the first two difficulties listed above. The experimental signal of such a decay is rather clear: monochromatic photons associated with heavy particle decay products. The probability o f this decay increases quadratically with the V-meson mass and becomes equal to the V -+ e+e decay rate for M V = 110 GeV. The only problem is to find the appropriate V-meson. If the Higgs boson is lighter than the T-meson, it could be detected in T -+ H3' decay. It is supposed that the b-quark must have an SU(2)w partner - the t-quark. The following lower limit for Mvti-was obtained at PETRA: M V >~ 30 GeV [5]. The upper limit o f m t comes from investigation of the Higgs potential: m t ~< 100 GeV [6]. VtT with MVt T ~< 80 GeV may be discovered soon and the process e+e - -+ Vti- -+ H7 will work as a H-boson factory. The aim of this paper is to calculate the strong interaction radiative correction to the V -* H3' decay rate. Recent ly the one-loop c~s correction to the r~c --, gg decay rate has been computed and the coefficient of c~s turned out to be rather large [7]. This fact casts a shadow on the application of perturbation theory in a s to this decay. It was interesting for me to learn the impact of gluon corrections on the bare amplitude V -+ H3' fohnd in ref. [4]. My result is the following. The coefficient in front of c~s is rather large. In particular for T, ifM~t/M T2 <~0.8, the correction to F ( T -+ H3,)/F(T -+ e+e - ) amounts to 35% of the original result (I normalize as to its value in J / ~ decay: as(M 2) = 0.2). The correction for the toponium decay rate is smaller because of reduction of c~s and equals 25% for MVt i- = 80 GeV. The sign of the correction is negative, so it diminishes the decay rate. What we are going to calculate is the radiative correction to the bound-state decay rate. The relevant technique was developed in various details by investigators of the positronium system [8]. One must compute the one-loop correction to free e+e - pair annihilation at rest and then multiply the result by ~ (0). The Breit hamiltonian which describes the positronium dynamics produces an o~2 correction to ~ (0), and we neglect this since we are interested only in O(a) corrections. For the quark gluon perturbation theory the situation is just the same, but 4(0) is now mostly due to nonperturbative confinement effects. Hence a reliable computation of ~ (0) seems impossible. By considering the ratio F ( V -+ HT)/F(V -+ e+e - ) we avoid this problem since in this ratio t)(0) cancels out. Non159
Volume 97B, number 1
PHYSICS LETTERS
17 November 1980
universal corrections to A(V -, HT) due to confinement effects are supposed to be of order tl2/M2v where # ~ 250 MeV. Clearly, they are much less important numerically than 0 ( % ) corrections. The 0 ( % ) correction to the decay rate V -+ e+e was computed earlier [9]. Let us write out the expression for T M
FV~e+e - taking into account the ~s terms: [,V.,,e+e_ = 4 rro~Z(eQgV)2Mv [1 - 4CFO~s(M2v)/rr] •
(1)
The following definition o f g v is used: (01CyuC IV)gvM2euCV); eQ is the quark charge; C F = 4/3. To obtain the analogous correction to the decay rate V --* H')' we must compute the seven diagrams depicted in figs. l a - g as well as the seven diagrams with interchanged fermion legs. Let us discuss the computation of diagrams le and lf. Diagram le describes the Higgs-quark interaction dressed by the gluon. It contains In A where A is an ultraviolet cut-off. To understand this in detail let us consider diagrams 2 a - c , which describe scattering of a quark with momentum P and mass m on the Higgs boson with zero momentum transfer. The sum of diagrams 2 a - 2 c is equal to i 3E,/SPlf,= m + i 32;/3m [/;=m and contains In A because the scalar vertex is renormalized by vector interaction. For example, in dealing with the pion-nucleon interaction we must fix the interaction constant at some value of the external momenta and this gives us the rule for eliminating in A. Here we work in a strict WeinbergSalam model where the H~tq interaction is uniquely determined by the fact that the quark mass is given by the condensate value of the Higgs field. The corresponding term in the interaction lagrangian is the following: .C = rn ~ (1 + ~tt/r~), where ~l is the vacuum expectation value of the Higgs field. When the quark-gluon interaction is switched on we must add a quark mass renormalization counter-term to the lagrangian: /2 = [ m
iZ(m)]~(l+¢H/r/).
This counterterm makes the quark mass equal to its "observable" value and at the same time produces an extra term in the physical Higgs-quark interaction. We depict this term in fig. 2d. It is equal to - i )2(m)/rn. Writing out the divergent part of the quark mass operator ~div(P) = am In (A2/m 2) + b(/;
m) In (A2/m2),
we see that In A disappears in the sum of diagrams 2 a - 2 d . We hope that the reader is now convinced that the diagram of fig. l f m u s t be added to that of fig. le. Now let us dwell on the computation of diagram lg. The point is that there the well-known 1/u singularity appears when we compute this diagram (v is the quark-antiquark relative velocity). In the decay probability this term would result in the following contribution: CFoes rr/2v. But the nature of this term is, in fact, a Coulomblike interaction and it can be obtained by expanding I t~+(0)/@(0)l 2 = 2rr/k/(1
e -2w/k)
{a}
(d)
{b}
(e)
{c)
(fi
(gl!
Fig. l. The diagrams producing the O (a s) correction to the decay rate V ~ H% Permutations of the quark legs must also be taken into account. 160
Volume 97B, number 1
PttYSICS LETTERS
17 November 1980
I
_
_
±
-
-
I 1
!i. ×
(ai
/
,
15
(hi
I I I
~Z
~
10
i i
I l
I
,
x
¢c
~d
0
Fig. 2. Higgs quark interaction to first order in a s.
I 0,2
I 0.4
I 0.6
L 0.8
I~_ 1.0 K
Fig. 3. The function a (K).
in C~s(k = 2U/CFO~s). This means that it was already accounted for in the wave function and there is no need to account for the as~t) piece once more. Hence, the term ~ a s/u in the diagram of fig. lg must be subtracted. A final remark about the computation of the diagrams in fig. 1. Some diagrams contain an infrared divergency coming from the low gluon frequencies. To regularize it we give the guon a mass k. We are dealing with the decay of a colourless bound state o f nonrelativistic quarks, so in the sum of all graphs in k disappears. The decay probability is given by the expression:
PV-H-r a(K)=4
7r2 12(1 - K)
+ 4+
+I-K
~ - 1_ _ + 2 K ( K - - 2 ) F(1 - 2K) + __ 2(1 - / ~ ) 1 - 2K (l K)2
\ ~ !
arctg~l~-K!
+
[qb(K) + F ( 1 ) - F ( - 1 ) ]
1_~+2+il-2~
(3)
~
ln2(1-K),
2 v, 2 F(x) = f ~ y - a In (1 + y ) dy is the Spence function, where K = MH/M 1
q~(K) =
dy
In [1 - 4 y ( l ~ y ) K ]
0 q)(K) can be expressed as the sum of two Spence functions of complex arguments, a is regular at K = 1/2. a(0) = 7 + 6 in 2 - 7r2/8. a(K)l~:~l = 47r/3 (1 - t<)1/2. The a(K) dependence is presented in fig. 3. Formula (2) works when M V - M H ~ a2M v . I f M V - M H ~ c~2Mv, the Higgs mass is close to the masses of quarkonium bound states and the V ~ HI' decay must be studied in another way (in ref. [10] the case of the H-mass close to the levels o f the 1'system is discussed). To obtain the value of the ratio r ' ( v ~ H1')/I'(V ~ e+e - ) eq. (2) must be divided by eq. (1) and this slightly diminishes the magnitude o f the correction. Some numerical estimates were presented in the beginning o f the paper. The limit ~ -+ oo in expression (3) is very interesting. In this case we are dealing with the decay of a heavy scalar particle into a bound QQ state and a photon. This is an exclusive process. The description of exclusive processes within QCD is the subject of intensive study nowadays, a = 2 In 2 In (Q2/m2Q) in this limit where Q2 substitutes M 2. The (c~s In Q2)n summation in this process can be performed with the help o f OPE analogously to ref. [1 1]. The decay H ~ (QQ)v1' with the subsequent decay V -+ e+e - from the experimental point of view presents the clearest signal in searching for the H-boson. Unfortunately, the ratio o f this decay rate to the decay rate H -+ Q 0 161
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PHYSICS LETTERS
17 November 1980
is very small: F(H -+ VT)/P (H -+ QQ) = 0.S (gv eQ )2 (mQ/MH)2 , where (g# ec)2 = 6.5 X 10 - 3 , (gqceb) 2 = 6.2 X 10 - 4 . The author is indebted to L.B. Okun, M.I. Polikarpov, M.A. Shifman and M.B. Voloshin for useful discussions. 1 wish to thank M.A. Shifman for improving my bad English.
References [1] P.W. Higgs, Phys. Lett. 12 (1964) 132. [2] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, Proc. 8th Nobel Syrup. (Stockholm, 1968) ed. N. Svarthohn (Almquist and Wiksell, Stockholm, 1968) p. 367. [3] A.D. Linde, Pis'ma Zh. Eksp. Teor. Fiz. 23 (1976) 73; S. Weinberg, Phys. Rev. Lett. 36 (1976) 294. [4] F. Wilczek, Phys. Rev. Lett. 39 (1977) 1304. [5] W. Barrel et al., DESY 80/04 (1980). [6] A.A. Anselm, Pis'ma Zh. Eksp. Teor. Fiz 29 (1979) 645; ll.D. Politzer and S. Wolfram, Phys. Lett. 82B (1979) 242; Erratum 83B (1979) 421. [7] R. Barbieri, G. Curci, E. d'Emilio and E. Remiddi, Nucl. Phys. B154 (1979) 535. [8] W.E. Caswell, G.P. Lepage and J. Sapirstein, Phys. Rev. Lett. 38 (1977) 488; M.I. Vysotsky, Yad. Eiz. 29 (1978) 845. [9] R. Barbieri, R. Gatto, R. K6gerler and Z. Kunszt, Phys. Lett. 57B (1975) 455. [10] J. Ellis, M.K. Gaillard, D.V. Nanopoulos and.C.T. Sachrajda, Phys. Lett. 83B (1979) 339, [11] S. Brodsky and G. Lepage, SLAC-PUB-2294 (1979).
162
PIIYSICS LETTERS
17 November 1980
STRONG INTERACTION CORRECTIONS TO SEMIWEAK DECAYS: CALCULATION OF THE V --' Hy DECAY RATE TO ORDERer s M.I. VYSOTSKY
Institute of Theoretical and Experimental Physics, Moscow, USSR Received 2 June 1980
The decay of heavy onium V ~ [t3, was proposed by Wilczek as a source of Higgs bosons. The calculation of one-loop gluon corrections to this process is presented.
The Higgs boson [ 1] is one of the m o s t intriguing particles of modern physics. It is indispensable for the presentday theory o f electroweak interactions, the Weinberg-Salam model [2]. Peculiar features of the H-particle - an elementary scalar field whose coupling to the matter fields is proportional to their masses - makes the investigation of the properties of the Higgs boson interesting both in theoretical and experimental respects. It is extremely difficult to find them because: (1) the Higgs couplings to particles which can be accelerated to produce H-bosons (e, /a, d and g) are rather small; (2) even if the Higgs particle was Produced it would be difficult to detect it; (3) Higgs bosons are supposed to be rather heavy: Mtt ~> 10 GeV [3]. Recently Wilczek [4] has suggested detecting the H-boson in the decays V -+ H3,, where V is a vector particle consisting of a QQ pair (Q is a heavy quark). This way avoids the first two difficulties listed above. The experimental signal of such a decay is rather clear: monochromatic photons associated with heavy particle decay products. The probability o f this decay increases quadratically with the V-meson mass and becomes equal to the V -+ e+e decay rate for M V = 110 GeV. The only problem is to find the appropriate V-meson. If the Higgs boson is lighter than the T-meson, it could be detected in T -+ H3' decay. It is supposed that the b-quark must have an SU(2)w partner - the t-quark. The following lower limit for Mvti-was obtained at PETRA: M V >~ 30 GeV [5]. The upper limit o f m t comes from investigation of the Higgs potential: m t ~< 100 GeV [6]. VtT with MVt T ~< 80 GeV may be discovered soon and the process e+e - -+ Vti- -+ H7 will work as a H-boson factory. The aim of this paper is to calculate the strong interaction radiative correction to the V -* H3' decay rate. Recent ly the one-loop c~s correction to the r~c --, gg decay rate has been computed and the coefficient of c~s turned out to be rather large [7]. This fact casts a shadow on the application of perturbation theory in a s to this decay. It was interesting for me to learn the impact of gluon corrections on the bare amplitude V -+ H3' fohnd in ref. [4]. My result is the following. The coefficient in front of c~s is rather large. In particular for T, ifM~t/M T2 <~0.8, the correction to F ( T -+ H3,)/F(T -+ e+e - ) amounts to 35% of the original result (I normalize as to its value in J / ~ decay: as(M 2) = 0.2). The correction for the toponium decay rate is smaller because of reduction of c~s and equals 25% for MVt i- = 80 GeV. The sign of the correction is negative, so it diminishes the decay rate. What we are going to calculate is the radiative correction to the bound-state decay rate. The relevant technique was developed in various details by investigators of the positronium system [8]. One must compute the one-loop correction to free e+e - pair annihilation at rest and then multiply the result by ~ (0). The Breit hamiltonian which describes the positronium dynamics produces an o~2 correction to ~ (0), and we neglect this since we are interested only in O(a) corrections. For the quark gluon perturbation theory the situation is just the same, but 4(0) is now mostly due to nonperturbative confinement effects. Hence a reliable computation of ~ (0) seems impossible. By considering the ratio F ( V -+ HT)/F(V -+ e+e - ) we avoid this problem since in this ratio t)(0) cancels out. Non159
Volume 97B, number 1
PHYSICS LETTERS
17 November 1980
universal corrections to A(V -, HT) due to confinement effects are supposed to be of order tl2/M2v where # ~ 250 MeV. Clearly, they are much less important numerically than 0 ( % ) corrections. The 0 ( % ) correction to the decay rate V -+ e+e was computed earlier [9]. Let us write out the expression for T M
FV~e+e - taking into account the ~s terms: [,V.,,e+e_ = 4 rro~Z(eQgV)2Mv [1 - 4CFO~s(M2v)/rr] •
(1)
The following definition o f g v is used: (01CyuC IV)gvM2euCV); eQ is the quark charge; C F = 4/3. To obtain the analogous correction to the decay rate V --* H')' we must compute the seven diagrams depicted in figs. l a - g as well as the seven diagrams with interchanged fermion legs. Let us discuss the computation of diagrams le and lf. Diagram le describes the Higgs-quark interaction dressed by the gluon. It contains In A where A is an ultraviolet cut-off. To understand this in detail let us consider diagrams 2 a - c , which describe scattering of a quark with momentum P and mass m on the Higgs boson with zero momentum transfer. The sum of diagrams 2 a - 2 c is equal to i 3E,/SPlf,= m + i 32;/3m [/;=m and contains In A because the scalar vertex is renormalized by vector interaction. For example, in dealing with the pion-nucleon interaction we must fix the interaction constant at some value of the external momenta and this gives us the rule for eliminating in A. Here we work in a strict WeinbergSalam model where the H~tq interaction is uniquely determined by the fact that the quark mass is given by the condensate value of the Higgs field. The corresponding term in the interaction lagrangian is the following: .C = rn ~ (1 + ~tt/r~), where ~l is the vacuum expectation value of the Higgs field. When the quark-gluon interaction is switched on we must add a quark mass renormalization counter-term to the lagrangian: /2 = [ m
iZ(m)]~(l+¢H/r/).
This counterterm makes the quark mass equal to its "observable" value and at the same time produces an extra term in the physical Higgs-quark interaction. We depict this term in fig. 2d. It is equal to - i )2(m)/rn. Writing out the divergent part of the quark mass operator ~div(P) = am In (A2/m 2) + b(/;
m) In (A2/m2),
we see that In A disappears in the sum of diagrams 2 a - 2 d . We hope that the reader is now convinced that the diagram of fig. l f m u s t be added to that of fig. le. Now let us dwell on the computation of diagram lg. The point is that there the well-known 1/u singularity appears when we compute this diagram (v is the quark-antiquark relative velocity). In the decay probability this term would result in the following contribution: CFoes rr/2v. But the nature of this term is, in fact, a Coulomblike interaction and it can be obtained by expanding I t~+(0)/@(0)l 2 = 2rr/k/(1
e -2w/k)
{a}
(d)
{b}
(e)
{c)
(fi
(gl!
Fig. l. The diagrams producing the O (a s) correction to the decay rate V ~ H% Permutations of the quark legs must also be taken into account. 160
Volume 97B, number 1
PttYSICS LETTERS
17 November 1980
I
_
_
±
-
-
I 1
!i. ×
(ai
/
,
15
(hi
I I I
~Z
~
10
i i
I l
I
,
x
¢c
~d
0
Fig. 2. Higgs quark interaction to first order in a s.
I 0,2
I 0.4
I 0.6
L 0.8
I~_ 1.0 K
Fig. 3. The function a (K).
in C~s(k = 2U/CFO~s). This means that it was already accounted for in the wave function and there is no need to account for the as~t) piece once more. Hence, the term ~ a s/u in the diagram of fig. lg must be subtracted. A final remark about the computation of the diagrams in fig. 1. Some diagrams contain an infrared divergency coming from the low gluon frequencies. To regularize it we give the guon a mass k. We are dealing with the decay of a colourless bound state o f nonrelativistic quarks, so in the sum of all graphs in k disappears. The decay probability is given by the expression:
PV-H-r a(K)=4
7r2 12(1 - K)
+ 4+
+I-K
~ - 1_ _ + 2 K ( K - - 2 ) F(1 - 2K) + __ 2(1 - / ~ ) 1 - 2K (l K)2
\ ~ !
arctg~l~-K!
+
[qb(K) + F ( 1 ) - F ( - 1 ) ]
1_~+2+il-2~
(3)
~
ln2(1-K),
2 v, 2 F(x) = f ~ y - a In (1 + y ) dy is the Spence function, where K = MH/M 1
q~(K) =
dy
In [1 - 4 y ( l ~ y ) K ]
0 q)(K) can be expressed as the sum of two Spence functions of complex arguments, a is regular at K = 1/2. a(0) = 7 + 6 in 2 - 7r2/8. a(K)l~:~l = 47r/3 (1 - t<)1/2. The a(K) dependence is presented in fig. 3. Formula (2) works when M V - M H ~ a2M v . I f M V - M H ~ c~2Mv, the Higgs mass is close to the masses of quarkonium bound states and the V ~ HI' decay must be studied in another way (in ref. [10] the case of the H-mass close to the levels o f the 1'system is discussed). To obtain the value of the ratio r ' ( v ~ H1')/I'(V ~ e+e - ) eq. (2) must be divided by eq. (1) and this slightly diminishes the magnitude o f the correction. Some numerical estimates were presented in the beginning o f the paper. The limit ~ -+ oo in expression (3) is very interesting. In this case we are dealing with the decay of a heavy scalar particle into a bound QQ state and a photon. This is an exclusive process. The description of exclusive processes within QCD is the subject of intensive study nowadays, a = 2 In 2 In (Q2/m2Q) in this limit where Q2 substitutes M 2. The (c~s In Q2)n summation in this process can be performed with the help o f OPE analogously to ref. [1 1]. The decay H ~ (QQ)v1' with the subsequent decay V -+ e+e - from the experimental point of view presents the clearest signal in searching for the H-boson. Unfortunately, the ratio o f this decay rate to the decay rate H -+ Q 0 161
Volume 97B, number 1
PHYSICS LETTERS
17 November 1980
is very small: F(H -+ VT)/P (H -+ QQ) = 0.S (gv eQ )2 (mQ/MH)2 , where (g# ec)2 = 6.5 X 10 - 3 , (gqceb) 2 = 6.2 X 10 - 4 . The author is indebted to L.B. Okun, M.I. Polikarpov, M.A. Shifman and M.B. Voloshin for useful discussions. 1 wish to thank M.A. Shifman for improving my bad English.
References [1] P.W. Higgs, Phys. Lett. 12 (1964) 132. [2] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, Proc. 8th Nobel Syrup. (Stockholm, 1968) ed. N. Svarthohn (Almquist and Wiksell, Stockholm, 1968) p. 367. [3] A.D. Linde, Pis'ma Zh. Eksp. Teor. Fiz. 23 (1976) 73; S. Weinberg, Phys. Rev. Lett. 36 (1976) 294. [4] F. Wilczek, Phys. Rev. Lett. 39 (1977) 1304. [5] W. Barrel et al., DESY 80/04 (1980). [6] A.A. Anselm, Pis'ma Zh. Eksp. Teor. Fiz 29 (1979) 645; ll.D. Politzer and S. Wolfram, Phys. Lett. 82B (1979) 242; Erratum 83B (1979) 421. [7] R. Barbieri, G. Curci, E. d'Emilio and E. Remiddi, Nucl. Phys. B154 (1979) 535. [8] W.E. Caswell, G.P. Lepage and J. Sapirstein, Phys. Rev. Lett. 38 (1977) 488; M.I. Vysotsky, Yad. Eiz. 29 (1978) 845. [9] R. Barbieri, R. Gatto, R. K6gerler and Z. Kunszt, Phys. Lett. 57B (1975) 455. [10] J. Ellis, M.K. Gaillard, D.V. Nanopoulos and.C.T. Sachrajda, Phys. Lett. 83B (1979) 339, [11] S. Brodsky and G. Lepage, SLAC-PUB-2294 (1979).
162