Feynman Rules for QCD in Space-Cone Gauge

0.13 MB·English

by  Alexander Karlberg

Feynman Rules for QCD in Space-Cone Gauge
Alexander Karlberg and Thomas Søndergaard
Niels Bohr International Academy and Niels Bohr Institute,
Blegdamsvej 17, DK-2100 Copenhagen, Denmark
(Dated: January 25, 2012)
We present the Lagrangian and Feynman rules for QCD written in space-cone gauge and after
eliminating unphysical degrees of freedom from the gluonic sector. The main goal is to clarify
and allow for straightforward application of these Feynman rules. We comment on the connection
between BCFW recursion relations and space-cone gauge.
PACSnumbers: 11.15.Bt
2
I. INTRODUCTION where
1
0 = igTaFa , [Ta,Tb]=if Tc,
2 Calculating QCD amplitudes by means of Feynman Fµν − µν abc
n diagrams can be an extremely challenging task. The Tr[TaTb]= 21δab, (2)
a gauge-dependence of vertices and unphysical degrees of
and
J freedom often makes intermediate steps immensely com-

4 plicated. However, in 1998 Chalmers and Siegel showed Fa =∂ Aa ∂ Aa +gf AbAc. (3)
2 that the complexity of Feynman diagram calculations in µν µ ν − ν µ abc µ ν

Yang-Mills theory could be greatly reducedif a so-called The contraction gives
]
h space-cone gauge was used [1] (see also [2, 3] for similar
t simplifications in other theories). Fa Fµνa =(∂ Aa ∂ Aa)(∂µAνa ∂νAµa)
- µν µ ν − ν µ −
p By now several alternative approaches are also avail- +4gf (∂ Aa)AµbAνc
e able, such as the Britto-Cachazo-Feng-Witten (BCFW) abc µ ν
[h recursion relation [4, 5]. At tree-level the above men- +g2fabcfab′c′AbµAcνAµb′Aνc′. (4)
tioned space-cone constructionis closely relatedto these
2 relations [6]. Here Aµ is just the usual vector field with inner product
v given by
The main goal with this short paper is to write down
1
4 all Feynman rules for QCD when working in the space- A B =A0B0 A1B1 A2B2 A3B3. (5)
cone gauge. To our knowledge not allof these have been · − − −
4
1 explicitly presented in the literature, and it is therefore We now introduce the lightcone components
. ourhopethatthis paperwillallowforeasyandstraight-
1
forward application whenever such rules are needed. 1 1
0 A+ (A0+A3), A (A0 A3),
−
2 The paper is structured as follows; in section II we re- ≡ √2 ≡ √2 −
1 viewthe Yang-MillsLagrangianinspace-conegaugeand 1 1
: the elimination of unphysical degrees of freedom. Sec- A (A1+iA2), A¯ (A1 iA2), (6)
v ≡ √2 ≡ √2 −
i tion III introduces some notation and the spinor helicity
X
formalism. In section IV we give the Feynman rules fol- and in terms of these the inner product is
r lowing from section II. In section V we make some com-
a ments on the connection between BCFW relations and A B =A+B−+A−B+ AB¯ A¯B. (7)
· − −
the space-cone gauge. In section VI we add quarks to
the Lagrangian and show that effective four-point ver- Our first goal is to express eq. (4) in terms of the light-
tices involvingquark-antiquarkpairswillappear. Insec- cone components, however, we use the gauge freedom
tionVIIwegivetheFeynmanrulesforquarksandfinally to set A = 0, and hence discard all terms containing
in section VIII we have our conclusions. an A. Note that in many of the intermediate calcula-
tions we have used the fact that a term symmetric in
two color indices will vanish when contracted with the
anti-symmetric colorfactor.
II. YANG-MILLS LAGRANGIAN IN
Written in terms of the lightcone components the
SPACE-CONE GAUGE
quadratic part of eq. (4) becomes
(∂ Aa ∂ Aa)(∂µAνa ∂νAµa)=
We start from the standard Lagrangian of Yang-Mills µ ν − ν µ −
theory 4 ∂ A+a∂+A a ∂A+a∂¯A a ∂¯A+a∂A a
− − − −
− −
1 1 (cid:2) +∂A+a∂−A¯a+∂A−a∂+A¯a
LYM = 2g2Tr[FµνFµν]=−4FµaνFµνa, (1) 2 ∂−A+a∂−A+a+∂+A−a∂+A−a+∂A¯a∂A¯a(cid:3) ,(8)
−
(cid:2) (cid:3)
2
the three-point interaction III. SPINOR FORMALISM
(∂µAaν)AµbAνc =(∂−A+a)A+bA−c+(∂+A−a)A−bA+c WechoosetousethePaulimatriceswiththefollowing
(∂A+a)A¯bA c (∂A a)A¯bA+c, normalization
− −
− −
(9)
1 1 0 1 0 1
σ0 = , σ1 = − ,
and the four-point interaction √2(cid:18)0 1 (cid:19) √2(cid:18)−1 0 (cid:19)
1 0 i 1 1 0
AbµAcνAµb′Aνc′ =2A+bA−cA−b′A+c′. (10) σ2 = √2(cid:18)−i 0 (cid:19), σ3 = √2(cid:18)−0 1(cid:19), (15)
Collecting these expressions the Lagrangian takes the such that a contraction between these and a four-vector
form is
1 1 p0+p3 p1 ip2 p+ p¯
LYM = − 4FµaνFµνa Pa˙b =Pµσµ = √2 p1+ip2 p0− p3 = p p− ,
(cid:18) − (cid:19) (cid:18) (cid:19)
= ∂ A+a∂+A a+∂A+a∂¯A a+∂¯A+a∂A a (16)
− − − −
−
∂A+a∂−A¯a ∂A−a∂+A¯a and
− −
1
+ 2 ∂−A+a∂−A+a+∂+A−a∂+A−a+∂A¯a∂A¯a Pa˙b = pp¯− −pp+ . (17)
gf(cid:2) (∂ A+a)A+bA c+(∂+A a)A bA+c (cid:3) (cid:18)− (cid:19)
abc − − − −
−
(cid:2) (∂A+a)A¯bA−c (∂A−a)A¯bA+c Note that det(P) = 21P2, so if P2 = 0 this matrix only
− − hasonenon-vanishingeigenvalueandcanbedecomposed
1g2fabcfab′c′A+bA−cA−b′A+c′. (cid:3)(11) into a bispinor product
− 2
Following [1] we then use the equation of motion for A¯ Pa˙b =pa˙pb. (18)
to eliminate this “auxiliary” component from the La-
We will from now on use the following braket notation
grangian, that is, we use
pa p , pa˙ [p, pa p, pa˙ p]. (19)
∂ ∂ ≡| i ≡ | ≡h | ≡|
∂+ ∂(∂+LA¯a) +∂− ∂(∂ LA¯a) Notice that our convention differs from [1].
(cid:18) (cid:19) (cid:18) − (cid:19)
∂ ∂ ∂
+∂ L +∂¯ L L =0, (12)
∂(∂A¯a) ∂(∂¯A¯a) − ∂A¯a A. Reference frame
(cid:18) (cid:19) (cid:18) (cid:19)
and get the following expression for A¯a
As always when one wants to do amplitude calcula-
∂+ ∂ tionsbyFeynmanrulestherearesomeLorentzframesin
A¯a = A−a+ −A+a which the calculations are easier to perform. To set up
∂ ∂
this we introduce two (for now) arbitrarymassless refer-
1
−gfabc∂2 ((∂A+b)A−c)+((∂A−b)A+c) . ence momenta, one denoted with a ⊕ and one denoted
with . We then choose to work in the Lorentz frame
(cid:2) (cid:3)(13) ⊖
whereourtworeferencemomentahavethefollowingsim-
ple expressions
Pluggingthisbackintoeq.(11),anddoingabitofrewrit-
ing, we obtain
0
0 0
LYM =A+a∂µ∂µA−a P⊕ =(cid:18)0 p−⊕ (cid:19)=|−i[−|= p−⊕ !(cid:16) 0 qp−⊕ (cid:17),
∂ q (20)
+2gf −A+a A+b(∂A c)
abc −
+2gfabc(cid:18)∂∂∂+A−b(cid:19)A−c(∂A+a) P⊖ =(cid:18)p0+⊖ 00(cid:19)=|+i[+|= q0p+⊖ !(cid:16) qp+⊖ 0(cid:17)(2,1)
(cid:18) (cid:19)
+2g2fabcfa′bc′1 (∂A+a)A−c 1 (∂A−c′)A+a′ . that is the frame where they both move along the i = 3
∂ ∂
(cid:0) (cid:1) (cid:16) (14)(cid:17) apx−ips+, b=ut +in o[pp+os]i,teanddirtehcatitonw.e hNaovteectahlaletdPt⊕he·sPp⊖ino=r
This is the pure Yang-Mills Lagrangianwritten in terms of⊕th⊖e hmo−mie−ntum for and vice versa. Our choice
of two scalar fields A+ and A , consistent with massless of labe⊕lling will soon bec|o−mie apparent. Since any four-
−
vectorfieldsonlyhavingtwophysicaldegreesoffreedom. vector, contracted with the Pauli matrices, is written in
3
the form of eq. (16), using the following normalized ma- one of the external momenta of a + helicity gluon and
trices as basis P one of the external momenta of a helicity gluon,
th⊖e rules and explicit calculations simpl−ify greatly.
+ [+ [ + [ [+
| i |, |−i−|, | i−| , |−i | , The external non-reference legs will just contribute
p+ p− + [ +] + [ +] withtheplusorminuslightconecomponentofthepolar-
⊖ ⊕ h −i− h −i−
(22) ization vector, depending on the helicity (note that we
p p
always take external momenta to be outgoing), i.e.
four-vectors take the form
+ [+ [
P =p+| pi+ | +p−|−pi−−| P+ =(ǫ+(P))+, (29)
⊖ ⊕
+ [ [+
+p¯ | i−| +p |−i | , (23) or
+ [ +] + [ +]
h −i− h −i−
where the coefficiepnts are just the lpightcone components. P− =(ǫ (P))−. (30)
Since P = p [p the lightcone components are −
| i |
p [p ] +p [p+]
p+ = h−pi− − , p− = h pi+ , oHfotwheev[epr,f]oarntdhe+rpefeirnentcheelneugms tehreat(oǫr±.)T±hveasneifsahctboercsacuasne
⊕ ⊖ − h i
p [p+] +p [p ] only be countered in the three-point vertex and only if
p¯= −h− i , p= −h i − . (24) they sit on the ∂ A /∂ term, i.e.
+ [ +] + [ +] ± ∓
h −i− h −i−
We also chpoose to use our referpence momenta in the p−(ǫ+)+ = [p+] h+−i[−+] p→− [−+] ,
expression for the polarization vectors ǫ±(P) p hp+pi[−+] −−−→ h+−i[−+]
ǫ+(P)= |+i[p|, ǫ (P)= |pi[−|. (25) pp+(ǫ−)− = [p −]h−p+i[−+[ ]+] −p−→−+→ p +h−+[i +].
+p − [p ] − h −i− h −i−
h i − (31)
p p
That is, we have used our reference momenta to fix the
The product of these two contributions is 1, and since
gauge-freedomone has in polarization vectors.
−
everydiagramwillalwayscontainthisproduct(assuming
From eq. (14) it is evident that we are only concerned
one takes both reference momenta to correspond to ex-
with the (ǫ) components, andfromeq.(23) wesee that
±
ternallegs), we just write the externallines for reference
+ [p ] legs as
(ǫ+)+ = h− i − , (ǫ+)− =0, (26)
+p p−
h i ⊕
+p [ +]
(ǫ−)− = h[p i]−p+ , (ǫ−)+ =0. (27) Pr±ef =i. (32)
− ⊖
Hence, with this setup we have that only the positive
The term representing the propagatoris the usual bo-
helicitygluonshavethe+lightconecomponentandonly
son propagator (notice that we for simplicity discard all
the negative helicity gluons the component. For this
factors of i in the following rules)
−
reason the labels in eq. (14) is now actually denoting
±
thehelicityandnotjustthespecificlightconecomponent.
Q
1
= . (33)
Q2
IV. FEYNMAN RULES IN PURE YANG-MILLS
The three-point vertex splits into the case where one
In this section we present the color-ordered Feynman of the lines is a reference leg and the case in which non
rulesoneobtainsfromeq.(14),thatistherulesonecould of the lines are. With a reference leg present we have
use in calculating, for instance, the partial tree ampli- used up the ∂ A /∂ term for a cancellation like in
± ∓
tudes An in eq. (31) and are only left with the contribution from the
opposite helicity leg, through ∂A , i.e.
=2gn 2 Tr[Ta1Ta2 Tan]A (1,2,...,n), (28) ±
n − n
A ··· K
±
X
where the sum is over all non-cyclic permutations of ex-
ternal legs. Before we do so, remember that in the last
Q = 2q, (34)
section our massless reference momenta was just arbi- ∓
trarilychosen,however,if we makethe choicethat P is
⊕
Pr±ef
4
or exist and the simplified diagrams (32), (34) and (35)
Kr±ef should be discarded.
Before turning to the inclusion of quarks let us make
some comments on the connection between working in
Q = 2q, (35) space-cone gauge and the BCFW recursion relation.
∓
−
V. BCFW RELATIONS FROM SPACE-CONE
P
±
GAUGE
where the minus sign in the second diagram comes
from the antisymmetry of the colorfactor fabc. There Oneoftheinterestingfeaturesofthespace-conegauge
can only be one reference leg on a three-point vertex is, that a diagrammatic proof of the BCFW recursion
since there is only one term in these that can counter relation can be obtained from the above Feynman rules.
the vanishing of that leg. The main observation is that no vertex depends on p¯.
In the second case of no reference leg, the three-point Therefore, a shift in the + [ direction of a reference
| i−|
vertex is leg leaves no imprint on the vertices, but only on inter-
nal propagators. Using a propagator relation between
K
± shifted andunshifted propagators,it is possible to relate
amplitudes calculatedinspace-conegaugeto the BCFW
p k
Q = 2q ∓ ∓ . (36) result. For the four- and five-point case this calculation
∓ p − k was done in [6] where the reader may also find details
(cid:18) (cid:19)
regarding the propagator identity. However, these two
P examples are in a wayspecial, since no four-pointvertex
±
enters the calculations. The first case where the four-
Again the minus sign is a consequence of the anti- point vertex is manifest for all choices of reference mo-
symmetric colorfactor. menta is the NMHV six-point amplitude. In a very
We can never have a reference leg on a four-point condensed notation it is given by
vertexsincethese donotcontributewith asimilar“1/0”
counter-term. However, there are still two different
A6( ++ )= +
kinds of four-point vertices because of the two possible ⊕ −−⊖
helicity configurations ++ and + + (cyclically
−− − −
speaking). The first case is + + + (39)
K− Q−
where and denotes the positive and negative refer-
⊕ ⊖
pq+tk ence gluons respectively. When we use the propagator
= 2 , (37)
− (p+k)2 identity introduced by Vaman and Yao, we obtain a fac-
torization which has the structure of a three-point am-
plitudetimesafive-pointamplitude. Herethefour-point
P+ T+
vertexentersexplicitlyinthefive-pointamplitude,which
seems to be in conflict with [1], where it was shown ex-
and the second case
plicitly that the five-point amplitude is independent of
the four-point vertex.
K− T+ + ˆ ˆ +
⊖ ⊕ ⊖ ⊕
1
pk+tq pq+kt
+ +
= 2(p+q)2 +2(p+k)2 . (38) → P2
(40)
− − − −
P+ Q
−
+ + + +
− − − −
These rules reduce the number of diagrams contribut- 1
ing to a specific color-ordered amplitude significantly. (41)
→ P2
However,thereductionreliesheavilyonthechoiceofhav-
ingexternalmomentaasreferencemomentasuchthatdi- ˆ ˆ
⊕ ⊖
⊕ ⊖
agrams with both reference legs on the same three-point
vertex and diagrams with a reference leg on a four-point To explain this, we notice, that the result in [1] exploits
vertex all vanish. The rules are of course perfectly al- the fact that in the space-cone gauge, the number of di-
lowed without this choice, but then no such constraints agrams and the types of vertices that enters a specific
5
calculation, is highly dependent on the chosen reference to eq. (1), where q,0 is just the Lagrangianfor the free
L
lines. When twoofthe externalgluonsarechosenasref- Dirac field, and the interaction can be written out in
erences,thefour-pointvertexinthe five-pointamplitude terms of the lightcone components as
is absent. However, if we only choose one of the exter-
nal momenta as reference, the four-point vertex will still gAaµψ¯γ Taψ =g A+aψ¯γ Taψ+A aψ¯γ+Taψ
µ − −
be present in the calculation. The structure of the five-
h
point amplitude, with only one external gluon chosen as Aaψ¯γ¯Taψ A¯aψ¯γTaψ . (45)
− −
reference line, is shown below.
=0 i
+
⊕ Notethatwehavealsowr|itten{zthe}γ-matricesinthelight-
A5( ++ )= + + cone notation now, i.e. γ+ 1 (γ0+γ3), etc. The ex-
⊕ −− ≡ √2
pression for A¯a, following from eq. (12), should then be
replaced by
+ −+ −
∂+ ∂
+ (42) A¯a = A a+ −A+a
−
− ∂ ∂
1
gf ((∂A+b)A c)+((∂A b)A+c)
⊕ − − abc∂2 − −
Comparing this with (40) and (41) we notice, that the g 1 ψ¯γ(cid:2)Taψ , (cid:3)(46)
− ∂2
diagrams with four-point vertices exactly matches those
(cid:0) (cid:1)
of (42). The diagramswithonly three-pointverticescan and after substituting this back into the original La-
justaseasilybefoundwhenusingthepropagatoridentity grangianwe obtain
in the rest of the diagrams. We therefore conclude that
the amplitudes combine in exactly the right way as to = YM + q,0+ q,I, (47)
L L L L
reproduce the BCFW result, and that we should expect
the procedure to generalize to any number of external where
legs.
∂
Lastly,wewouldliketomakeacommentregardingthe q,I =g A+aψ¯γ−Taψ −A+a ψ¯γTaψ
L − ∂
propagator identity that is needed in order to conclude (cid:20) (cid:18) (cid:19) (cid:21)
the BCFW recursionrelationin [6]. It is shownthat the ∂+
+g A aψ¯γ+Taψ A a ψ¯γTaψ
identity is equivalent to − − ∂ −
(cid:20) (cid:18) (cid:19) (cid:21)
1 1
dz +g2f (∂A+a)A c ψ¯γTbψ
=0, (43) abc −
∂ ∂
z(z z1)(z z2) (z zn 1) (cid:20)
Z − − ··· − − (cid:0)1 (cid:1) (cid:0)1 (cid:1)
over a contour enclosing all poles. In the original proof (∂A−c)A+a ψ¯γTbψ
−∂ ∂
of the BCFW recursionrelation[5], two main properties (cid:21)
g2 1 (cid:0) 1 (cid:1) (cid:0) (cid:1)
of tree-levelamplitudes was needed. First that the poles ψ¯γTaψ ψ¯γTaψ . (48)
of the shifted amplitude A(z) all come from propagators − 2 ∂ ∂
going on-shell. Second, that A(z) 0 as z . These (cid:0) (cid:1) (cid:0) (cid:1)
→ →∞
statements are proved in [5], but the arguments become
VII. FEYNMAN RULES INVOLVING QUARKS
almost trivial when put into the light of the space-cone
formalism.
Thefirststatementisimmediatelyclearsincetheshifts The external states for quarks are given by
can be chosen such that the vertices of the amplitude
areunaffected. Hence, onlyinternalpropagatorschange.
Then the second statement follows from (43) since the P =u¯s(P), (49)
integrand obviously converges fast enough to zero.
and
VI. ADDING FERMIONS
P =vs(P), (50)
We will now add quarks to our Lagrangian. This
amounts to adding
=ψ¯(iγ Dµ)ψ mψ¯ψ where u¯s = (us)†γ0, and us and vs are positive- and
q µ
L − negativ-energysolutions,respectively,ofthe Dirac equa-
= q,0+gAaµψ¯γµTaψ, (44) tion, with s labelling spin up or down. The propagator
L
6
is just the usual Dirac propagator Like in the pure gluon case there can not be reference
legs on these four-point vertices either. The second one
Q is a double pair quark-antiquarkinteraction
Q/ +m
= . (51)
Q2 m2 K P
−
γ γ γ γ
From eq. (48) we can read off the color-ordered Feyn- = + ,
man rules involving vertices with quarks. (t+q)(k+p) (q+k)(p+t)
The three-point vertices involving one gluon and a
quark-antiquark pair is Q T (58)
and
P
±
K P
p
∓
= γ γ, (52)
∓− p γ γ
= . (59)
− (q+k)(p+t)
and Q T
P Notice that the γ-matrices in eq. (58) and (59) are not
±
multipliedtogether. Ina realamplitude calculationthey
will appear between corresponding external spinors, for
p
∓
= γ∓ γ . (53) example like
− − p
(cid:18) (cid:19)
2 3
or in case of a reference leg u¯(P2)γv(P1)u¯(P4)γv(P3)
=
p1+p2 p4+p3
Pr±ef
1 4 +(2 4). (60)
↔
= γ, (54) ThiscoversalltheFeynmanrulesoneobtainsforQCD
−
in the space-cone gauge.
and
VIII. CONCLUSIONS
Pr±ef
InthispaperwehaveexplicitlywrittendownallFeyn-
man rules for QCD in space-cone gauge when unphysi-
= γ. (55)
cal degrees of freedom in the gluonic sector have been
removed. Combined with a clever choice of reference
frame this reduces the amount of Feynman diagrams
Due to the elimination of the A¯ component from needed for gluon amplitude calculations considerably.
the Lagrangian, we have instead introduced four-point We then made some comments about the close connec-
vertices involving quarks. The first one is an effective tion between BCFW recursion relations and the space-
gluon-gluon-quark-antiquarkinteraction cone gauge, especially concerning the role played by the
four-pointvertex. Wehavealsoseenthatinthepresence
K P
± ∓ of quarks the former manipulations of the Lagrangian
lead to the introduction of effective four-point interac-
(p k)
tion terms involving quark-antiquarkpairs.
= − γ, (56)
(k+p)2
Q T ACKNOWLEDGMENTS
and
K P We would like to thank Poul Henrik Damgaard, Emil
± ∓
Bjerrum-Bohr and Diana Vaman for useful discussions
and comments. TS would also like to thank Simon Bad-
(p k)
= − γ. (57) ger for helpful comments.
− (k+p)2
Q T
7
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