Changes between Version 2 and Version 3 of HiggsEffective


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Timestamp:
04/04/12 14:28:19 (8 years ago)
Author:
trac
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  • HiggsEffective

    v2 v3  
     1= Higgs Effective couplings to gluons (and photons) =
    12
     3The Higgs effective field theory ({{{
     4heft
     5}}}) model is an 'extension' of the Standard Model, where the Higgs boson couples directly to gluons (and photons). In the SM these couplings are present through a heavy (top) quark loop. For a not too heavy Higgs (m,,h,,<2 m,,t,,), it is a good approximation to take the mass of the heavy quark in the loop to infinity (For this approximation to hold, not only should the Higgs mass be smaller than twice the top mass, also all other kinematic variables, such as the transverse momentum of the Higgs boson, should be smaller than 2m,,t,,.) This results in effective couplings between gluons and Higgs bosons.
    26
    3 == Higgs Effective couplings to gluons (and photons) ==
     7The effective vertices can be derived from the effective dimension five Lagrangian:
     8{{{
     9#!latex
     10$\mathcal{L}_{h}=-\frac{1}{4}g_hG_{\mu\nu}^aG_{\mu\nu}^a \Phi,$
     11\texttt{{\small where}} $G^a_{\mu\nu}=\partial_{\mu}A^a_{\nu}-\partial_{\nu}A^a_{\mu}+gf^{abc}A^b_{\mu}A^c_{\nu}$.
     12}}}
     13The coupling constant g,,h,, is given by:
     14{{{
     15#!latex
     16$g_h=\frac{\alpha_s}{3\pi v}\Big(1+ \frac{7}{30}\tau + \frac{2}{21}\tau^2+ \frac{26}{525}\tau^3\Big),$ \texttt{{\small with}} $\tau=m_h^2/(4m_t^2)$
     17}}}
     18and higher orders in τ have been neglected. Due to the non-abelian nature of the SU(3),,C,, color group the effective vertices do not only include two, but also three and four gluons coupling to the Higgs boson. Since !MadGraph can work only with three- and four-point vertices, the four-gluon interactions in the {{{
     19heft
     20}}} model are obtained by rewriting the QCD four-gluon interaction in terms of three-point vertices with an extra non-propagating internal tensor particle, T. This trick can be easily understood by noting that the usual (text-book) form of the four-gluon interaction is the sum of three terms, whose color and Lorentz structure correspond to 2 → 2 diagrams where a color octet tensor is exchanged in the s, t, u channels. With the introduction of this extra particle, the four-gluon-Higgs vertices can be reduced to diagrams with at most four-point vertices. To get the standard diagrammatic visualization of four-gluon and four-gluon-Higgs vertices it is sufficient to contract the T particle lines to a single point.
    421
    5 The Higgs effective field theory ( {{{
    6 heft
    7 }}} ) model is an `extension'
    8 of the Standard Model, where the Higgs boson couples directly to
    9 gluons (and photons). 
    10 In the SM these couplings are present through a
    11 heavy (top) quark loop. For a not too heavy Higgs ($m_h<2m_t$), it is
    12 a good approximation to take the mass of the heavy quark in the loop
    13 to infinity (For this approximation to hold, not only should the Higgs mass
    14 be smaller than twice the top mass, also all other kinematic variables,
    15 such as the transverse momentum of the Higgs boson, should be smaller than
    16 %$2m_t$.)  This results in effective couplings between gluons and
    17 Higgs bosons.
     22The gluon couplings to a pseudo-scalar Higgs are also implemented. The name of the pseudo-scalar Higgs in !MadGraph is {{{
     23h3
     24}}} ( i.e. , the same as in the 2HDM and MSSM models). The effective dimension five Lagrangian for the pseudo-scalar Higgs coupling to the gluons is
     25{{{
     26#!latex
     27$\mathcal{L}_{A}=\frac{1}{2}g_AG_{\mu\nu}^a\tilde{G}_{\mu\nu}^a \Phi_A,$ \texttt{{\small where}} $\tilde{G}_{\mu\nu}^a$ \texttt{{\small is the dual of}} $G_{\mu\nu}^a$, $\tilde{G}_{\mu\nu}^a=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}G_{\rho\sigma}^a$.
     28}}}
     29The effective coupling constant g,,A,, is given by
     30{{{
     31#!latex
     32$g_A=\frac{\alpha_s}{2\pi v}\Big(1+ \frac{1}{3}\tau+ \frac{8}{45}\tau^2 + \frac{4}{35}\tau^3\Big)$,
     33}}}
     34where the higher orders in τ have been neglected. The pseudo-scalar Higgs has only effective couplings to two or three gluons. The four-gluon-pseudo-scalar Higgs vertex is absent due to the anti-symmetry of the epsilon tensor ε^μνρσ^. If a mixed Higgs with no definite CP parity is needed, it sufficient to change the couplings of the Higgs to the gluons. First generate the process with the SM Higgs, then, after downloading the code, change the coupling in the {{{
     35./Source/Model/couplings.f
     36}}} file. The coupling constant is defined as a two-dimensional object, where the first and second elements are the CP-even and CP-odd couplings of the Higgs to the gluons, respectively. The HELAS subroutines automatically use the correct kinematics for odd-, even- or mixed CP Higgs's coupling to the gluons. At present, the implementation allows production of only one Higgs-boson. The effective couplings of two Higgs bosons to gluons are available in HELAS, but not yet included in the HEFT model.
    1837
    19 The effective vertices can be derived from the effective dimension five Lagrangian
     38== The non-propagating auxiliary particle T ==
    2039
    21 where $G^a_{\mu\nu}=\partial_{\mu}A^a_{\nu}-\partial_{\nu}A^a_{\mu}+gf^{abc}A^b_{\mu}A^c_{\nu}$. The coupling constant $g_h$ is given by
    22 %\[  g_h=\frac{\alpha_s}{3\pi v}\Big(1+ \frac{7}{30}\tau + \frac{2}{21}\tau^2+ \frac{26}{525}\tau^3\Big),\]% with $\tau=m_h^2/(4m_t^2)$ and higher orders in $\tau$ have been neglected.
    23 Due to the non-abelian nature of the $SU(3)_C$ color group the
    24 effective vertices do not only include two, but also three and four
    25 gluons coupling to the Higgs boson. Since MadGraph can work only with
    26 three- and four-point vertices, the four-gluon interactions in the
    27 {{{
    28 heft
    29 }}} model are obtained by rewriting the QCD four-gluon
    30 interaction in terms of three-point vertices with an extra
    31 ''non-propagating'' internal tensor particle,
    32 %$T$. This trick can be easily
    33 understood by noting that the usual (text-book) form of the four-gluon
    34 interaction is the sum of three terms, whose color and Lorentz
    35 structure correspond to $2 \to 2 $ diagrams where a color octet tensor
    36 is exchanged in the $s,t,u$ channels.  With the introduction of this
    37 extra particle, the four-gluon-Higgs vertices can be reduced to
    38 diagrams with at most four-point vertices. To get the standard
    39 diagrammatic visualization of four-gluon and four-gluon-Higgs
    40 vertices it is sufficient to contract the $T$ particle lines to
    41 a single point.
    42 
    43 The gluon couplings to a pseudo-scalar Higgs are also implemented. The
    44 name of the pseudo-scalar Higgs in MadGraph is {{{
    45 h3
    46 }}} ( ''i.e.'' ,
    47 the same as in the 2HDM and MSSM models). The effective dimension five
    48 Lagrangian for the pseudo-scalar Higgs coupling to the gluons is
    49 
    50 where $\tilde{G}''{\mu\nu}^a$ is the dual of $G''{\mu\nu}^a$, $\tilde{G}''{\mu\nu}^a=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}G''{\rho\sigma}^a$.
    51 The effective coupling constant $g_A$ is given by
    52 %\[  g_A=\frac{\alpha_s}{2\pi v}\Big(1+ \frac{1}{3}\tau+ \frac{8}{45}\tau^2 + \frac{4}{35}\tau^3\Big),\]% where the higher orders in $\tau$ have been neglected.
    53 The pseudo--scalar Higgs has only effective couplings to two or three
    54 gluons. The four-gluon-pseudo-scalar Higgs vertex is absent due to
    55 the anti-symmetry of the epsilon tensor $\epsilon^{\mu\nu\rho\sigma}$.
    56 If a mixed Higgs with no definite CP parity is needed,
    57 it sufficient  to change the couplings of
    58 the Higgs to the gluons. First  generate the process with
    59 the SM Higgs, then, after downloading the code,
    60 change the coupling in the =./Source/Model/couplings.f= file. The
    61 coupling constant is defined as a two-dimensional object, where the
    62 first and second elements are the CP-even and CP-odd couplings
    63 of the Higgs to the gluons, respectively. The
    64 HELAS subroutines automatically use the correct kinematics for
    65 odd-, even- or mixed CP Higgs's coupling to the gluons.
    66 At present, the implementation allows production of only one
    67 Higgs-boson.  The effective couplings of two Higgs bosons to gluons are
    68 available in HELAS, but not yet included in the HEFT model.
    69 
    70 
    71 === The non-propagating auxiliary particle {{{
    72 T
    73 }}} ===
    74 To describe the four-gluon-Higgs coupling a vertex with 5 external lines is needed. This cannot be done with MadGraph. However, there is a way to circumvent this problem by introducing non-propagating auxiliary particles.
    75 It is possible to rewrite the four-gluon interactions as two three-point interactions connected by the new auxiliary tensor particle (called {{{
    76 tn
    77 }}} internally, and shows as {{{
    78 T
    79 }}} in the MG diagrams, the PDG code is 99). The troublesome five-point interaction (between the four gluons and the Higgs) reduces then to three three-point interactions.
    80 
     40To describe the four-gluon-Higgs coupling a vertex with 5 external lines is needed. This cannot be done with !MadGraph. However, there is a way to circumvent this problem by introducing non-propagating auxiliary particles. It is possible to rewrite the four-gluon interactions as two three-point interactions connected by the new auxiliary tensor particle (called tn internally, and shows as T in the MG diagrams, the PDG code is 99). The troublesome five-point interaction (between the four gluons and the Higgs) reduces then to three three-point interactions.
    8141
    8242Notice that the color part of the four-gluon vertex is exactly like the sum of the s-, t-, and u-channel exchange diagram, where the exchanged particle is in the adjoint representation of color, i.e. an octet.
    8343
    8444Second, notice that the 'Feynman part' (i.e. the part with the metric tensors) is almost (except for a factor of two) an projection operator
    85 
    86 This means that we have the following new Feynman rules.
    87 
    88 The gluon-gluon-tensor vertex is given by
    89 
    90 and the tensor propagator is
    91 
     45{{{
     46#!latex
     47$(g^{\mu\rho}g^{\nu\sigma}-g^{\mu\sigma}g^{\nu\rho})(g^{\rho\tau}g^{\sigma o}-g^{\rho o}g^{\sigma\tau})=2(g^{\mu\tau}g^{\nu o}-g^{\mu o}g^{\nu\tau}).$
     48}}}
     49This means that we have the following new Feynman rules.
     50The gluon-gluon-tensor vertex is given by
     51{{{
     52#!latex
     53$V_2^T=gf^{abc}(g^{\mu\rho}g^{\nu\sigma}-g^{\mu\sigma}g^{\nu\rho})/\sqrt{2}$
     54}}}
     55and the tensor propagator is
     56{{{
     57#!latex
     58$\Delta^{\mu\nu,\rho\sigma}_{ab}=-i g^{\mu\rho}g^{\nu\sigma} \delta_{ab}.$
     59}}}
    9260Notice that the new particle is a rank-2 tensor (hence the name 'tensor'). It does not propagate, because there is no momentum dependence in the propagator. And it only exist as an internal particle, it can never be seen as an external one.
    9361
    9462With these new rules we can make six tree-level diagrams with four external gluon lines: three with two gluon-gluon-gluon vertices and three with two gluon-gluon-tensor vertices. The sum of these diagrams is exactly the same as the four diagrams in the original QCD: the three diagrams with two gluon-gluon-gluon vertices have not changed, and the four-gluon interaction is equal to the sum of the three diagrams with the two gluon-gluon-tensor vertices.
    9563
    96 The only thing needed to implement the four-gloun-Higgs coupling is a coupling between the tensor particle and the Higgs boson. A short derivations leads to the following tensor-tensor-Higgs vertex
     64The only thing needed to implement the four-gloun-Higgs coupling is a coupling between the tensor particle and the Higgs boson. A short derivations leads to the following tensor-tensor-Higgs vertex
     65{{{
     66#!latex
     67$V^{TTH}= ig_H g^{\mu\rho}g^{\nu\sigma} \delta^{ab}.$
     68}}}
    9769
    9870
     71== Higgs coupling to photons ==
     72
     73The coupling between the scalar Higgs to photons is mediated by a top quark loop and by a W boson loop. In the limit of small Higgs masses (lower than approx 150 GeV), the loop induced interaction can approximately described by the Lagrangian
     74{{{
     75#!latex
     76$$L_{\textrm{heft}}=-\frac{1}{4}gF_{\mu\nu} F_{\mu\nu} H.$$
     77}}}
     78Due to the abelian nature of QED there is only one effective vertex between photons and Higgs bosons. The value for the coupling constant in the {{{
     79heft
     80}}} model is given by
     81{{{
     82#!latex
     83$$g=-\frac{\alpha}{\pi v} \frac{47}{18}\Big( 1+ \frac{66}{235}\tau_w +\frac{228 }{1645}\tau_w^2+\frac{696}{8225}\tau_w^3+\frac{5248}{90475} \tau_w^4+\frac{1280}{29939}\tau_w^5+\frac{54528}{1646645}\tau_w^6-\frac{56}{705} \tau_t-\frac{32}{987}\tau_t^2\Big),$$
     84}}}
     85where
     86{{{
     87#!latex
     88$\tau_t=m_h^2/(4m_t^2)$ \texttt{{\small and}} $\tau_w=m_h^2/(4m_W^2)$.
     89}}}
     90Higher order in τ,,t,, and τ,,w,, have been neglected.
    9991
    10092
    101 === Higgs coupling to photons ===
    102 The coupling between the scalar Higgs to photons is mediated by a top quark loop and by a W boson loop. In the limit of small Higgs masses (lower than approx 150 GeV), the loop induced interaction can approximately described by the Lagrangian %\[L_{\textrm{heft}}=-\frac{1}{4}gF_{\mu\nu} F_{\mu\nu} H.\]%
    103 Due to the abelian nature of QED there is only one effective vertex between photons and Higgs bosons.
    104 The value for the coupling constant in the {{{
    105 heft
    106 }}} model is given by
    107 %\[g=-\frac{\alpha}{\pi v} \frac{47}{18}\Big( 1+ \frac{66}{235}\tau_w +\frac{228 }{1645}\tau_w^2+\frac{696}{8225}\tau_w^3+\frac{5248}{90475} \tau_w^4+\frac{1280}{29939}\tau_w^5+\frac{54528}{1646645}\tau_w^6-\frac{56}{705} \tau_t-\frac{32}{987}\tau_t^2\Big),\]% where $\tau_t=m_h^2/(4m_t^2)$ and $\tau_w=m_h^2/(4m_W^2)$. Higher order in $\tau_t$ and $\tau_w$ have been neglected.
    108 
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    111 -- Main.RikkertFrederix - 25 Oct 2007
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     93-- Main.!RikkertFrederix - 25 Oct 2007