Changes between Version 1 and Version 2 of HiggsEffective


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Timestamp:
Apr 4, 2012, 2:28:08 PM (13 years ago)
Author:
trac
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  • HiggsEffective

    v1 v2  
    1 = Higgs Effective couplings to gluons (and photons) =
    21
    3 The Higgs effective field theory ({{{
     2
     3== Higgs Effective couplings to gluons (and photons) ==
     4
     5The Higgs effective field theory ( {{{
    46heft
    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.
     7}}} ) model is an `extension'
     8of the Standard Model, where the Higgs boson couples directly to
     9gluons (and photons). 
     10In the SM these couplings are present through a
     11heavy (top) quark loop. For a not too heavy Higgs ($m_h<2m_t$), it is
     12a good approximation to take the mass of the heavy quark in the loop
     13to infinity (For this approximation to hold, not only should the Higgs mass
     14be smaller than twice the top mass, also all other kinematic variables,
     15such as the transverse momentum of the Higgs boson, should be smaller than
     16%$2m_t$.)  This results in effective couplings between gluons and
     17Higgs bosons.
    618
    7 The effective vertices can be derived from the effective dimension five Lagrangian:
     19The effective vertices can be derived from the effective dimension five Lagrangian
     20
     21where $G^a_{\mu\nu}=\partial_{\mu}A^a_{\nu}-\partial_{\nu}A^a_{\mu}+gf^{abc}A^b_{\mu}A^c_{\nu}$. 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.
     23Due to the non-abelian nature of the $SU(3)_C$ color group the
     24effective vertices do not only include two, but also three and four
     25gluons coupling to the Higgs boson. Since MadGraph can work only with
     26three- and four-point vertices, the four-gluon interactions in the
    827{{{
    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 }}}
    13 The 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 }}}
    18 and 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 {{{
    1928heft
    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.
     29}}} model are obtained by rewriting the QCD four-gluon
     30interaction in terms of three-point vertices with an extra
     31''non-propagating'' internal tensor particle,
     32%$T$. This trick can be easily
     33understood by noting that the usual (text-book) form of the four-gluon
     34interaction is the sum of three terms, whose color and Lorentz
     35structure correspond to $2 \to 2 $ diagrams where a color octet tensor
     36is exchanged in the $s,t,u$ channels.  With the introduction of this
     37extra particle, the four-gluon-Higgs vertices can be reduced to
     38diagrams with at most four-point vertices. To get the standard
     39diagrammatic visualization of four-gluon and four-gluon-Higgs
     40vertices it is sufficient to contract the $T$ particle lines to
     41a single point.
    2142
    22 The gluon couplings to a pseudo-scalar Higgs are also implemented. The name of the pseudo-scalar Higgs in !MadGraph is {{{
     43The gluon couplings to a pseudo-scalar Higgs are also implemented. The
     44name of the pseudo-scalar Higgs in MadGraph is {{{
    2345h3
    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 }}}
    29 The 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 }}}
    34 where 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.
     46}}} ( ''i.e.'' ,
     47the same as in the 2HDM and MSSM models). The effective dimension five
     48Lagrangian for the pseudo-scalar Higgs coupling to the gluons is
    3749
    38 == The non-propagating auxiliary particle T ==
     50where $\tilde{G}''{\mu\nu}^a$ is the dual of $G''{\mu\nu}^a$, $\tilde{G}''{\mu\nu}^a=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}G''{\rho\sigma}^a$.
     51The 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.
     53The pseudo--scalar Higgs has only effective couplings to two or three
     54gluons. The four-gluon-pseudo-scalar Higgs vertex is absent due to
     55the anti-symmetry of the epsilon tensor $\epsilon^{\mu\nu\rho\sigma}$.
     56If a mixed Higgs with no definite CP parity is needed,
     57it sufficient  to change the couplings of
     58the Higgs to the gluons. First  generate the process with
     59the SM Higgs, then, after downloading the code,
     60change the coupling in the =./Source/Model/couplings.f= file. The
     61coupling constant is defined as a two-dimensional object, where the
     62first and second elements are the CP-even and CP-odd couplings
     63of the Higgs to the gluons, respectively. The
     64HELAS subroutines automatically use the correct kinematics for
     65odd-, even- or mixed CP Higgs's coupling to the gluons.
     66At present, the implementation allows production of only one
     67Higgs-boson.  The effective couplings of two Higgs bosons to gluons are
     68available in HELAS, but not yet included in the HEFT model.
    3969
    40 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. 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.
     70
     71=== The non-propagating auxiliary particle {{{
     72T
     73}}} ===
     74To 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.
     75It is possible to rewrite the four-gluon interactions as two three-point interactions connected by the new auxiliary tensor particle (called {{{
     76tn
     77}}} internally, and shows as {{{
     78T
     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
    4181
    4282Notice 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.
    4383
    4484Second, notice that the 'Feynman part' (i.e. the part with the metric tensors) is almost (except for a factor of two) an projection operator
    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 }}}
    49 This means that we have the following new Feynman rules.
    50 The 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 }}}
    55 and the tensor propagator is
    56 {{{
    57 #!latex
    58 $\Delta^{\mu\nu,\rho\sigma}_{ab}=-i g^{\mu\rho}g^{\nu\sigma} \delta_{ab}.$
    59 }}}
     85
     86This means that we have the following new Feynman rules.
     87
     88The gluon-gluon-tensor vertex is given by
     89
     90and the tensor propagator is
     91
    6092Notice 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.
    6193
    6294With 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.
    6395
    64 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
    65 {{{
    66 #!latex
    67 $V^{TTH}= ig_H g^{\mu\rho}g^{\nu\sigma} \delta^{ab}.$
    68 }}}
     96The 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
    6997
    7098
    71 == Higgs coupling to photons ==
    72 
    73 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
    74 {{{
    75 #!latex
    76 $$L_{\textrm{heft}}=-\frac{1}{4}gF_{\mu\nu} F_{\mu\nu} H.$$
    77 }}}
    78 Due 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 {{{
    79 heft
    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 }}}
    85 where
    86 {{{
    87 #!latex
    88 $\tau_t=m_h^2/(4m_t^2)$ \texttt{{\small and}} $\tau_w=m_h^2/(4m_W^2)$.
    89 }}}
    90 Higher order in τ,,t,, and τ,,w,, have been neglected.
    9199
    92100
    93 -- Main.!RikkertFrederix - 25 Oct 2007
     101=== Higgs coupling to photons ===
     102The 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 %\[L_{\textrm{heft}}=-\frac{1}{4}gF_{\mu\nu} F_{\mu\nu} H.\]%
     103Due to the abelian nature of QED there is only one effective vertex between photons and Higgs bosons.
     104The value for the coupling constant in the {{{
     105heft
     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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