Decays of the Higgs Boson

Transcription

1 Decays of te Higgs Boson Daniel Odell April 8, 5 We will calculate various decay modes of te Higgs boson. We start wit te decay of te Higgs to a fermion-antifermion pair. f f Figure : f f First, we ll calculate te decay rate for a single particle to two particles in general as it will prove useful in later parts. In te center of mass frame k i = (m i,,, ) k = (E,,, p) k = (E,,, p) and te decay rate is d 3 k d 3 k Γ = (π) (π) (π) δ (k i k k ) A m i E E

2 Te delta function can be expressed as δ(m i E E )δ 3 ( k + k ) Te spatial part allows us to perform te integral over k. Leaving Γ = π d 3 k m i (π) 3 δ(m i E E ) A E E We now ave δ(f(k )). Using te appropriate properties of te delta function, and canging from d 3 k to k dk dω, we ave Γ = 8πm k A i were k is te root of f(k), or E m, and A is independent of te angles wen te incoming particle is a scalar. In te case were te two outgoing particles ave identical mass k = m i m Now, all we need is te amplitude for te process of te Higgs decaying to a fermion-antifermion pair. At leading order, we ave only one vertex, as sown in Figure. Tis gives us ( ) imf ia = ū (p )v(p ) v were te term in parentesis is te vertex factor and te v in te denominator is te ground state expectation value of φ, m W sin(θ w) e. Now we ave ( ) imf e ia = ū (p )v(p ) m W sin(θ w ) were Taking te square amplitude and summing over te spins, we get A = m f e m W sin θ w tr [(p γ + m f )(p γ m f )] u s (p)ū s (p) = p γ + m s v s (p) v s (p) = p γ m s Tere are four terms inside te trace argument. Te two tat ave only one γ are zero because te γ matrices are traceless. Tis leaves tr [ (p γ)(p γ) m ] f

3 Let s look at te first term inside te argument. We ave tr(p µ γ µ p ν γ ν ) But leaving or So, we ave or A = tr(γ µ γ ν ) = η µν η µν p µ p ν p p m f e m W sin θ w tr[p p m f ] A m f = e m W sin θ w ( ) m m f were tr() = and p p = m + p = m + m m f (in te center-of-mass frame). Inserting tis expression into te decay rate, we get m Γ = m f m ( ) f e m 8π m m W sin θ w m f Simplifying, we arrive at Γ( f f) = Or in terms of te fine structure constant Γ( f f) = W + W ( ) e m 3/ f 3π sin θ w m m f W m α 8π sin θ w m f m W ( m f m ) 3/ Next, we ll consider te decay of te Higgs to a W + boson and a W boson. Again, we ave only one vertex. Using te Feynman rule, we ave ia = im W e η µν ɛ sinθ µ(p)ɛ ν(k) w Now, we ave to take te square magnitude and sum over te polarizations. Tis gives ( A = m W e sin η µν + p ) µp ν ( η µν θ w m + kµ k ν ) W MW 3

4 Figure : W + W were summing over te polarizations gives ɛ µ (p)ɛ ν(p) = η µν + p µp ν m W Continuing, we get A = m W e ( p sin θ w = m W ( e sin + θ w m W k m W ) (p k) m W + ) (p k) m W were p = k = m W. If we expand p k were p = ( m, p ), k = ( m, p ), and p = m m W, we get A = e m ( sin θ w m m W W m If we use tis in our decay rate equation, ) + m W m Γ( W + W ) = ( ) m / 8πm e m ( m W sin θ w m m W W m αm 3 = 6πsin θ w m W 3 Z Z ( m W m ) ( ) / + m W m m W m ) + m W m Because te vertex factor for te Z Z only differs by te mass, we can swap m Z for m W in te terms wit parentesis to get te rate for te decay to

5 Figure 3: Z Z Z Z. We also need to divide by because te Z bosons are indistinguisible. After tat, we ave Γ( Z Z αm 3 ( ) ( ) / ) = 3πsin θ w m m Z W m + m Z m m Z m gg Te decay of te Higgs to a pair of gluons is a little more complicated. Because tere is no direct interaction between te Higgs and te gluons, we ave to consider a single quark loop. Figure : gg Te fermion-gluon vertex is igγ µ t a 5

6 and te fermion propagator is So te amplitude looks like ia = im q d q i(p + q) γ + m q v (π) (p + q) m q ip γ + m p m (igγ µ t a ) iq γ + m q q m q (igγ ν t b ) i(q k) γ + m q (q k) m ɛ µ(p)ɛ ν(k) q We ll simplify te numerator for now and express te denominator in terms of Feynman parameters. ia = im qg d q v (π) N µν (p + q) m q q m q (q k) m q were N µν is as were T r [ (i(p + q) γ + m q )γ µ t a (iq γ + m q )γ ν t b (i(q k) γ + m q ) ] Using te Feynman parameters, we can rewrite everyting else in te integral x (q ) 3 q = q + xp yk = m q xym keeping in mind tat p = ( m, p), k = ( m, p), and p = k =. So, p k = m + p and p = m. Now, we can evaluate N µν and express it in terms of q. First, we can split it up into two traces, N µν = T r [(i(p + q) γ + m q )γ µ (iq γ + m q )γ ν (i(q k) γ + m q )] T r[t a t b ] leaving only T r[t a t b ] = δab T r [(i(p + q) γ + m q )γ µ (iq γ + m q )γ ν (i(q k) γ + m q )] to be evaluated. We expand te argument of te trace wic gives us 8 terms. We discard terms wit an odd number of γ matrices as te trace of tose terms is zero. Using te following two identities, T r[γ µ γ ν γ ρ γ σ ] = (η µν η ρσ η µρ η νσ + η µσ η νρ ) 6

7 we can simplify N µν to T r[γ µ γ ν ] = η µν m [ p ν k µ p µ k ν + p µ q ν k ν q µ + q µ q ν + (m q p k q )η µν] Wen multiplied by ɛ µ(p)ɛ ν(k), terms wit p µ and k ν vanis because gluons ave transverse polarization. We can also ignore terms linear in q as tey vanis wen te integration is performed. We will use dimensional regularization to evaluate te integral, so q µ q ν d q η µν wic simplifies te expression to ( N µν ɛ µ(p)ɛ ν(k) = m q [m q + xy ) ( ) ] m + d q ɛ (p) ɛ (k) Now, our amplitude looks like ia = im qg v x dy d d q [ m q + ( ) xy m + ( d ) q ] (π) (q ) 3 We ll perform te momentum integration in two steps. First, d d ( ) q (π) d q ( ) (q ) 3 = d d Γ(3 d ) ( ) 3 d (π) d/ Γ(3) If we consider d = ɛ, we get (π) ɛ Γ(ɛ) Γ(3) ( ) ɛ π Expanding Γ(ɛ) as Γ(ɛ) = ɛ γ + O(ɛ) and taking ɛ, we re left wit were Next, 3π d d q C (π) (q ) 3 = C Γ(3 d ) ( ) 3 d (π) d/ Γ(3) ( C = m q + xy ) m 7

8 Te rigt and side converges for d =, so we ave C 3π Combining te two results back into te integrals over te Feynman parameters, ia = i α sm x πv δab ɛ (p) ɛ ( xy) (k) dy xyτ q were and = i α sm 6πv δab ɛ (p) ɛ (k)i f (m /m q) I f (m /m q) = 3 τ q = m m q x dy ( xy)m m q xym If we square te amplitude and sum over polarizations and color, we get A = α sm 9π v I f (m /m q) Using our decay rate equation from earlier and summing over te quark flavors, ( ) Γ( αm m gg) = α s 8sin θ w 9π I f (m /m q) 5 gg Te cross section is found similarly to te total decay rate, we multiply by ( 8 to average over te initial spins and colors of gluons, and divide by te relative velocity of te incoming particles. σ = d 3 q 56 v p v k (π) 3 (π) δ (p + k q) A E q E p E k If we cange te integration measure to d q, te integral can easily be done (wit te valid restriction tat E q >. σ = d q(π)δ(q m 56 v p v k )(π) δ (p + k q) A E p E k m W Using te second delta function to perform te integration, σ = π 5 E p E k v p v k δ((p + k) m ) A 8 q )

9 We now turn to te parton model. We calculate te cross section for protonproton scattering as follows σ(p(p ) + p(p ) + X) = f g (x )f g (x )σ were f g (x) is te gluon distribution function and x and x are te fraction of te proton s momenta tat te gluons carry. In order to evaluate tis expression, we must express σ in terms of x and P (te proton s momentum). Figure 5: Gluon fusion in te parton model. If we consider te scattering of two ig energy protons (were we can neglect teir masses), we ave P = (E,,, E) P = (E,,, E) in te center-of-mass frame. We parameterize te gluons momenta as p = x P k = x P Now, to express σ in terms of x s and P s, we ll look at (p + k), wic becomes (p + k) = (x P + x P ) = x P + x x P P + x P = x x E Te energies of te gluons are similarly parameterized as E p = x E 9

10 E k = x E giving te following expressions for te velocities Now, σ becomes v p = p E p = x E x E = v k = k E k = x E x E = σ = π x x E δ(x x E m ) A We are asked to find a relation between Γ and te parton model cross section, so we ll express A in terms of Γ, Now, we ave A = m 8πΓ σ = π x x E δ(x x E m )m 8πΓ = π 6 x x E δ(x x E m )m Γ Inserting tis into te cross section for proton-proton collisions, σ(p(p )+p(p ) +X) = π m Γ 6E f g (x )f g (x ) δ(x x E m x x ) We ll use te properties of te delta function to make tings a little more manageable. δ(x x E m ) = δ(g(x )) g(x ) as a root at χ x were χ = ( m E ) and g (χ) = x E. So, we ave σ(p(p )+p(p ) +X) = π m Γ 6E χ δ(x χ x f g (x )f g (x ) ) x x x E were we ve canged te lower limit of x. Initially, we were integrating over a square [, ] in bot x and x. But evaluating x at χ x, and x being limited to, we lose everyting less te χ on te x axis. Te gluon distribution function is given in te problem as f g (x) = 8( x)7 x

11 wic gives σ(p(p )+p(p ) +X) = π m Γ E = π m Γ E χ χ ( x ) 7 ( x ) 7 δ(x χ x ) x x x x x E ( x ) 7 ( χ x ) 7 χ x x Now, we can insert te expression for Γ, and plot te cross section as a function of te center-of-mass energy. Figure 6: Gluon fusion cross section as a function of te center-of-mass energy 6 γγ (via quark/lepton loop) It s fairly simple to arrive at te γγ decay rate from te gg rate we ve already calculated. Figure represents te same process. We only ave to include a factor for te electric carges of te fermions, generalize te color factor to N c (f), swap e wit g s, and sum over all carged fermions. ( ) Γ( αm m γγ) = 8sin θ w m W α 8π f Q f N c (f)i f (m /m f ) 7 γγ (via W boson loop) Tere are only ( if you count te crossed potons separately) diagrams for tis process tat only involve W bosons. Unfortunately, to obtain a gauge-invariant result, one must consider te contributions from te Goldstone bosons, te

12 gosts, and te various combinations between temselves and te W bosons. In all, tere are 3 (6, again wen counting crossed potons separately). Many of tem are similar to diagrams calculated earlier, only wit different vertex factors and propagators. I m going to coose one tat is topologically different and rely on te work of my sources for te rest. Figure 7: γγ wit point vertex Tere are two vertices in tis diagram. Te first one connecting te Higgs boson to te W boson loop gives im W gη λµ Te second is a four point vertex, were two W bosons anniilate to produce two potons. It gives ie (η νρ η µλ η µρ η νλ η µν η λρ ) Including te two W boson propagators, te amplitude looks like ia = m W ge η λµ (η νρ η µλ η µρ η νλ η µν η λρ )ɛ ρ(p)ɛ ν(k) d d q (π) d We can use Feynman parameters to rewrite te integrand as i i q m W (p + k q) m W q m W (p + k q) m W = [x(( q) m W ) + ( x)(q m )] were = p + k. We can rearrange and write it as [q ]

13 were q = q x and = x(x ) + m W ( = m in te center-of-mass frame). Now, we can perform te integration over q, leaving ia = 6im W ge ɛ (p) ɛ Γ( d (k) ) (π) d/ Γ() [ m x(x ) + m W ] d Te biggest problem is tat te Γ function can not be evaluated for zero (or wen d =, as is te case ere). Fortunately, tere are twelve oter diagrams to consider. Wen tey are all summed togeter, te infinite terms drop, and we are left wit quite an elegant result. Xianyu summarizes te results as ia x = i (π) d / e m W g ɛ (p)ɛ (k)[aγ( d/) + B] were x runs over te tirteen different diagrams. He ten tabulates te coefficients as follows Diagrams A B a -6J c, d - e 8J J 3 g J, i 3J J j (m W /m ) J 5 k, l J m J 5 b (m /m W ) J f (m /m W ) J were te J x functions are defined below. J = = ɛ = ɛ J = [m W x( x)m ] d/ log ( m W x( x)m ) + O(ɛ ) x x dy (m W xym ) d/ dylog ( m W x( x)m 3 ) d/

14 For te purposes of sowing tat te divergences cancel, we ll concentrate on J and J. If we expand te Γ function as Γ( ɛ ) = ɛ γ + log(π) it s straigtforward to see tat as ɛ, Γ( ɛ ) diverges. However, J and J go to and respectively in tat same limit. Wen we insert tose values into te A column in our table, te sum of te coefficients goes to zero, and te divergences cancel. Te surviving terms can be found by evaluating J 3, J, and J 5, wic are similar integrals over Feynman parameters. Tey give iαm g m W π ɛ (p) ɛ (k)i W (m /m W ) for te γγ amplitude via a W boson loop. Here, I W (τ W ) = τ W [6I (τ W ) 8I (τ W ) + (τ W )(I (τ W ) I (τ W )) + I 3 (τ W )] were τ W = m /m W and I (τ W ) = I (τ W ) = I 3 (τ W ) = log[ x( x)τ W ] x x dylog[ xyτ W ] dy (8 3x + y + xy)τ W xyτ W Combining tis wit te expression from te γγ via quark loop, we get te total decay rate for γγ Γ( γγ) = αm 8sin θ w m m W α 8π f Q f N c (f)i f (τ f ) I W (τ W ) 8 Form Factor Wen calculating te gg decay, we included a form factor I f (τ q ) = 3 x dy ( xy) xyτ q

15 were τ q = m /m q. We can reduce tis to a single paramter integral, evaluate it numerically, and plot it for a top quark loop as a function of te Higgs mass. Performing te integration over dy, we get + ( ) ln ( x( x)τ q) τ q τ q τ q x Figure 8: Form factor I f (τ q ) as a function of te Higgs mass Te argument of te natural log function above as to be greater tan zero. If it wasn t, I ad to return an arbitrarily small negative number until te relative decay widts were even comparable. However, it did give me results fairly consistent wit Scwan. Using tis form factor (see Figure 8), I was able to get a reasonable gg decay widt (see Figure 9). Unfortunately, te gluon decay mode is still disproportionately likely. I m including it because its beavior across te range of te Higgs mass does follow te trend of Xianyu s results. 9 Brancing Ratios Figure 3 reflects te inclusion of te Higgs to f f, W + W, and Z Z decay. Te five major decay modes are plotted. I cut te bottom of te y axis off at 3 so tat te beavior of eac widt can be seen. Oterwise, te gluon and weak boson decay mode beaviors are not clearly visible. One effect of zooming in is tat te b b decay widt goes out muc furter tan can be seen. 5

16 Figure 9: gg decay widt Figure : Brancing ratios as a function of in GeV. t t (green), b b (tick blue), gg (tin blue), W + W (red), Z Z (yellow) 6

17 References. An Introduction to Quantum Field Teory Micael E. Peskin and Daniel V. Scroeder. Solutions to Peskin and Scroeder Final Project III Zong-Zi Xianyu ttp://learn.tsingua.edu.cn:88/38/webpage/files/peskin_ Scroeder/Peskin_FP3.pdf.3 Final project: Higgs decay Cristoper Scwan ttp:// 7